Add files using upload-large-folder tool

-9E0T4oBgHgl3EQfxAH-/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:36f4efa43ea50d329d29f00f5d9c3da7d14da7b558a6535236c4cd5c9b489436
+size 112302

-9E5T4oBgHgl3EQfRw7N/content/tmp_files/2301.05524v1.pdf.txt

@@ -0,0 +1,843 @@
+Veltman Criteria in Beyond Standard Model Effective Field Theory
+of Complex Scalar Triplet
+aJaydeb Das 1, bNilanjana Kumar2
+a Department of Physics and Astrophysics, University of Delhi, Delhi-110007, India
+bCentre For Cosmology and Science Popularization (CCSP),
+SGT University, Gurugram, Haryana-122006, India
+Abstract
+The Standard Model Higgs mass, not being protected by any symmetry, suffers from large
+correction terms due to quadratic divergence coming from the self energy corrections. Veltman
+Condition (V.C.) ensures that the coefficient of the quadratic divergent term either vanishes or
+becomes negligible. If the Standard Model (SM) is valid upto a scale (Λ) and new physics exists
+after that, V.C. demands Λ ≲ 760 GeV. But the non-observation of new physics has pushed the scale
+to be ≥ 1 TeV already, making it impossible to satisfy V.C. in the Standard Model without very large
+fine tuning. Attempts has been made to satisfy the V.C. in many Beyond Standard Model (BSM)
+theories but they fail to satisfy V.C. at large Λ including the scenario of complex triplet scalar
+with hypercharge 1. Hence, alternate scenario can be considered where the new physics appears
+much above the Electroweak scale, and at low energy it emerges as Standard Model Effective Field
+Theory (SMEFT). In this literature, we consider some specific BSM scenarios to appear at a large
+scale such that at low energy we get the Beyond Standard Model Effective Field Theory (BSM-
+EFT). We found that the V.C. satisfies easily if the BSM is type-II seesaw model with complex
+triplet scalar (Y = 1) compared to the other extensions in the BSM-EFT framework. We examine
+the model parameter dependence of the Wilson Coefficients (W.C.) in detail and show that the
+cancellation of the Wilson Coefficients appearing in the V.C. is highly dependent on some specific
+values of the model parameters.
+1
+Introduction
+The smallness of the observed Higgs mass is confirmed by the experiments [1,2] at the Large Hadron
+Collider (LHC). However, in the Standard Model (SM) of particle physics, the scalar mass (mass of
+Higgs boson) is not protected by any symmetry. Hence, if SM is valid upto a large scale, Planck scale,
+the Higgs mass suffers from quadratic divergence (∼ Λ2). In order to ensure that the mass of the Higgs
+boson is small, one has to consider a very large fine tuning in the SM. A way to ensure that the Higgs
+mass does not get large correction at a higher scale is coined as Veltman Condition (V.C.) [3]. V.C.
+checks if the sum of all quadratically divergent terms coming from the self energy diagrams of the
+Higgs boson are either zero or very small. On the other hand, experiments such as the Large Hadron
+Collider (LHC) is pushing the New Physics (NP) scale towards > 1 TeV and the Veltman Criteria is
+not possible to satisfy in the SM, as it demands Λ to be less than 760 GeV [4].
+Simple extensions of SM has been studied in literature [4–10], where the V.C. is valid but only in
+some region of parameter space. Overall, there are two main concerns in these models: (1) These
+theories encounter different problems at large scale, such as the potential becomes unstable leading
+[email protected]
+[email protected]
+1
+arXiv:2301.05524v1  [hep-ph]  13 Jan 2023
+
+to the invalidity of the theory beyond that scale. (2) The non observation of the Beyond Standard
+Model (BSM) particles at LHC is pushing their masses above TeV scale [11].
+One may assume that SM is valid upto certain scale (Λ) and above that scale some unknown symmetry
+appears to protect the Higgs mass, then the Higgs mass can be stabilised and the fine tuning problem
+can also be addressed. For examples, in the Composite Higgs Scenario [12], where the Higgs is dissolved
+in higher degrees of freedom above the symmetry breaking scale or in Supersymmetric theories [13],
+where the bosonic and fermionic degrees of freedom cancels out exactly – the Higgs mass is maintained
+to be finite and small. However these theories also can not avoid certain amount of fine tuning [14,15]
+coming from several sources. However, the none of these theories are observed at LHC and other
+experiments so far.
+These observations raise the question that what if the new physics lie at a very large scale. In such a
+scenario, SM can emerge as an Effective Field Theory (SMEFT) [16] by integrating out the dynamics
+of the larger theory. The Information of the heavy particles appearing in the loop are absorbed in
+the higher dimensional operators in the Effective Field Theory (EFT) and the theory is invariant
+under SM symmetries. Ref [17] has shown that the V.C. can be satisfied in the SMEFT framework by
+including the higher dimensional operators and their Wilson Coefficients. Only a few of the operators
+are relevant to the V.C. and they play a major role in satisfying the V.C.
+In this paper, we take one step forward and ask this question what if the theory at a very high
+scale (Λ) is governed by the larger symmetries, and how they affect the V.C.? We adopt the Beyond
+Standard Model Effective Field theory (BSM-EFT) [18] approach, which has been studied previously
+in Ref [18–24].
+In BSM-EFT the Lagrangian becomes invariant under the particular BSM model
+in consideration. The motivation to study the V.C. in BSM-EFT framework is two fold. 1) We can
+specifically check how many higher dimensional operators are allowed by the model. 2) We can express
+the Wilson Coefficients in terms of the model parameters. Hence, the sign dependence of the W.C.’s
+come naturally.
+We begin with simple BSM scenarios (in BSM-SMEFT) such as, scalar singlet, doublet and triplets
+(real or complex). Remarkably, we found that among these models, it is possible to generate all the
+SMEFT operators that contributes to the V.C. in the complex scalar triplet model with Y = 1. This
+model makes the cancellation easier in V.C. with less fine tuning than the other scenarios3. This
+particular model is well motivated in literature from different aspects such as: 1) Neutrino mass
+generation through the see-saw mechanism [25], 2) type-II Leptogenesis scenario [26] 3) Enhancement
+of the h → γγ branching ratio [27] etc, among many other [28]. Hence we pursue this model only in
+detail.
+In Section 2, we show how the V.C. depends on the SMEFT operators in the WARSAW basis [29].
+In Section 3, we discuss some specific models in the BSM-EFT scenario, and express the Wilson
+coefficients of Y = 1 complex scalar triplet model in terms of the model parameters. In Section 4, we
+show how it is possible to satisfy the V.C. by exact cancellation of the Wilson Coefficients at different
+scale and interpret the result in terms of the model parameter space. Then in Section 5 we conclude.
+3Even type I and type II seesaw models does not generate all these operators. Moreover a recent study [22] has also
+shown that these models are also not favored from the fact that the radiative electroweak symmetry breaking can not
+be triggered even at the Planck scale.
+2
+
+2
+SMEFT operators in Veltman Criteria
+The physical mass of the Higgs in the Standard Model can be written in terms of the bare mass term
+mh(0) and the higher-order self-energy corrections:
+m2
+h = m2
+h(0) + δm2
+h = m2
+h(0) + Log Div. Term + Quadratic Div. Term + Finite terms,
+(1)
+where the assumption is that the SM is valid upto the scale Λ and the correction terms are coming
+from the loop diagrams involving scalars, fermions and bosons in the loop. The d = 4 potential in the
+Standard Model in terms of Higgs doublet (H) is
+V (H) = −m2
+HH†H + λ(H†H)2.
+(2)
+This leads to the correction to the higgs mass and the quadratic divergent contribution is,
+(δm2
+h)SM
+=
+Λ2
+16π2 (6λ + 9
+4g2
+W + 3
+4g2
+Y − 6y2
+t ),
+(3)
+where, gY and gW are the U(1)Y and SU(2)L gauge couplings respectively and gt = 2mt/v is the top
+quark Yukawa coupling. Here we neglect couplings of the lighter quarks and Λ is the cut-off scale. The
+Veltman Condition (V.C.) demands that δm2
+h ∼ 0 or at least controllably small. With the observed
+Higgs mass at 125 GeV, the condition to make δmh ∼ 0 demands Λ < 760 GeV, which is already ruled
+out by LHC. One way to solve this problem is to introduce new particles, which can contribute in the
+loops and soften the fine tuning by ensuring exact cancellation or partial as we have already discussed
+in the introduction.
+A popular way to address this problem is to consider the effects of the higher dimensional operators in
+the EFT framework. Let us assume that the New Physics (NP) exists at a very high scale Λ. The effect
+of NP can be effectively integrated out at Λ and this will effectively give us SM, plus some effective
+operators involving only the SM fields, which holds through out the low energy scale, otherwise known
+as the Standard Model Effective Field Theory (SMEFT) [16]. The Lagrangian, which incorporates
+dimension six SMEFT operators in addition to the Standard Model dimension four operators, can be
+expressed as,
+L = ∑
+i
+C4iQ4i + 1
+Λ2 ∑
+i
+C6iQ6i.
+(4)
+In contrast to C4i, which is the only function of the parameters linked to the degrees of freedom in the
+Standard Model, C6i are the Wilson Coefficients, which are functions of the integrated out dynamics
+at Λ. These operators can be expanded at any choice of basis, for example, HISZ basis [30], Warsaw
+basis [29,32], SILH basis [31] etc. The set of dimension six operators that involves Higgs in Warsaw
+basis are:
+QH = (H†H)3, QHD = (H†DµH)∗(H†DµH), QH◻ = (H†H) ◻ (H†H)
+QHB = (H†H)BµνBµν, QHW = (H†H)W a
+µνW a,µν, QGG = (H†H)GA
+µνGA,µν
+QHWB = (H†τaH)BµνW a
+µν
+(5)
+It can be shown that the last operator does not contribute Higgs self energy correction [17]. The first
+operator will also not contribute at one-loop level as the Higgs does not develop a vev at Λ. There can
+be the appearance of the operators involving the gluons of the form QGG = (H†H)GA
+µνGA,µν. However,
+while considering BSM-EFT framework with heavy scalars, this operator does not contribute as scalars
+3
+
+do not carry any color charge. Note that, these operators can be written in any basis, for example
+Ref [17] choose the HISZ basis.
+We choose the Warsaw basis because it is self consistent at one
+loop [32,33] and easier to check the running of the Wilson coefficients in Warsaw basis.
+The correction to the Higgs mass from the higher order terms in the Lagrangian is given by
+(δm2
+h)total
+=
+Λ2
+16π2 ∑
+i
+fi(C4i,C6i) +
+Λ2
+(16π2)2 ∑
+i
+gi(C4i,C6i)
+(6)
+Here fi and gi are one loop and two loop correction to the Higgs mass. The V.C., δm2
+h ∼ 0 translates
+into
+f(C4i,C6i),g(C4i,C6i) ∼ 0
+(7)
+The coefficients, C4i and C6i are function of Λ and the model parameters. Hence Eq:6 can be written
+in terms of the SM and higher dimension operators contribution as,
+(δm2
+h)total ≡ (δm2
+h)SM(fi(C4i),gi(C4i)) + (δm2
+h)HO(fi(C6i),gi(C6i))
+(8)
+Also, it has been shown in Ref [17] that at d ≥ 8, the SMEFT operators are not able to produce any Λ6
+divergence, which will produce any effective Λ2 divergence while calculating the self energy correction
+of Higgs mass. There are studies in the literature, where the V.C in terms of EFT has been studied
+in detail [17,34,35]. In particular it has been shown in Ref: [17] that it is possible to satisfy the V.C
+for appropriate values and sign of the Wilson coefficients at large Λ.
+3
+BSM-EFT with Complex Scalar Triplet
+In the above section, we see that only four operators are involved in the V.C. Now, we assume that the
+new physics at a high scale follow certain symmetries of a BSM model which effectively produces SM as
+an EFT. In this BSM-EFT framework, these 4 operators may or may not be possible to generate at one
+loop, depending on the underlying symmetry of the model at scale Λ. In Table: 1 we present if these
+4 operators can be generated at one loop in some simple BSM-EFT cases with additional scalar(s) or
+not 4. For the calculation, we have implemented the Lagrangian of each model in CoDEx [36,37] and
+generated the Wilson coefficients as an output 5.
+Among all popular SM extensions, we have found that BSM-EFT with complex scalar triplet will
+easily address the V.C, as it generates all four Wilson Coefficients at one loop. In other models, the
+cancellation will be harder to achieve as the number of operators are less than four. For example
+in 2HDM scenario and real scalar singlet + triplet model, only three operators can be generated.
+Whereas, in complex scalar singlet model, only 2 operators are generated and the cancellation will be
+hard to obtain (hence large fine tuning) in these models compared to the complex scalar triplet model.
+Even larger fine tuning will be unavoidble for the real scalar singlet model as it generates only one
+operator. The complex scalar triplet with additional doublet also can generate these four operators
+but we examine the minimal scenario only with complex scalar triplet model in the following.
+4Note that we are not checking non scalar extensions of SM because, the sign of the top-loop contribution (dominant
+contribution) or rather fermionic contribution is opposite to the other diagrams with a gauge boson or a scalar in the
+loop. Therefore, V.C. is hard to solve by adding non scalar particles such as vector-like quarks or fermions, additional
+gauge bosons etc.
+5We have also cross checked our result with Matchmakereft [38].
+4
+
+Model
+Quantum No
+QHD
+QHB
+QHW
+QH◻
+Real Scalar Singlet
+(1,1,0)
+
+
+
+
+Real Scalar Triplet
+(1,3,0)
+
+
+
+
+Complex Scalar Triplet
+(1,3,1)
+
+
+
+
+Complex scalar doublets (2HDM)
+(1,2,±1/2)
+
+
+
+
+Real Scalar Singlet +
+(1,1,0)
+
+
+
+
+Real Scalar Triplet
+(1,3,0)
+Complex Scalar Triplet +
+(1,3,1)
+
+
+
+
+Complex Scalar Doublet
+(1,2,1/2)
+Table 1: SMEFT operators in Warsaw basis in different BSM-EFT scenarios.
+Let us consider that beyond the scale Λ, there exists a heavy complex triplet, ∆, with weak hypercharge
+Y = 1. The most general renormalizable tree-level scalar potential of such a model is given by
+V (H,∆)
+=
+−m2
+H (H†H) + M2Tr[∆†∆] + (µ∆HT iσ2∆+H + h.c) + λ(H†H)
+2 + λ1 (H†H)Tr[∆†∆]
++
+λ2 (Tr[∆†∆])
+2 + λ3Tr[(∆†∆)2] + λ4 (H†∆∆†H).
+(9)
+The extra Yukawa term for neutrino mass generation is,
+LY
+=
+y∆ℓT iCiσ2∆ℓ + h.c.
+(10)
+Here the trilinear coupling µ∆ can be taken as positive by absorbing its phase into Φ and ∆. The
+total Lagrangian is,
+L = LY − V (H,∆),
+(11)
+The detail of this model is summarized in ‘Model Description’ part of the Appendix.
+The dimension six operators the Warsaw basis, as listed in Eq:5, can be expanded and the calculation
+of the symmetry factors are shown in the ‘Calculation’ part of the Appendix. Hence, the Higgs mass
+correction in terms of W.C.’s is found to be,
+(δm2
+h)BSM
+=
+Λ2
+16π2 ( − 3CHD + 12CH◻ + 9CHW + 3CHB)
++
+Λ2
+(16π2)2 (54CH − 9
+2(g2
+Y + 3g2
+W )CHD + 108g2
+W CHW )
+(12)
+Which leads to the total correction to the Higgs mass to be,
+δm2
+h = (δm2
+h)SM + (δm2
+h)BSM
+(13)
+5
+
+We found the the following expressions of the Wilson Coefficients appearing in one loop contribution
+in (δm2
+h)BSM:
+CHD
+=
+−
+g4
+Y
+320π2 + 4µ2
+∆
+M2 −
+λ2
+4
+24π2 + 11g2
+Y µ2
+∆
+24π2M2 −
+8µ4
+∆
+3π2M4 + λ4µ2
+∆
+6π2M2 + 3λµ2
+∆
+8π2M2
+(14)
+CH◻
+=
+−
+g4
+W
+1920π2 + 2µ2
+∆
+M2 −
+λ2
+1
+16π2 − λ1λ4
+16π2 −
+λ2
+4
+192π2 − g2
+W µ2
+∆
+96π2M2 +
++
+11g2
+Y µ2
+∆
+96π2M2 −
+49µ4
+∆
+12π2M4 + λ1µ2
+∆
+8π2M2 +
+λ4µ2
+∆
+48π2M2 + 3λµ2
+∆
+4π2M2
+(15)
+CHB
+=
+g2
+Y λ1
+32π2 + g2
+Y λ4
+64π2 + 11g2
+Y µ2
+∆
+64π2M2
+(16)
+CHW
+=
+g2
+W λ1
+48π2 + g2
+W λ4
+96π2 + 25g2
+W µ2
+∆
+192π2M2 .
+(17)
+Here, M is the mass of the heavy triplet. For the theory to be valid, it is sufficient to assume that M
+is greater than Λ. We assume the order of magnitude to be the same for M and Λ in our calculation
+as a limiting scenario. For M >> Λ, the W.C.’s will obtain smaller values.
+4
+Result
+We consider the one loop correction to the Higgs mass at first and fix two benchmark scenarios at
+large scales, such as 100 TeV, 106 TeV. In Fig: 1 we show the model parameter space of λ1 and
+λ4, for which quadratic divergence cancels out exactly, making δm2
+h = 0. The SM input parameters,
+such as (gW , yt, gY , λ) are determined at the benchmark scales by solving two loop Renormalized
+Group Equation (RGE)’s. λ1 and λ4 are varied in such a way that the Wilson Coefficients obey the
+perturbative limit and the running of the Wilson Coefficients from Λ to the EW scale is not varied
+much. Note that, the tree level couplings (λ and µH) also get shifted due to the higher dimensional
+operators. The parameter λ can not be more than O(1) and this puts an upper limit on the quantity,
+µ2
+∆
+2M2 < O(1), where µ∆ =
+√
+2v∆M2
+v2
+H
+, in the limit of large masses of the triplet (M). Also, recent precission
+measurements of the ρ parameter gives ρ = 1.00038 ± 0.00020, resulting in v∆ < 2.56 [39].
+Figure 1: Variation of λ1 and λ4 with µ∆ at two benchmark values of Λ.
+6
+
+6
+=100 GeV
+ = 5*104 GeV
+3
+ = 105 GeV
+0
+-3
+A=100TeV
+-6
+-50
+-25
+0
+25
+50
+入46
+=100 GeV
+=5*108GeV
+3
+ = 109 GeV
+0
+-3
+Λ=10%
+Tev
+-6
+-50
+-25
+0
+25
+50
+入4From Fig: 1, we can see that the parameter space of (λ1, λ4) is very much constrained from the V.C.
+Note that, both positive and negative values of λ1 and λ4 are allowed. The green line represents the
+highest possible value of µ∆, which comes from
+µ2
+∆
+2M2 ∼ O(1). The V.C. only satisfies over the thin lines
+for different values of µ∆. The nature of the plots is highly dependent on the values of µ∆, because the
+Wilson coefficients have (µ∆/M)2 and (µ∆/M)4 dependence with additional suppression of 1/16π2.
+The freedom to choose the W.C’s in terms of parameters λ1, λ4 and µ∆, allows the exact cancellation
+even at a very large scale ( Λ = 106 TeV). The small change in parameter space is due to the running
+of the SM parameters. Hence we found that the V.C is insensitive to the scale Λ, and the fine tuning
+is appearing only due to the precession of the numbers that the parameters take, which is negligible6.
+Figure 2: Variation of the Wilson Coefficients with model parameter λ4
+In Fig: 2, we show the variation of the Wilson Coefficients with the model parameter λ4 at 100 TeV.
+The corresponding value(s) of λ1 can be inferred form Fig: 1. The Wilson coefficients show similar
+behaviour for the othe benchmark case. For CHD and CH◻, negative values are more preferred, whereas,
+for CHW and CHB, both positive and negative values are allowed. However, when λ4 is negative, all
+coefficients are negative mostly, except for some values of the CHD and CH◻. Again, when λ4 is positive,
+CHW and CHB are always positive but CHD and CH◻ are mostly negative except for some values as
+shown in Fig: 2. Thus, it is clearly visible that the cancellation among the Wilson coefficients are not
+ad-hoc in V.C., but are controlled by the model parameters. We have also implemented the two loop
+6Note that, for other models, where the number of Wilson Coefficient is less than 4 can be generated, the exact
+cancellation will be harder to obtain and the amount of fine tuning will also be very high compared to this model.
+7
+
+10
+= 100 GeV
+ = 5104 GeV
+5
+μ = 105 GeV
+CHD
+-
+-5
+Λ= 100 TeV
+-10
+-50
+-25
+0
+25
+50
+入42.5
+=100 GeV
+ = 5*104 GeV
+1.5
+μ = 105 GeV
+0.5
+CH
+0.5
+1.5
+A=100TeV
+2.5
+50
+-25
+0
+25
+50
+入40.02
+=100 GeV
+M = 5*104 GeV
+0.01
+M = 105 GeV
+0.00
+0.01
+Λ= 100 TeV
+0.02
+-50
+-25
+0
+25
+50
+入40.050
+=100GeV
+= 5*104GeV
+0.025
+M = 105 GeV
+0.000
+0.025
+Λ = 100 TeV
+-0.050
+-50
+-25
+0
+25
+50
+入4contribution to Higgs mass correction and in V.C. Due to the extra suppression by (1/16π2), the effect
+is not visible, hence we do not show that.
+HD
+H 
+100
+104
+106
+108
+-8
+-6
+-4
+-2
+0
+Λ(GeV)
+λ1 = 4, λ4 = 40,
+μΔ = 1000 GeV
+HW
+HB
+100
+104
+106
+108
+0.000
+0.005
+0.010
+0.015
+0.020
+Λ(GeV)
+λ1 = 0.01, λ4 = 40,
+μΔ = 1000 GeV
+Figure 3: Running of the Wilson Coefficients from Λ = 106 TeV to the cut-off scale for one set of model
+parameters. We have kept the value of µ∆ to be fixed at 1 TeV.
+We have also checked the running of the Wilson coefficients from the effective scale Λ to the electroweak
+scale. We show the running of the Wilson coefficients in Fig: 3 for a particular choice of the model
+parameters, λ1 and λ4. We choose λ1 = 4.0 and λ4 = 40 as an input parameter. This particular
+choice of parameter represents the maximum possible value of the model parameters as can be seen
+in Fig: 2. We found that, the values of these W.C.’s do not change much and also the sign does not
+change. The conclusion remains same for other allowed values of λ1 and λ4. The values of W.C’s.
+(Ci(1TeV)2/Λ2) are highly constrained at the EW scale [40] from various experiments. The values
+of Wilson coefficients (as in Fig: 2,Fig: 3), for which the V.C. is satisfied, is well within the current
+experimental limits.
+5
+Conclusion
+The Veltman Condition can not be satisfied within the framework of the Standard Model because of
+significant quadratic divergences to the Higgs self-energy correction if the cutoff scale Λ is ∼ 1 TeV
+or higher. However, in addition to the dimension four operators from the Standard Model, we have
+also included dimension six operators whose contributions to the Higgs mass correction result from
+integrating out the heavy triplet scalar with hypercharge one in terms of the SMEFT operators. We
+show how the quadratic divergence of the Higgs self-energy vanishes in this particular model due to
+the cancellation among the SM parameters and the Wilson Coefficients.
+We have shown the relevant SMEFT operators which contributes in the V.C., and expressed them in
+terms of the model parameters. Hence, the sign of the Wilson Coefficients are not ad-hoc, it is driven
+by the larger theory, which is a heavy triplet scalar in our case. We found that, in other models the
+cancellation is harder to achieve because some of the operators are absent. In other words, one has
+to allow for a minimum fine tuning in order to generate the model parameter space which is allowed
+by the V.C. However, the values of the Wilson Coefficients will be different in every model, as it is
+controlled by the specific model parameters.
+In order to achieve the Veltman Condition, it should be noted that the contributions from two par-
+ticular dimension six operators QHD and QH◻ play a dominating role in cancelling out the quadratic
+8
+
+divergences. However, this may or may not be the case in other models. We have observed that
+for energy scales Λ = 100 TeV and 106 TeV, the cancellation is almost similar, when the W.C’s are
+expressed in terms of λ1 and λ4 for a given µ∆. The minimal change in the parameter space is mainly
+due to the running of the SM parameters. If we introduce some relaxation in the V.C., by allowing
+some amount of fine tuning, the model parameter space will surely enlarge, but it will get narrower
+with the increasing values of Λ. Thus, the Veltman Condition can be easily satisfied in the framwork
+of effective field theory, when a scalar triplet exists at a very large scale. The study of this model
+as an Effective Field Theory can also be useful to revisit the Type II leptogenesis scenario, where it
+will be possible to generate specific dimension six terms which are allowed by the symmetries of the
+model.
+Acknowledgements: JD acknowledges the Council of Scientific and Industrial Research (CSIR),
+Government of India, for the SRF fellowship grant with File No. 09/045(1511)/ 2017-EMR-I. JD also
+would like to acknowledge Research Grant No. SERB/CRG/004889/SGBKC/2022/04 of the SERB,
+India, for partial financial support. The work of NK is supported by Department of Science and
+Technology, Government of India under the SRG grant, Grant Agreement Number SRG/2022/000363.
+We also thank Prof. Anirban Kundu and Dr. Supratim Das Bakshi for useful discussion.
+6
+Appendix
+Model Description
+In the type-II seesaw model, the scalar sector is extended by a complex scalar
+triplet(∆) with hypercharge 1, in addition to the Higgs doublet (H). Explicitly,
+H (1,2,+1/2) = (φ+
+φ0), ∆(1,3,+1) = (∆+/
+√
+2
+∆++
+∆0
+−∆+/
+√
+2)
+(18)
+with the neutral components:
+φ0 = vH + h + iφ3
+√
+2
+, ∆0 = v∆ + δ + iξ
+√
+2
+(19)
+The numbers in the parentheses represent the charges of SU(3)C × SU(2)L × U(1)Y gauge group of
+the SM. The kinetic terms corresponding to the scalar fields are given as
+Lkin ⊃ (DµH)†DµH + Tr[(Dµ∆)†(Dµ∆)],
+(20)
+with the covariant derivatives
+DµH
+=
+∂µH − igY
+2 W a
+µσaH − igW
+2 BµH,
+Dµ∆
+=
+∂µ∆ − igY
+2 Tr[W a
+µσa,∆] − igW
+2 Bµ∆.
+(21)
+Here σa (a = 1, 2, 3) are the Pauli spin matrices and gW and gY are the gauge couplings associated
+with SU(2)L and U(1)Y gauge group respectively.
+9
+
+Calculation
+The dimension six SMEFT operators which contribute Higgs mass correction either at
+one-loop or two-loop level in this model can be written upto a total derivative as,
+QHD
+=
+(H+DµH)∗(H+DµH) ⊃ (∂µH†)HH† (∂µH†) + [g2
+W
+4 σaσbH†W a
+µHH†W µbH
++
+g2
+Y
+4 H†BµHH†BµH]
+QH◻
+=
+(H+H) ◻ (H+H) = −∂µ (H†H)∂µ (H†H)
+QHW
+=
+(H+H)W a
+µνW a,µν ⊃ 2H†[σa (∂µW a
+ν )σb (∂µW νb) − σa (∂µW a
+ν )σb (∂νW µb)]H
++
+g2
+W σafabcσpfpqrH†W b
+µW c
+νW µqW νrH
+QHB
+=
+(H+H)BµνBµν ⊃ 2H†[∂µBν∂µBν − ∂µBν∂νBµ]H
+QH
+=
+(H†H)3.
+(22)
+Note that only momentum dependent vertices can generate quartic divergence at one-loop level. Pos-
+sible Feynman diagrams originating from these terms are similar to Ref. [17].
+References
+[1] G. Aad et al. [ATLAS], “Observation of a new particle in the search for the Standard
+Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B 716, 1-29 (2012)
+doi:10.1016/j.physletb.2012.08.020 [arXiv:1207.7214 [hep-ex]].
+[2] S. Chatrchyan et al. [CMS], “Observation of a New Boson at a Mass of 125 GeV with the CMS
+Experiment at the LHC,” Phys. Lett. B 716, 30-61 (2012) doi:10.1016/j.physletb.2012.08.021
+[arXiv:1207.7235 [hep-ex]].
+[3] M. J. G. Veltman, “The Infrared - Ultraviolet Connection,” Acta Phys. Polon. B 12, 437 (1981).
+[4] I. Chakraborty and A. Kundu, “Triplet-extended scalar sector and the naturalness problem,”
+Phys. Rev. D 89, no.9, 095032 (2014) doi:10.1103/PhysRevD.89.095032 [arXiv:1404.1723 [hep-
+ph]].
+[5] A. Kundu and S. Raychaudhuri, “Taming the scalar mass problem with a singlet higgs boson,”
+Phys. Rev. D 53, 4042 (1996) [arXiv:hep-ph/9410291 [hep-ph]].
+[6] A. Drozd, B. Grzadkowski and J. Wudka, “Multi-Scalar-Singlet Extension of the Standard Model -
+the Case for Dark Matter and an Invisible Higgs Boson,” JHEP 1204, 006 (2012) [arXiv:1112.2582
+[hep-ph]].
+[7] A. Drozd, “RGE and the Fine-Tuning Problem,” arXiv:1202.0195 [hep-ph].
+[8] I. Chakraborty and A. Kundu, “Controlling the fine-tuning problem with singlet scalar dark
+matter,” Phys. Rev. D 87, 055015 (2013) [arXiv:1212.0394 [hep-ph]].
+[9] I. Chakraborty and A. Kundu, “Two-Higgs doublet models confront the naturalness problem,”
+Phys. Rev. D 90, 115017 (2014) [arXiv:1404.3038 [hep-ph]].
+[10] I. Chakraborty and A. Kundu, “Naturalness problem: Off the beaten track,” Pramana 87, no.
+3, 38 (2016).
+10
+
+[11] R. L. Workman et al. [Particle Data Group], “Review of Particle Physics,” PTEP 2022, 083C01
+(2022) doi:10.1093/ptep/ptac097
+[12] R.
+Contino,
+“The
+Higgs
+as
+a
+Composite
+Nambu-Goldstone
+Boson,”
+doi:10.1142/9789814327183 0005 [arXiv:1005.4269 [hep-ph]].
+[13] P. Fayet, “Supersymmetry and Weak, Electromagnetic and Strong Interactions,” Phys. Lett. B
+64, 159 (1976) doi:10.1016/0370-2693(76)90319-1
+[14] J.
+Barnard,
+D.
+Murnane,
+M.
+White
+and
+A.
+G.
+Williams,
+“Constraining
+fine
+tuning
+in Composite Higgs Models with partially composite leptons,”
+JHEP 09,
+049 (2017)
+doi:10.1007/JHEP09(2017)049
+[15] M. van Beekveld, S. Caron and R. Ruiz de Austri, “The current status of fine-tuning in super-
+symmetry,” JHEP 01, 147 (2020) doi:10.1007/JHEP01(2020)147
+[16] I. Brivio and M. Trott, “The Standard Model as an Effective Field Theory,” Phys. Rept. 793,
+1-98 (2019) doi:10.1016/j.physrep.2018.11.002 [arXiv:1706.08945 [hep-ph]].
+[17] A. Biswas, A. Kundu and P. Mondal, “Hierarchy problem and dimension-six effective operators,”
+Phys. Rev. D 102, no.7, 075022 (2020) doi:10.1103/PhysRevD.102.075022 [arXiv:2006.13513 [hep-
+ph]].
+[18] S. Adhikari, I. M. Lewis and M. Sullivan, “Beyond the Standard Model effective field
+theory:
+The singlet extended Standard Model,” Phys. Rev. D 103, no.7, 075027 (2021)
+doi:10.1103/PhysRevD.103.075027 [arXiv:2003.10449 [hep-ph]].
+[19] S. Karmakar and S. Rakshit, “Relaxed constraints on the heavy scalar masses in 2HDM,” Phys.
+Rev. D 100, no.5, 055016 (2019) doi:10.1103/PhysRevD.100.055016 [arXiv:1901.11361 [hep-ph]].
+[20] T. Alanne and F. Goertz, “Extended Dark Matter EFT,” Eur. Phys. J. C 80, no.5, 446 (2020)
+doi:10.1140/epjc/s10052-020-7999-2 [arXiv:1712.07626 [hep-ph]].
+[21] S. Bar-Shalom, J. Cohen, A. Soni and J. Wudka, “Phenomenology of TeV-scale scalar Lepto-
+quarks in the EFT,” Phys. Rev. D 100, no.5, 055020 (2019) doi:10.1103/PhysRevD.100.055020
+[arXiv:1812.03178 [hep-ph]].
+[22] Y. Du, X. X. Li and J. H. Yu, “Neutrino seesaw models at one-loop matching: discrimination
+by effective operators,” JHEP 09, 207 (2022) doi:10.1007/JHEP09(2022)207 [arXiv:2201.04646
+[hep-ph]].
+[23] X. Li, D. Zhang and S. Zhou, “One-loop matching of the type-II seesaw model onto the
+Standard Model effective field theory,” JHEP 04, 038 (2022) doi:10.1007/JHEP04(2022)038
+[arXiv:2201.05082 [hep-ph]].
+[24] D. Zhang and S. Zhou, “Complete one-loop matching of the type-I seesaw model onto the
+Standard Model effective field theory,” JHEP 09, 163 (2021) doi:10.1007/JHEP09(2021)163
+[arXiv:2107.12133 [hep-ph]].
+[25] J. Schechter and J. W. F. Valle, “Neutrino Masses in SU(2) x U(1) Theories,” Phys. Rev. D 22,
+2227 (1980);
+R. N. Mohapatra and G. Senjanovic, “Neutrino Masses and Mixings in Gauge Models with
+Spontaneous Parity Violation,” Phys. Rev. D 23, 165 (1981);
+C. -S. Chen and C. -M. Lin, “Type II Seesaw Higgs Triplet as the inflaton for Chaotic Inflation
+11
+
+and Leptogenesis,” Phys. Lett. B 695, 9 (2011) [arXiv:1009.5727 [hep-ph]];
+A. Chaudhuri, W. Grimus and B. Mukhopadhyaya, “Doubly charged scalar decays in a type II
+seesaw scenario with two Higgs triplets,” JHEP 1402, 060 (2014) [arXiv:1305.5761 [hep-ph]].
+[26] See, e.g. E. Ma and U. Sarkar, “Neutrino masses and leptogenesis with heavy Higgs triplets,”
+Phys. Rev. Lett. 80, 5716 (1998) [hep-ph/9802445];
+T. Hambye, E. Ma and U. Sarkar, “Supersymmetric triplet Higgs model of neutrino masses and
+leptogenesis,” Nucl. Phys. B 602, 23 (2001) [hep-ph/0011192];
+D. Aristizabal Sierra, M. Dhen and T. Hambye, “Scalar triplet flavored leptogenesis: a systematic
+approach,” arXiv:1401.4347 [hep-ph].
+[27] A. Arhrib, R. Benbrik, M. Chabab, G. Moultaka and L. Rahili, “hγγ Coupling in Higgs Triplet
+Model,” arXiv:1202.6621 [hep-ph];
+A. G. Akeroyd and S. Moretti, “Enhancement of H to gamma gamma from doubly charged scalars
+in the Higgs Triplet Model,” Phys. Rev. D 86, 035015 (2012) [arXiv:1206.0535 [hep-ph]].
+[28] I. Gogoladze, N. Okada and Q. Shafi, “Higgs boson mass bounds in a type II seesaw model with
+triplet scalars,” Phys. Rev. D 78, 085005 (2008) [arXiv:0802.3257 [hep-ph]];
+H. E. Logan and M. -A. Roy, “Higgs couplings in a model with triplets,” Phys. Rev. D 82, 115011
+(2010) [arXiv:1008.4869 [hep-ph]];
+F. Arbabifar, S. Bahrami and M. Frank, “Neutral Higgs Bosons in the Higgs Triplet Model with
+nontrivial mixing,” Phys. Rev. D 87, 015020 (2013) [arXiv:1211.6797 [hep-ph]]. P. S. Bhupal Dev,
+D. K. Ghosh, N. Okada and I. Saha, “125 GeV Higgs Boson and the Type-II Seesaw Model,”
+JHEP 1303, 150 (2013) [Erratum-ibid. 1305, 049 (2013)] [arXiv:1301.3453]. C. Englert, E. Re
+and M. Spannowsky, “Triplet Higgs boson collider phenomenology after the LHC,” Phys. Rev. D
+87, no. 9, 095014 (2013) [arXiv:1302.6505 [hep-ph]]; “Pinning down Higgs triplets at the LHC,”
+Phys. Rev. D 88, 035024 (2013) [arXiv:1306.6228 [hep-ph]].
+[29] B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, “Dimension-Six Terms in the Standard
+Model Lagrangian,” JHEP 10, 085 (2010) doi:10.1007/JHEP10(2010)085 [arXiv:1008.4884 [hep-
+ph]].
+[30] K. Hagiwara, T. Hatsukano, S. Ishihara and R. Szalapski, “Probing nonstandard bosonic in-
+teractions via W boson pair production at lepton colliders,” Nucl. Phys. B 496, 66-102 (1997)
+doi:10.1016/S0550-3213(97)00208-3 [arXiv:hep-ph/9612268 [hep-ph]].
+[31] G. F. Giudice, C. Grojean, A. Pomarol and R. Rattazzi, “The Strongly-Interacting Light Higgs,”
+JHEP 06, 045 (2007) doi:10.1088/1126-6708/2007/06/045 [arXiv:hep-ph/0703164 [hep-ph]].
+[32] E. E. Jenkins, A. V. Manohar and M. Trott, “Renormalization Group Evolution of the Standard
+Model Dimension Six Operators I: Formalism and lambda Dependence,” JHEP 10, 087 (2013)
+doi:10.1007/JHEP10(2013)087 [arXiv:1308.2627 [hep-ph]].
+[33] R. Alonso, E. E. Jenkins, A. V. Manohar and M. Trott, “Renormalization Group Evolution of the
+Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology,”
+JHEP 04, 159 (2014) doi:10.1007/JHEP04(2014)159 [arXiv:1312.2014 [hep-ph]].
+[34] G. Passarino, “Veltman, Renormalizability, Calculability,” Acta Phys. Polon. B 52, no.6-7, 533
+(2021) doi:10.5506/APhysPolB.52.533 [arXiv:2104.13569 [hep-ph]].
+[35] F. Abu-Ajamieh, “Model-independent Veltman condition, naturalness and the little hier-
+archy problem *,” Chin. Phys. C 46, no.1, 013101 (2022) doi:10.1088/1674-1137/ac2ffa
+[arXiv:2101.06932 [hep-ph]].
+12
+
+[36] S. Das Bakshi, J. Chakrabortty and S. K. Patra, “CoDEx: Wilson coefficient calculator connecting
+SMEFT to UV theory,” Eur. Phys. J. C 79, no.1, 21 (2019) doi:10.1140/epjc/s10052-018-6444-2
+[arXiv:1808.04403 [hep-ph]].
+[37] Anisha, S. Das Bakshi, S. Banerjee, A. Biek¨otter, J. Chakrabortty, S. Kumar Patra and
+M. Spannowsky, “Effective limits on single scalar extensions in the light of recent LHC data,”
+[arXiv:2111.05876 [hep-ph]].
+[38] A. Carmona, A. Lazopoulos, P. Olgoso and J. Santiago, “Matchmakereft: automated tree-level
+and one-loop matching,” SciPost Phys. 12, no.6, 198 (2022) doi:10.21468/SciPostPhys.12.6.198
+[arXiv:2112.10787 [hep-ph]].
+[39] R. Ghosh, B. Mukhopadhyaya and U. Sarkar, “The ρ parameter and the CDF W-mass anomaly:
+observations on the role of scalar triplets,” [arXiv:2205.05041 [hep-ph]].
+[40] J. Ellis,
+M. Madigan,
+K. Mimasu,
+V. Sanz and T. You,
+“Top,
+Higgs,
+Diboson and
+Electroweak Fit to the Standard Model Effective Field Theory,” JHEP 04, 279 (2021)
+doi:10.1007/JHEP04(2021)279 [arXiv:2012.02779 [hep-ph]].
+13
+

-NE3T4oBgHgl3EQfrQru/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:13507e0c5117a4d559c65a09a1c5780c45d64f9dad545934d8e62167fc624373
+size 370513

-dAzT4oBgHgl3EQfSvvG/content/tmp_files/2301.01238v1.pdf.txt

@@ -0,0 +1,554 @@
+1 
+EL ELIXIR DE LA ENERGÍA ETERNA 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Abstract 
+ 
+  The recent announcement of a purported breakthrough result in inertial nuclear fusion at NIF 
+(Lawrence Livermore Laboratory, USA) has aroused a tide of media and public interest. The 
+excitement has been generalized to the whole field of research in fusion energy with,  in its wake,  
+announcements of an imminent advent of the cure for the energetic crisis and the aggravating 
+influence in the climate change associated to the fossil fuels.  This opinion article is intended to 
+show that such expectations are not founded on sound scientific bases and that there is a long way 
+until  the practical production of electricity from nuclear fusion is achieved, if ever.        
+ 
+Resumen 
+ 
+  El reciente anuncio de un supuestamente trascendental resultado en fusión nuclear  inercial en  
+NIF (Lawrence Livermore Laboratory, EEUU de Norteamérica) ha desatado un enorme interés en el 
+público y los medios de comunicación. El entusiasmo se ha trasladado a todo el campo de la 
+investigación en fusión para la producción de energía con, a su estela, anuncios de la llegada 
+inminente de la solución a la crisis energética y al efecto agravante del cambio climático asociado a 
+los combustibles fósiles. Este artículo de opinión pretende poner de manifiesto que tales 
+expectativas no están fundadas en bases científicas sólidas y que hay un largo camino por recorrer 
+hasta que se logre, a niveles prácticos,  la producción de electricidad a partir de la fusión nuclear, si  
+se consigue alguna vez.    
+ 
+ 
+ Introducción   
+ 
+  Recientemente se anunció con extraordinario aparato mediático un nuevo hito alcanzado en la 
+fusión nuclear, muy oportunamente publicitado en el contexto actual de crisis energética.  En 
+principio no hay nada que objetar a ello: en la comunidad científica es bien conocida la necesidad 
+de financiación que tienen los grupos de investigación, y no digamos los grandes laboratorios como 
+el Lawrence Livermore Laboratory, que deben anunciar sus logros científicos a fin de despertar el 
+interés de la opinión pública, que busca soluciones tangibles a sus problemas inmediatos, y, con 
+ello, captar la atención de las instancias políticas, siempre ávidas  de réditos demoscópicos.   
+ 
+   No obstante, opino que se está vertiendo con demasiada frecuencia información sesgada que 
+induce a confusión.  Comencemos con el detonante del presente texto,  la noticia que ha 
+desencadenado el frenesí mediático: el DOE (Department of Energy)  norteamericano anunció a 
+bombo y platillo el pasado 13 de diciembre que en la instalación NIF (National Ignition Facility) 
+del Lawrence Livermore Laboratory  se acababa de conseguir   superar el breakeven en un 
+experimento de fusión,  lo cual  simplemente consiste en conseguir más energía que la suministrada. 
+Suena muy bien y prometedor, pero conviene puntualizar. En primer lugar, esto se ha conseguido 
+mediante la compresión con 192 láseres de altísima potencia sincronizados en un brevísimo pulso 
+del orden de varios nanosegundos1 (la mayor concentración energética mediante láser jamás 
+José Manuel Quesada Molina 
+Departamento de Física Atómica, Molecular y Nuclear 
+Universidad de Sevilla 
+
+2 
+conseguida) de una diminuta cápsula de diamante  conteniendo un pellet ultracongelado con dos 
+isótopos del hidrógeno, deuterio y tritio (DT)2. Ello es un mecanismo completamente diferente al 
+confinamiento magnético que se utiliza en los llamados tokamaks3, donde el confinamiento se 
+consigue mediante campos magnéticos con geometría toroidal (donut). En la actualidad el diseño 
+tipo tokamak  es el  más ampliamente utilizado, particularmente en  Europa,  donde bajo estas 
+premisas desde comienzos de los años 90 se realizan experimentos en  el JET (Joint European 
+Torus) en Gran Bretaña, con financiación de la Unión Europea (al menos era sí hasta el Brexit, ya 
+que actualmente está en fase terminal),  y actualmente está en construcción el ITER (acrónimo de 
+International Thermonuclear Experimental Reactor) en Cadarache, Francia. Por lo tanto, el logro 
+alcanzado en el NIF es muy difícilmente extrapolable a la apuesta científico-técnica mayoritaria en 
+curso, que es el confinamiento magnético;  es más, dicha vinculación se me escapa por completo. 
+Por lo tanto, considero que ésta es una matización clave, que se debe realizar claramente  desde el 
+principio: ambos mecanismos (compresión inercial – mediante láser u otro tipo de haces de 
+partículas - y confinamiento magnético), aunque comparten finalidad, difieren esencialmente y los 
+pretendidos logros en uno no son trasladables al otro.  
+ 
+¿Es todo esto como se anuncia y promete? 
+ 
+  A raíz de la noticia, y aprovechando su tirón mediático, he podido leer nuevamente (para mi gran  
+sorpresa) algo que lleva repitiéndose desde los orígenes del desarrollo de la fusión nuclear: se 
+trataría de reproducir en la Tierra el proceso que tiene lugar en el Sol y permite la vida; es más, se le 
+llega a poner fecha: un reactor de fusión conectado a la red eléctrica en 10 años. Lo primero no es  
+cierto en sentido estricto y lo segundo es, simplemente, un despropósito que prefiero atribuir a una 
+mala interpretación periodística (que, lamentablemente, cala en el imaginario del público no 
+avisado).  El tiempo dirá y la hemeroteca lo reflejará.  
+ 
+  En el Sol se quema hidrógeno, un recurso  inagotable a nuestra escala, para producir partículas 
+alfa, que son estables, es decir sin producir residuos radiactivos4. Todo extraordinariamente 
+prometedor, salvo por un detalle: no es posible realizarlo en la Tierra, ya que entra dentro del 
+dominio de la ciencia ficción. El motivo: la probabilidad de que dos núcleos de hidrógeno (es decir, 
+dos protones) superen su repulsión eléctrica mutua y se fundan es tan pequeña que ni siquiera ha 
+podido medirse experimentalmente en un laboratorio. En el Sol ello se produce debido a  las 
+monstruosas (a escala terrestre) densidades de masa que se alcanzan en su interior por la atracción 
+gravitatoria de su enorme masa (también comparada con la de la Tierra); pero en la Tierra  no son 
+alcanzables tales densidades5. Este es el motivo de que haya que recurrir  a otras mezclas de 
+fusionantes : deuterio-deuterio (DD),  la ya mencionada deuterio-tritio (DT), etc.. Es decir, lo que se 
+pretende realizar en la Tierra es parecido a lo que ocurre en el Sol, pero no es lo mismo; en ambos 
+casos hay un mecanismo común, pero el combustible es diferente. En particular, la única que se 
+vislumbra con posibilidades de permitir la fusión para producir energía eléctrica es la combinación 
+DT6, que es la adoptada en todos los proyectos vigentes  que pretenden conducir a ese objetivo.  
+ 
+  El deuterio es abundante (constituye una pequeña fracción del hidrógeno natural) y estable. Pero el 
+tritio no es ninguna de las dos cosas: es radiactivo  y, por lo tanto, no existe naturalmente; es decir, 
+hay que producirlo. Esto cambia bastante el panorama de supuestas bondades del combustible (casi 
+infinito, según se anuncia): el tritio, como isótopo del hidrógeno, se comporta químicamente (y, por 
+lo tanto, biológicamente) exactamente igual que el hidrógeno normal ; es decir, dado el papel 
+central del hidrógeno en el ciclo de la vida, el tritio se incorpora al mismo sin que haya forma de 
+separarlo químicamente, porque es hidrógeno (aunque radiactivo). Su vida media de 12.6 años hace 
+que en ese tiempo su cantidad se reduzca a la mitad, pero  en un reactor de fusión ha de producirse 
+continuamente, para lo cual se coloca en su  borde exterior una manta de litio (que debe 
+
+3 
+enriquecerse en su minoritario isótopo adecuado), que al ser bombardeada con neutrones 
+procedentes de las fusiones DT produce el tritio que regenera el consumido. Al menos ese es el 
+objetivo que se pretende alcanzar (sobre todo, en la tasa suficiente).  
+ 
+  Las consecuencias  de la infiltración y fuga del tritio a través de las paredes del reactor a las 
+enormes temperaturas a las que se pretende que funcione se conocen sólo parcialmente, ya que los 
+valores que se manejan  se basan en extrapolaciones. Por lo tanto el combustible previsto no es casi 
+infinito, ni es limpio ni seguro ni  barato. El tritio es tan problemático7 que en JET se ha trabaja en 
+la medida de lo posible sólo con  hidrógeno o sólo con deuterio, extrapolándose   las tasas de 
+reacción a la mezcla DT. De este modo se obtiene el factor Q (o eficiencia energética, que es el 
+cociente entre la potencia conseguida y la consumida) extrapolado, con el llamativo resultado de 
+que cuando se realizó el experimento con la mezcla DT  el valor de Q  obtenido fue 
+aproximadamente la mitad; lo cual muestra algo bien conocido en física e ingeniería, que es el 
+riesgo de las extrapolaciones, a la vez que pone en evidencia la problemática asociada al uso del 
+tritio. Lo anterior añade otra incógnita más a la pretendida limpieza  radiológica de la fusión para 
+producir energía eléctrica a partir de combustible limpio, accesible e inagotable, según otro  lugar 
+común en muchas declaraciones leídas en la prensa: “porque se extrae del agua del mar”.  
+ 
+Jugando con las definiciones 
+ 
+ Otro aspecto a destacar de la citada noticia tiene que ver con el ya mencionado logro del breakeven 
+en el NIF. Este consistió en  alcanzar  un factor Q de ganancia energética  de fusión  de 1.54, es 
+decir que se obtuvo 1.54 veces más energía que la se invirtió . Al margen de otras consideraciones 
+en las que entraré más adelante, resulta llamativo (por expresarlo suavemente) el cambio de 
+definición que ha conducido a este anunciado éxito. El NIF cambió hace algunos años la definición 
+del Q para colocar en el denominador (potencia que hay que suministrar a los láseres para 
+comprimir y calentar el plasma)  sólo la fracción que éstos devuelven en forma de radiación 
+ultravioleta para comprimir y calentar, es decir sólo la fracción aprovechable. Teniendo en cuenta 
+que la eficiencia de los láseres es muy baja (en torno al 1%), en rigor hay que dividir por toda la 
+energía invertida, es decir dividir el aunciado factor de ganancia Q = 1.54  por 100, con lo cual se 
+está aún muy lejos de recuperar lo invertido. Muy lejos.  Obviamente, esta redefinición unilateral 
+del factor Q por parte del NIF recibió severas críticas8, pero el hecho de que no se haya reflejado en 
+las noticias esta matización (¡de un factor 100!) por parte de sus voceros (o al menos yo no la he 
+encontrado) permite hacerse una idea del poder del lobby que hay detrás.  
+ 
+  Toda la discusión anterior se ha realizado omitiendo  un detalle  adicional que considero  
+fundamental para tener una visión clara de la situación real: no toda la energía liberada en la fusión 
+(el numerador del factor Q) es aprovechable para producir calor y, con ello, la energía eléctrica que 
+se pretende obtener.  En la fusión DT el 80% de la energía producida se la llevan los neutrones en 
+forma de energía cinética, siendo las partículas alfa (que se llevan el 20% restante) las responsables 
+de la mayor parte del calentamiento9.  Por el contrario, en una fisión del combustible típico de las 
+centrales nucleares de fisión (U235),  sólo en torno al 5% de la energía se la llevan los neutrones, 
+mientras que el resto corresponde a los fragmentos de fisión, núcleos de tamaño medio muy 
+cargados eléctricamente, que son los responsables del calentamiento de las barras de combustible, 
+que a su vez calientan el refrigerante (normalmente agua) encargado de transportarlo. Por lo tanto, 
+incluso sin la redefinición del NIF, el factor Q dista  de ser una medida realista de la rentabilidad 
+energética del proceso de fusión, ya que no sólo una fracción minoritaria de la energía liberada en la 
+fusión es aprovechable para producir calor, que es lo que interesa.  
+ 
+
+4 
+  En la misma línea de información sesgada por parte de los gabinetes de comunicación, ITER hizo 
+oficialmente pública una información que claramente conducía a error de interpretación. 
+Concretamente, se afirmaba que ITER sería capaz de producir 500 MW10 de potencia a partir de 50 
+MW de potencia suministrada. De ahí se infería lógicamente que esos 50 MW suministrados se 
+referían a la potencia total eléctrica invertida, no a la  calorífica finalmente suministrada al plasma 
+(es decir, igual que en el caso de NIF con los láseres). Ante las críticas recibidas11, tuvieron que 
+rectificar. Debido a la la eficiencia del proceso de conversión (siempre menor que la unidad,  
+usualmente mucho menor, al igual que en el caso del NIF),  la primera es  muy superior, 
+estimándose  en más de 300 MW necesarios para mantener la fusión12. Además, los 500 MW 
+producidos son totales, de los cuales, como ya se ha indicado, aproximadamente  el 80% 
+corresponde a los neutrones rápidos, que son mucho menos eficientes produciendo calor (sólo son 
+capaces de transferir una parte del mismo al medio antes de escapar), que, al no ser utilizable para 
+mantener la temperatura del plasma,  ITER  propone13 aprovechar calentando el agua del circuito 
+refrigerante de la  manta que envolverá la cámara de vacío para producir electricidad. Toda la 
+responsabilidad del mantenimiento de la temperatura del plasma recaerá sobre las partículas alfa,  
+que depositarán directamente en el mismo toda su  energía  (que, recordemos, es sólo el 20 % de la 
+energía en cada fusión). La regeneración del tritio (caro, escaso y que debe producirse en reactores 
+nucleares de fisión, principalmente) necesario para mantener la fusión se pospone para para una 
+fase posterior de ITER, donde se experimentará con la capacidad del isótopo minoritario Li6 del 
+litio natural (que debe ser enriquecido para ello) para producirlo en la suficiente cantidad para 
+mantener la reacción14. Nuevamente nos movemos en el campo de las expectativas, sólo la 
+experimentación demostrará la viabilidad de la propuesta.  
+ 
+Un poco de historia  
+ 
+  Las radicales diferencias entre los procesos de fusión y fisión nuclear son la causa de que entre la 
+primera reacción nuclear explosiva en cadena (Trinity , 16 de julio de 1945 en Alamogordo, Nuevo 
+México, EEUU) y la primera producción comercial de energía eléctrica mediante la fisión 
+controlada15 (18 de diciembre  1957, en Shippingport, Pennsylvania, EEUU) transcurriesen 
+solamente 12 años, mientras que tras  la primera explosión termonuclear16 (Ivy Mike, 1 de 
+noviembre de 1952 en el atolón Enewetak, en las Islas Marshall) aún no se ha conseguido 
+domesticar la fusión para mantenerla bajo control y producir energía aprovechable.  
+ 
+  En el caso de la fisión se pretende (y consigue desde el año 1942) mantener controlada una 
+reacción en cadena, donde los garantes de esa  continuidad son los neutrones producidos en cada 
+reacción. En el caso de la fusión los neutrones no juegan ningún papel en mantenimiento de la 
+misma, sino que el  agente garante de la  reacción en cadena es el calor producido, que se debe 
+traducir en temperatura (manteniendo la densidad). Cuando no se pretende el control de la misma 
+sino todo lo contrario (bombas), ello se hace por fuerza bruta (nunca mejor dicho) recurriendo  a la 
+extraordinaria presión de radiación originada por el fulminante de fisión17, sin que ésta escape antes 
+de conseguir instantáneamente su objetivo (todo ocurre en unos pocos microsegundos18). En 
+cambio, para mantener la reacción de fusión en un reactor no se puede, obviamente,  recurrir a ese 
+mecanismo explosivo y se debe conseguir que el calor generado por las reacciones de fusión se 
+recicle sin escapar para mantener la temperatura, al tiempo que la densidad se mantenga 
+temporalmente. Una empresa formidable, que aún está por conseguirse.     
+paral   
+ 
+ 
+
+5 
+ 
+¿Por qué se persigue conseguir tan elevadas densidades y temperaturas en un futuro reactor 
+nuclear de fusión?  
+ 
+  Porque es preciso conseguir que núcleos atómicos ligeros superen la repulsión debida a su carga 
+eléctrica y se fundan en un núcleo mayor y  alguna otra partícula emergente; y además que lo hagan 
+en la tasa (velocidad a la que se producen las reaccciones)  suficiente. La clave radica en que la 
+suma de las masas de los productos de la reacción es ligeramente inferior a la masa de los 
+reaccionantes, convirtiéndose esa diferencia de masa m en energía E, según la archiconocida 
+fórmula de Einstein E=m c2, donde c es la velocidad de la luz.  Este mecanismo es el opuesto al de 
+la fisión nuclear, aunque la consecuencia es la misma: conversión de masa en energía. En la fisión 
+un núcleo pesado captura un neutrón y se rompe en dos fragmentos de aproximadamente la mitad 
+de su masa y  varios neutrones. Aquí tenemos por tanto una gran diferencia cualitativa: a diferencia 
+de la fusión, en la fisión el agente desencadenante (el neutrón, que como su nombre indica, carece 
+de carga eléctrica) no tiene que superar en primer lugar la repulsión eléctrica por parte del núcleo 
+(donde hay protones y neutrones que, en todo caso, lo atraen por la llamada interacción nuclear o 
+fuerte, que es de corto alcance). Por ello en la fisión, si se dan las circunstancias adecuadas 
+(características del núcleo progenitor y energía del neutrón incidente) el núcleo compuesto 
+resultante, que se forma en un estado excitado, se rompe espontáneamente en  busca de una mayor 
+estabilidad del sistema, es decir fisiona. Nos encontramos ante una situación radicalmente diferente 
+a la de la fusión, donde los dos intervinientes han de superar su repulsión mutua (ambos están 
+cargados positivamente), lo cual implica enormes temperaturas para conseguirlo19. Además la 
+densidad ha de ser altísima para que la tasa de reacción sea la suficiente, como comentaré a 
+continuación.  
+ 
+La tasa de reacción es la clave 
+ 
+  La tasa de una reacción nuclear (es decir, el número de reacciones por unidad de tiempo) es 
+proporcional a la densidad de blancos20, al flujo de proyectiles que los bombardean y  a la 
+probabilidad de que la reacción se produzca una vez que proyectil y blanco colisionan. Esta última 
+cantidad es a su vez proporcional a una magnitud llamada sección eficaz21, que viene determinada 
+por la estructura nuclear intrínseca de cada pareja proyectil-blanco y en la cual nuestro margen de 
+maniobra está limitado a la velocidad relativa, es decir, a la  temperatura. Por lo tanto, para cada 
+pareja de proyectil y blanco reaccionantes (fusionantes o fisionantes), conseguir una tasa de 
+reacción suficiente exige unos valores adecuados de densidad y temperatura.  
+ 
+  La tasa de reacción es la clave para producir energía aprovechable, porque las reacciones de fusión  
+se producen  rutinariamente en laboratorio mediante el uso de aceleradores (controladas, pero no 
+automantenidas ya que exigen aporte continuo de energía). Una de las fuentes habituales de 
+neutrones es la llamada DT (deuterio-tritio), la misma mezcla prevista en ITER,  en la cual 
+mediante un acelerador se bombardea con deuterones un blanco de tritio gaseoso, en cada una de 
+cuyas reacciones de fusión se liberan unos 17.6 MeV22 de energía, que se reparten entre una 
+partícula alfa (núcleo de helio, que se lleva aproximadamente el 80% de diche energía) y un neutrón 
+(de alta energía en la jerga especializada,  que se lleva el 20% restante). Pero la producción 
+energética en forma de calor (debido  mayoritariamente a la energía que transportan las partículas 
+alfa23) es ínfima debido a los valores   de las tasas de reacción implicadas. Es decir,  esta fusión DT 
+( y lo mismo se puede decir de las fuentes de neutrones DD) no sirve para producir energía  
+eléctrica aprovechable, su finalidad es producir  neutrones rápidos.    
+ 
+ 
+
+6 
+ 
+ 
+El balance energético 
+ 
+  En principio, un argumento  en favor de la fusión nuclear frente a la fisión es la  energía específica  
+(o energía por unidad de masa). Vamos a explicarlo. Una característica de los núcleos atómicos, 
+aunque no exclusiva, ya que lo siguiente es aplicable a cualquier sistema regido por las leyes de la 
+Física Cuántica (es decir a todos), es que su masa es menor que  la suma de las masas de sus 
+constituyentes por separado. Esa diferencia, traducida en energía por la fórmula de Einstein,  es lo 
+que se conoce como energía de ligadura. Por lo tanto si en una reacción nuclear pasamos de una 
+situación con menos ligadura (más masa) a otra de más ligadura (menor masa), la diferencia se 
+transforma en energía cinética de los productos y radiación. Y esa es la energía que, en forma de 
+calor, se utiliza (en un reactor nuclear de fisión) o debería algún día poder  utilizarse (en un reactor 
+nuclear de fusión)  para producir energía eléctrica.   Si colocamos los isótopos conocidos (es decir 
+tipos de núcleos atómicos) en orden creciente con sus masas, comenzando en el hidrógeno, 
+encontramos que la ligadura por nucleón va aumentando en promedio  hasta el Fe56 (núcleo de 
+hierro con 26 protones y 30 neutrones, donde alcanza casi 9 MeV por nucleón); a partir de ese 
+punto comienza a disminuir suavemente hasta el final de la tabla , donde llega a unos 7.5  MeV por 
+nucleón en la región de isótopos que nos interesa (la llamada zona de los actínidos). Ello quiere 
+decir que, cuando dos isótopos ligeros (por debajo de la masa del Fe56) se fusionan, el resultado 
+está  más ligado en general, es decir  tiene menos masa, y esa pérdida de masa se transforma en 
+energía. Por ejemplo en la típica reaccción DT la ligadura del deuterón son unos 2.2 MeV, es decir 
+aproximadamente 1.1 MeV por nucleón ; la ligadura del tritio son unos 8.5 MeV, es decir unos 2.8 
+MeV por nucleón; la ligadura de la partícula alfa son unos 28.3 MeV, es decir aproximadamente 7.1 
+MeV por nucleón. Por lo tanto, se pasa de una ligadura incial en el sistema DT de aproximadamente 
+10.7 MeV a los 28.3 MeV de la partícula alfa, es decir hay una ganancia de energía ligadura de unos 
+3.5 MeV por nucleón inicial (~17.6/5)   eV.  La situación opuesta se presenta en el otro extremo de 
+la tabla de isótopos cuando un núcleo pesado, por ejemplo el U235, captura un neutrón y fisiona. La 
+energía de ligadura del U235  son unos 1786.7 MeV, es decir aproximadamente  unos 7.6 MeV por 
+nucleón,  y la de dos típicos productos de fisión (recordemos que es un proceso probabilístico) está 
+próxima a la máxima del Fe56, digamos que en torno a los 8.5 MeV por nucleón; por lo tanto se ha 
+ganado aproximadamente 0.9 MeV por nucleón en energía de ligadura. Multiplicando esta cantidad 
+por los 236 nucleones del núcleo compuesto inicial (el de U235  más el neutrón absorbido) resultan 
+unos 212 MeV de energía liberada en una fisión típica, cantidad bastante próxima a los valores 
+medidos experimentalmente. Desde este punto de vista, en principio, la fusión DT es claramente 
+más interesante, ya que la ganancia neta  de ligadura por nucleón del combustible  es casi 4 veces 
+mayor (lo cual se traduce en una energía específica unas 4 veces mayor  del combustible DT 
+respecto del U235), pero hay que considerar que la mayor parte de esa energía cinética corresponde 
+a los neutrones (el ya mencionado 80% típicamente en el caso de la fusión DT, frente al 
+aproximadamente  5% en el caso de la fisión del U235), que es muy poco  aprovechable para 
+producir calor, es decir, en última instancia energía eléctrica. En resumidas cuentas, un reactor de 
+fusión es una magnífica fuente de neutrones muy energéticos, otro asunto muy diferente es cómo 
+aprovechar la energía producida (de la que esos neutrones se llevan lel 80%), ya que sólo una 
+pequeña parte de ella se podrá transformar en calor y, aún menos, en energía eléctrica a partir de él.  
+ 
+¿Por qué es tan difícil conseguir la fusión nuclear automantenida? 
+ 
+  En el caso de la fisión, en la que se basan las centrales nucleares actuales,  la  densidad de núcleos  
+está fijada por ser el combustible  un medio sólido (aunque puede ser líquido, que para el caso es lo 
+mismo) y  la sección eficaz (es decir, la probabilidad de que se produzca la reacción) se hace 
+
+7 
+enorme en los núcleos fisionables para una energía adecuada de los neutrones (recordemos que los 
+neutrones no tienen carga eléctrica, por lo que  se cuelan sin obstáculo en los núcleos blanco, y  que 
+la sección eficaz varía con su energía – es decir con la temperatura del medio que los  termaliza, es 
+decir que los frena- debido a los detalles de la estructura nuclear). Por este motivo, para  mantener 
+bajo control la tasa de reacciones nucleares basta con controlar con precisión la población 
+neutrónica.  Con ello se consigue una tasa de reacción que  libera  la cantidad de calor suficiente 
+para ser transformado comercialmente en energía eléctrica.   
+ 
+ Por el contrario, en el caso de la fusión (la que nos anuncian como fuente de energía limpia e 
+inagotable del futuro) la sección eficaz  es extraordinariamente pequeña comparada con la de fisión. 
+Concretamente, en la fusión DT la sección eficaz  a la temperatura prevista de unos 150-200 
+millones de grados24  es  aproximadamente una cienmillonésima  de la sección eficaz de fisión de 
+un núcleo de U235 bombardeado por neutrones termalizados (es decir con energía óptima para 
+fisionar eficazmente este isótopo) en un reactor convencional  refrigerado por agua ligera a presión 
+(LWR-PWR), que opera a  unos 350ºC. Conseguir la temperatura necesaria en el plasma de fusión  
+(que es la sopa de núcleos y electrones en que se transforma la materia esas temperaturas), es ya en 
+sí misma una empresa formidable, pero a eso hay que añadir la necesidad de alcanzar  una densidad 
+suficiente de dicho plasma  y , además, que ambos parámetros se mantengan durante el tiempo 
+suficiente para mantener una tasa de reacción que la haga utilizable para producir energía.   En las 
+bombas  de fusión (empleadas con fines bélicos) se  utilizan una o varias bombas de fisión para 
+conseguir  simultáneamente los objetivos anteriores (compresión y calentamiento), y en ellas, 
+obviamente, ni el control ni el mantenimiento temporal son necesarios, sino todo lo contrario, 
+desafortunadamente. Pero evidentemente  este mecanismo está excluido para aplicaciones pacíficas, 
+de forma que el  objetivo de mantener el plasma en esas condiciones sólo se plantea  de forma 
+pulsada, ya sea mediante compresión con láseres  o confinamiento magnético (con la que se 
+pretende llegar a varios centenares de segundos).  Ello explica el ya mencionado  hecho de que 
+desde Ivy Mike en 1952 hayan transcurrido 70 años sin alcanzar la fusión controlada para la 
+producción comercial de energía eléctrica, siendo las predicciones más optimistas de unos 30, 40, 
+¿50? años adicionales para conseguirlo. Porque ITER , cuando funcione, está destinado ser la 
+prueba de concepto  científica  de la fusión controlada y automantenida para producir energía 
+eléctrica. Habrá que esperar a DEMO (como su nombre indica) para la prueba de concepto de 
+ingeniería, que demuestre que es posible verter energía neta a la red eléctrica.  Y  mientras tanto 
+debe continuarse investigando exhaustivamente en los efectos que el extraordinario bombardeo con 
+neutrones de alta energía  a semejantes temperaturas induce en las propiedades de los materiales 
+estructurales del reactor , en particular la fragilización,  aparición de fallas y deformaciones25. Y tras 
+todo ello, si se alcanza ese punto (cosa que en el mejor de los casos podrán ver nuestros nietos o 
+bisnietos, porque ninguno de nosotros tendrá la oportunidad de sonreir consultando la  hemeroteca) 
+habrá de demostrarse la viabilidad económica de esta fuente de energía, que dados los enormes 
+costes de desarrollo y su descomunal consumo energético previo acumulado en forma de consumo 
+de combustibles fósiles y energía eléctrica de origen nuclear de fisión  (hay estudios al respecto) 
+dista mucho de estar claro. 
+ 
+El tamaño importa 
+ 
+  Considero también pertinente mencionar el previsible tamaño de una central de fusión para 
+producción de energía eléctrica, si algún día llega a construirse. Una de las muchas críticas que  se 
+han realizado en contra de las centrales nucleares de fisión es la gran concentración de 
+infraestructuras y capital que implican y su tamaño,  que van radicalmente en contra de una 
+producción distribuida y cercana a los puntos de consumo. Ello sin olvidar los riesgos inherentes a 
+dicha concentración provenientes de posibles ataques terroristas. A estas alturas del texto, creo que 
+
+8 
+resulta evidente que en una central de fusión estos aspectos criticados en una central de fisión 
+aumentan  hasta dimensiones desconocidas hasta la fecha. No hay más que comparar el tamaño de 
+ITER con su precursor JET y, aún más, con el previsto para DEMO. Los gabinetes de comunicación 
+de los proyectos de fusión (NIF, ITER) nos inundan con informaciones grandilocuentes donde 
+siempre aparece  lo más de lo más: los láseres más potentes del mundo (en el caso del NIF), los 
+imanes superconductores mayores del mundo, la vasija de vacío mayor del mundo, la soldadura 
+electrónica más sofisticada  del mundo, etc .. (en el caso de ITER).  Son innegables logros de 
+ingeniería a gran escala (y puede que ya eso de por sí justifique el esfuerzo y la energía  invertidos),  
+pero no deberían hacernos perder la visión de conjunto: de lo que se trata es de producir energía 
+aprovechable en un futuro no demasiado lejano. Además, tampoco conviene olvidar que dicho 
+desarrollo en busca de cuanto más grande mejor (porque esa es la única manera conocida de 
+alcanzar las extremas condiciones descritas anteriormente) va en el sentido opuesto al seguido en 
+los modernos prototipos de centrales de fisión modulares, destinados a su instalación a escala local, 
+de los cual hay ya uno en fase operacional en Rusia26 y otro en China  en fase avanzada de 
+construcción27; hay muchos otros diseños  avanzados y prometedores en Japón, Europa y EEUU, 
+que hasta ahora no se han podido llevar a la práctica. El hecho de que hayan sido precisamente 
+Rusia y China los países que primero hayan llevado a la práctica esta idea innovadora, dice mucho 
+del panorama geopolítico actual, donde la segunda (a Rusia aún le quedan rescoldos científicos y 
+tecnológicos de la época soviética) se ha convertido a pasos agigantados en un referente mundial en 
+ciencia y tecnología en todas las áreas estratégicas. Igualmente, se continúa investigando 
+exhaustivamente en el ciclo de fisión del torio28,29, desarrollando la tecnología para reactores más 
+pequeños (llegando a la escala del MW), más seguros y con menos producción de residuos. No 
+olvidemos que ITER , cuando entre en funcionamiento, consumirá del orden de 300 MW sólo para 
+mantener la temperatura  del plasma.  
+ 
+Epílogo 
+ 
+  El proyecto ITER, al igual que la Estación Espacial Internacional (ISS, de sus siglas en inglés)  
+surgieron en las mismas fechas (años 90) y con los mismos loables propósitos (fomentar la 
+colaboración científico-técnica internacional), inmediatamente  tras el derrumbe del bloque 
+soviético y el comienzo de una época de absoluto dominio del bloque llamado occidental  (aunque 
+incluye también a Japón, Corea del Sur y, por supuesto, Australia y Nueva Zelanda) liderado por los 
+EEUU de Norteamérica y  la postración absoluta de la otra antigua potencia hegemónica, Rusia; 
+China, aunque despegando, aún contaba poco.  Era la época del famoso Final de la Historia de 
+Francis Fukuyama. No creo necesario resaltar cómo ha cambiado el panorama internacional. La 
+ISS, con Rusia retirándose, además de la poca relevancia de los resultados científicos obtenidos, 
+está abocada a convertirse pronto en un trozo más de chatarra orbital  destinada a desintegrarse en 
+unos 10 años (si no antes, el silencio mediático es poco prometedor en ese sentido). Opino que, 
+aparte de los innegables avances tecnológicos asociados a su desarrollo,  ese es su principal ( y 
+probablemente) único éxito.   
+ 
+  Los plazos han ido alargándose sin cesar: De la fecha inicialmente prevista de las primeras pruebas 
+con plasma en ITER, 2016, se pasó a 2025 y hasta 2035 para las pruebas con la mezcla real DT. Los 
+rumores sugieren insistentemente un nuevo alargamiento y la situación geopolítica  mundial (al 
+margen de los enormes problemas científico-técnicos asociados al proyecto) apunta a ello. Para 
+DEMO ya ni siquiera se dan fechas concretas, sólo se habla de que será una realidad en la segunda 
+mitad de la centuria. Vienen a la mente las palabras del Quijote: “Cuán largo me lo fiais, amigo 
+Sancho”.   
+ 
+
+9 
+1 Un nanosegundo es una milésima de una millonésima de segundo. 
+ 
+2 El núcleo de hidrógeno, que consiste en un único protón, es el más simple de la Naturaleza y, obviamente es estable, es 
+decir, no radiactivo. El de deuterio o deuterón consiste en un protón más un neutrón y es también estable. El de 
+tritio consiste en un protón más dos neutrones y es radiactivo, con una semivida de 12.6 años (el tiempo en que 
+tardan en desintegrarse la mitad de sus núcleos a partir de una población incial). 
+ 
+3 Inicialmente la idea del tokamak, que es una  palabra rusa porque  fue un concepto propuesto en la Unión Soviética en 
+los años 50, fue descartada por los investigadores norteamericanos, que llevaban desde los primeros años 50 
+trabajando en la Universidad de Princeton y el Laboratorio Nacional de Los Alamos tratando de controlar la fusión 
+nuclear para usos civiles mediante confinamiento magnético con la configuración llamada stellarator  (de aspecto 
+exterior muy parecido al tokamak, pero con importantes diferencias conceptuales entre ambos), en paralelo al 
+desarrollo del programa militar; de hecho ambos proyectos  compartían inicialmente gran parte de su nombre en 
+clave: Matterhorn-B para el militar (la B de bomb) y Matterhorn-S para la civil (con la S de stellarator) . A 
+comienzos de los años 60 los científicos norteamericanos tuvieron que rendirse a la evidencia de que la idea del 
+tokamak funcionaba mejor en muchos aspectos y comenzaron a investigar mayoritariamente  basándose en ella, 
+aunque varios grupos continuaron   la investigación en la línea original (stellarator), que aún se mantiene hoy en 
+día. Como puede fácilmente deducirse, la íntima interconexión en el entramado científico-militar norteamericano 
+era evidente desde los inicios en estas investigaciones. Abundando en esta idea, cabe resaltar que el Livermore 
+National Laboratory tiene una bien conocida vinculación íntima con la industria militar estadounidense en forma de 
+contratos de investigación  orientada tanto a las armas de fisión nuclear como a las basadas en láseres de alta 
+potencia (sobre todo a partir de la llamada Guerra de las Galaxias, que promovió su presidente Ronald Reagan en 
+los años 80).   
+ 
+4 La explicación del mecanismo de producción de energía en el Sol fue propuesta por Hans Bethe en su genial y 
+clarividente artículo (uno más entre tantos de los suyos)  “Energy Production in Stars”, Physical Review 55 (1939) 
+p. 434 , considerado uno de  los diez mejores de la Historia de la Física moderna en una clasificación del Instituto 
+Niels Bohr de Copenhague. Como curiosidad, este artículo fue inicialmente retirado por el autor para poder 
+presentarlo a un concurso de ideas científicas inéditas (que obviamente ganó), con cuyo premio costeó la mudanza 
+de su madre (judía perseguida en Alemania) a los Estados Unidos de América del Norte.  En el mismo, entre otras 
+muchas especulaciones basadas en las evidencias entonces disponibles, el autor propone una cadena que se inicia 
+con la fusión de dos núcleos de hidrógeno, es decir dos protones, y que globalmente se traduce en  que a partir de 4 
+protones se forma una partícula alfa con gran liberación de energía. 
+ 
+5 A no ser que se consiga crear algún día en la Tierra algo parecido a  una estrella de neutrones (en este caso de 
+protones, vamos, un Sol .. pero  necesitaríamos su masa para tener la compresión gravitacional suficiente, es decir, 
+su tamaño, con lo cual nos quedaríamos sin Tierra ..).  Mi imaginación no da para tanto y esto entra dentro del 
+campo de la ciencia ficción.  
+ 
+6 Es la combinación que presenta mayor probabilidad  de fusión a las temperaturas que,  aunque enormes (del orden de 
+los cien millones de grados), pueden alcanzarse  en una central de fusión. 
+ 
+7 En los experimentos realizados hasta la fecha en JET, la única instalación operativa por confinamiento magnético 
+capaz  de utilizarlo,  se ha evitado  en lo possible su utilización por las complicaciones que acarrea; de hecho, según 
+la información de que dispongo, no se utiliza desde 1997. 
+ 
+8 Clery, Daniel (10 October 2013). "Fusion "Breakthrough" at NIF? Uh, Not Really …". Science. 
+ 
+ 
+9 Por  su ausencia de carga eléctrica, los neutrones son mucho menos eficientes que las partículas cargadas, como las 
+alfas, para transformar su energía cinética en calor. 
+ 
+10 1 MW es un millón de vatios. Para hacernos una idea: uno de los últimos reactores nucleares instalados en España 
+produce del orden de 1000 MW de potencia eléctrica 
+ 
+11 Steven B. Krivit, “The ITER Power Amplification Myth”. En New Energy Times, 6 Oct. 2017 
+ 
+12 Ya que nos movemos en el ámbito de las estimaciones,  no encuentro  motivo a priori valorar unas más que otras: sólo 
+la experiencia dará su veredicto. 
+                                                 
+
+10 
+                                                                                                                                                                  
+ 
+13 https://www.iter.org/mach/Blanket 
+ 
+14 T. Giegerich et al, Development of a viable route for lithium-6 supply of DEMO and future fusion power plants, 
+Fusion Engineering and Design, Volume 149, December 2019, 111339 
+ 
+15  Los primeros reactores nucleares de fisión estuvieron vinculados al programa militar norteamericano (proyecto 
+Manhattan) . En concreto, la primera  reacción en cadena controlada fue  en el Pile-1 (“CP-1”, acónimo de Chicago 
+Pile 1) situado bajo el graderío oeste del campo de fútbol americano de la Universidad de Chicago, bajo la 
+dirección científica de Enrico Fermi, el 2 de diciembre de 1942.  Por ello se puede afirmar que la fisión nuclear 
+controlada y la explosiva se desarrollaron en paralelo. 
+ 
+16 El nombre que inicalmente se le dio fue el de bomba de hidrógeno, o simplemente bomba H porque utilizaba isótopos 
+de hidrógeno para producir la fusión, aunque el detonante fuera una bomba atómica (varias en los diseños 
+modernos). Como dato se aporta el hecho de que aproximadamente las tres cuartas partes de la energía liberada en 
+una explosiión termonulear proviene de la fisión, mientras que sólo la cuarta parte restante proviene de la fusión. 
+Ello es debido a que un  detonante central (sparkplug, o bujía, en su denominación inicial)  de fisión convencional 
+(Pu239 en el caso de Ivy Mike), que fisiona bajo el efecto de bombardeo con neutrones lentos,  comprime y calienta 
+la mezcla de isótopos de hidrógeno deuterio y tritio, que fusionan. En cada una de dichas fusiones se produce una 
+partícula alfa  (núcleo de helio) y un neutrón  de alta energía que se utiliza para fisionar otro isótopo (U238 en Ivy 
+Mike, dispuesto en la parte exterior del dispositivo), que precisamente fisiona muy eficientemente  bajo el 
+bombardeo con neutrones muy energéticos (no lo hace con neutrones lentos, por lo cual no es útil en las bombas 
+atómicas convencionales) . Por lo tanto puede afirmarse que una bomba termonuclear,de hidrógeno o de fusión  es 
+en realidad un amplificador de la fisión mediante la fusión. De hecho, a pesar de su nombre,  aproximadamente el 
+75% de su energía liberada proviene de la fisión. 
+ 
+17 La energía transportada por la radiación a temperaturas ordinarias es despreciable frente a la asociada a la agitación 
+cinética a nivel microscópico, pero la primera aumenta con la cuarta ptencia de la temperatura, mientras que la 
+segunada lo hace linealmente. A las enormes temperaturas alcanzadas en una reacción en una bomba de fisión la 
+presión de la radiación se comporta como un gigantesco mazo que se utiliza para comprimir y calentar el plasma de 
+fusión.  Ello está descartado, obviamente, para aplicaciones pacíficas.   
+ 
+18 Un microsegundo es una millonésima de segundo. 
+ 
+19 La temperatura es una medida de la agitación a nivel microscópico, es decir de la energía cinética (aproximadamente 
+proporcional a la velodidad al cuadrado en este contexto) de los constituyentes de la materia.  
+ 
+ 
+20 Número de partículas blanco por unidad de volumen. Obviamente en el caso de la fusión, proyectil y blanco son 
+intercambiables, ya que ambos están en movimiento en el sistema del laboratorio, a diferencia de la fisión donde los 
+blancos (usualmente  núcleos de U235) están en reposo y son bombardeados por los neutrones.  
+ 
+21 El cálculo teórico y medida experimental de las secciones eficaces de las diferentes reacciones  es, en última 
+instancia,  a lo que nos dedicamos los físicos nucleares. 
+ 
+22 El MeV es la unidad de energía típica de Física nuclear, siendo  la energíacinética que adquiere un electrón acelerado 
+por una diferencia de potencial de un millón de voltios. 
+ 
+23 Las particulas cargadas (en este caso las partículas alfa) son las responsables de la generación de la mayor parte del 
+calor. El proceso es como sigue: al estar cargadas, interaccionan eléctricamente con los átomos del medio, 
+arrancándoles electroles, es decir produciendo parejas iones positivos y electrones. Estos posteriormente se 
+recombinan para formar nuevamente átomos neutros, liberándose  la energía de ligadura correspondiente en forma 
+de  energías de vibración y rotación de dichos átomos, lo cual  macroscópicamente se traduce en el aumento de 
+temperatura. Los neutrones, por el contrario, son muy poco eficientes para producir calor a partir de su energía 
+cinética (es decir aumento de temperatura del medio) debido a su carencia de carga eléctrica, que obliga a producir 
+la ionización sólo indirectamente, a través de partículas cargadas secundarias. 
+ 
+24 El máximo se alcanza a unos 600 milllones de grados y aumenta unos dos órdenes de magnitud, pero esa temperatura 
+excede con mucho lo previsiblemente alcanzable. 
+ 
+
+11 
+                                                                                                                                                                  
+25 La instalación IFMIF, asociada a la futura instalación  DEMO, para la cual la ciudad de Granada es una firme 
+candidata, estará destinada a investigar los efectos de un  bombardeo masivo de neutrones (producidos mediante un 
+intenso haz de deuterio sobre un blanco de litio)  en diferentes materiales que se pretende utilizar en el reactor.   
+ 
+26 http://fnpp.info/ 
+ 
+27 https://www.world-nuclear-news.org/Articles/Linglong-One-reactor-pit-installed-at-Changjiang 
+ 
+28 http://ithec.org 
+ 
+29 https://www.thmsr.com/en/ 
+

-dAzT4oBgHgl3EQfSvvG/content/tmp_files/load_file.txt

@@ -0,0 +1,270 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf,len=269
+page_content='1 EL ELIXIR DE LA ENERGÍA ETERNA  Abstract  The recent announcement of a purported breakthrough result in inertial nuclear fusion at NIF (Lawrence Livermore Laboratory, USA) has aroused a tide of media and public interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' The excitement has been generalized to the whole field of research in fusion energy with,  in its wake, announcements of an imminent advent of the cure for the energetic crisis and the aggravating influence in the climate change associated to the fossil fuels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' This opinion article is intended to show that such expectations are not founded on sound scientific bases and that there is a long way until  the practical production of electricity from nuclear fusion is achieved, if ever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Resumen  El reciente anuncio de un supuestamente trascendental resultado en fusión nuclear  inercial en NIF (Lawrence Livermore Laboratory, EEUU de Norteamérica) ha desatado un enorme interés en el público y los medios de comunicación.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' El entusiasmo se ha trasladado a todo el campo de la investigación en fusión para la producción de energía con, a su estela, anuncios de la llegada inminente de la solución a la crisis energética y al efecto agravante del cambio climático asociado a los combustibles fósiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Este artículo de opinión pretende poner de manifiesto que tales expectativas no están fundadas en bases científicas sólidas y que hay un largo camino por recorrer hasta que se logre, a niveles prácticos,  la producción de electricidad a partir de la fusión nuclear, si se consigue alguna vez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Introducción  Recientemente se anunció con extraordinario aparato mediático un nuevo hito alcanzado en la fusión nuclear, muy oportunamente publicitado en el contexto actual de crisis energética.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En principio no hay nada que objetar a ello: en la comunidad científica es bien conocida la necesidad de financiación que tienen los grupos de investigación, y no digamos los grandes laboratorios como el Lawrence Livermore Laboratory, que deben anunciar sus logros científicos a fin de despertar el interés de la opinión pública, que busca soluciones tangibles a sus problemas inmediatos, y, con ello, captar la atención de las instancias políticas, siempre ávidas  de réditos demoscópicos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  No obstante, opino que se está vertiendo con demasiada frecuencia información sesgada que induce a confusión.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Comencemos con el detonante del presente texto,  la noticia que ha desencadenado el frenesí mediático: el DOE (Department of Energy)  norteamericano anunció a bombo y platillo el pasado 13 de diciembre que en la instalación NIF (National Ignition Facility) del Lawrence Livermore Laboratory  se acababa de conseguir   superar el breakeven en un experimento de fusión,  lo cual  simplemente consiste en conseguir más energía que la suministrada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Suena muy bien y prometedor, pero conviene puntualizar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En primer lugar, esto se ha conseguido mediante la compresión con 192 láseres de altísima potencia sincronizados en un brevísimo pulso del orden de varios nanosegundos1 (la mayor concentración energética mediante láser jamás José Manuel Quesada Molina Departamento de Física Atómica, Molecular y Nuclear Universidad de Sevilla  2 conseguida) de una diminuta cápsula de diamante  conteniendo un pellet ultracongelado con dos isótopos del hidrógeno, deuterio y tritio (DT)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Ello es un mecanismo completamente diferente al confinamiento magnético que se utiliza en los llamados tokamaks3, donde el confinamiento se consigue mediante campos magnéticos con geometría toroidal (donut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En la actualidad el diseño tipo tokamak  es el  más ampliamente utilizado, particularmente en  Europa,  donde bajo estas premisas desde comienzos de los años 90 se realizan experimentos en  el JET (Joint European Torus) en Gran Bretaña, con financiación de la Unión Europea (al menos era sí hasta el Brexit, ya que actualmente está en fase terminal),  y actualmente está en construcción el ITER (acrónimo de International Thermonuclear Experimental Reactor) en Cadarache, Francia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Por lo tanto, el logro alcanzado en el NIF es muy difícilmente extrapolable a la apuesta científico-técnica mayoritaria en curso, que es el confinamiento magnético;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  es más, dicha vinculación se me escapa por completo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Por lo tanto, considero que ésta es una matización clave, que se debe realizar claramente  desde el principio: ambos mecanismos (compresión inercial – mediante láser u otro tipo de haces de partículas - y confinamiento magnético), aunque comparten finalidad, difieren esencialmente y los pretendidos logros en uno no son trasladables al otro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  ¿Es todo esto como se anuncia y promete?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  A raíz de la noticia, y aprovechando su tirón mediático, he podido leer nuevamente (para mi gran sorpresa) algo que lleva repitiéndose desde los orígenes del desarrollo de la fusión nuclear: se trataría de reproducir en la Tierra el proceso que tiene lugar en el Sol y permite la vida;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' es más, se le llega a poner fecha: un reactor de fusión conectado a la red eléctrica en 10 años.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Lo primero no es cierto en sentido estricto y lo segundo es, simplemente, un despropósito que prefiero atribuir a una mala interpretación periodística (que, lamentablemente, cala en el imaginario del público no avisado).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' El tiempo dirá y la hemeroteca lo reflejará.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  En el Sol se quema hidrógeno, un recurso  inagotable a nuestra escala, para producir partículas alfa, que son estables, es decir sin producir residuos radiactivos4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Todo extraordinariamente prometedor, salvo por un detalle: no es posible realizarlo en la Tierra, ya que entra dentro del dominio de la ciencia ficción.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' El motivo: la probabilidad de que dos núcleos de hidrógeno (es decir, dos protones) superen su repulsión eléctrica mutua y se fundan es tan pequeña que ni siquiera ha podido medirse experimentalmente en un laboratorio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En el Sol ello se produce debido a  las monstruosas (a escala terrestre) densidades de masa que se alcanzan en su interior por la atracción gravitatoria de su enorme masa (también comparada con la de la Tierra);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' pero en la Tierra  no son alcanzables tales densidades5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Este es el motivo de que haya que recurrir  a otras mezclas de fusionantes : deuterio-deuterio (DD),  la ya mencionada deuterio-tritio (DT), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='. Es decir, lo que se pretende realizar en la Tierra es parecido a lo que ocurre en el Sol, pero no es lo mismo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' en ambos casos hay un mecanismo común, pero el combustible es diferente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En particular, la única que se vislumbra con posibilidades de permitir la fusión para producir energía eléctrica es la combinación DT6, que es la adoptada en todos los proyectos vigentes  que pretenden conducir a ese objetivo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  El deuterio es abundante (constituye una pequeña fracción del hidrógeno natural) y estable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Pero el tritio no es ninguna de las dos cosas: es radiactivo  y, por lo tanto, no existe naturalmente;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' es decir, hay que producirlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Esto cambia bastante el panorama de supuestas bondades del combustible (casi infinito, según se anuncia): el tritio, como isótopo del hidrógeno, se comporta químicamente (y, por lo tanto, biológicamente) exactamente igual que el hidrógeno normal ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' es decir, dado el papel central del hidrógeno en el ciclo de la vida, el tritio se incorpora al mismo sin que haya forma de separarlo químicamente, porque es hidrógeno (aunque radiactivo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Su vida media de 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='6 años hace que en ese tiempo su cantidad se reduzca a la mitad, pero  en un reactor de fusión ha de producirse continuamente, para lo cual se coloca en su  borde exterior una manta de litio (que debe  3 enriquecerse en su minoritario isótopo adecuado), que al ser bombardeada con neutrones procedentes de las fusiones DT produce el tritio que regenera el consumido.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Al menos ese es el objetivo que se pretende alcanzar (sobre todo, en la tasa suficiente).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Las consecuencias  de la infiltración y fuga del tritio a través de las paredes del reactor a las enormes temperaturas a las que se pretende que funcione se conocen sólo parcialmente, ya que los valores que se manejan  se basan en extrapolaciones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Por lo tanto el combustible previsto no es casi infinito, ni es limpio ni seguro ni  barato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' El tritio es tan problemático7 que en JET se ha trabaja en la medida de lo posible sólo con  hidrógeno o sólo con deuterio, extrapolándose   las tasas de reacción a la mezcla DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' De este modo se obtiene el factor Q (o eficiencia energética, que es el cociente entre la potencia conseguida y la consumida) extrapolado, con el llamativo resultado de que cuando se realizó el experimento con la mezcla DT  el valor de Q  obtenido fue aproximadamente la mitad;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' lo cual muestra algo bien conocido en física e ingeniería, que es el riesgo de las extrapolaciones, a la vez que pone en evidencia la problemática asociada al uso del tritio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Lo anterior añade otra incógnita más a la pretendida limpieza  radiológica de la fusión para producir energía eléctrica a partir de combustible limpio, accesible e inagotable, según otro  lugar común en muchas declaraciones leídas en la prensa: “porque se extrae del agua del mar”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Jugando con las definiciones  Otro aspecto a destacar de la citada noticia tiene que ver con el ya mencionado logro del breakeven en el NIF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Este consistió en  alcanzar  un factor Q de ganancia energética  de fusión  de 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='54, es decir que se obtuvo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='54 veces más energía que la se invirtió .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Al margen de otras consideraciones en las que entraré más adelante, resulta llamativo (por expresarlo suavemente) el cambio de definición que ha conducido a este anunciado éxito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' El NIF cambió hace algunos años la definición del Q para colocar en el denominador (potencia que hay que suministrar a los láseres para comprimir y calentar el plasma)  sólo la fracción que éstos devuelven en forma de radiación ultravioleta para comprimir y calentar, es decir sólo la fracción aprovechable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Teniendo en cuenta que la eficiencia de los láseres es muy baja (en torno al 1%), en rigor hay que dividir por toda la energía invertida, es decir dividir el aunciado factor de ganancia Q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='54  por 100, con lo cual se está aún muy lejos de recuperar lo invertido.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Muy lejos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Obviamente, esta redefinición unilateral del factor Q por parte del NIF recibió severas críticas8, pero el hecho de que no se haya reflejado en las noticias esta matización (¡de un factor 100!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=') por parte de sus voceros (o al menos yo no la he encontrado) permite hacerse una idea del poder del lobby que hay detrás.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Toda la discusión anterior se ha realizado omitiendo  un detalle  adicional que considero fundamental para tener una visión clara de la situación real: no toda la energía liberada en la fusión (el numerador del factor Q) es aprovechable para producir calor y, con ello, la energía eléctrica que se pretende obtener.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En la fusión DT el 80% de la energía producida se la llevan los neutrones en forma de energía cinética, siendo las partículas alfa (que se llevan el 20% restante) las responsables de la mayor parte del calentamiento9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Por el contrario, en una fisión del combustible típico de las centrales nucleares de fisión (U235),  sólo en torno al 5% de la energía se la llevan los neutrones, mientras que el resto corresponde a los fragmentos de fisión, núcleos de tamaño medio muy cargados eléctricamente, que son los responsables del calentamiento de las barras de combustible, que a su vez calientan el refrigerante (normalmente agua) encargado de transportarlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Por lo tanto, incluso sin la redefinición del NIF, el factor Q dista  de ser una medida realista de la rentabilidad energética del proceso de fusión, ya que no sólo una fracción minoritaria de la energía liberada en la fusión es aprovechable para producir calor, que es lo que interesa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  4 En la misma línea de información sesgada por parte de los gabinetes de comunicación, ITER hizo oficialmente pública una información que claramente conducía a error de interpretación.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Concretamente, se afirmaba que ITER sería capaz de producir 500 MW10 de potencia a partir de 50 MW de potencia suministrada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' De ahí se infería lógicamente que esos 50 MW suministrados se referían a la potencia total eléctrica invertida, no a la  calorífica finalmente suministrada al plasma (es decir, igual que en el caso de NIF con los láseres).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Ante las críticas recibidas11, tuvieron que rectificar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Debido a la la eficiencia del proceso de conversión (siempre menor que la unidad, usualmente mucho menor, al igual que en el caso del NIF),  la primera es  muy superior, estimándose  en más de 300 MW necesarios para mantener la fusión12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Además, los 500 MW producidos son totales, de los cuales, como ya se ha indicado, aproximadamente  el 80% corresponde a los neutrones rápidos, que son mucho menos eficientes produciendo calor (sólo son capaces de transferir una parte del mismo al medio antes de escapar), que, al no ser utilizable para mantener la temperatura del plasma,  ITER  propone13 aprovechar calentando el agua del circuito refrigerante de la  manta que envolverá la cámara de vacío para producir electricidad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Toda la responsabilidad del mantenimiento de la temperatura del plasma recaerá sobre las partículas alfa, que depositarán directamente en el mismo toda su  energía  (que, recordemos, es sólo el 20 % de la energía en cada fusión).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' La regeneración del tritio (caro, escaso y que debe producirse en reactores nucleares de fisión, principalmente) necesario para mantener la fusión se pospone para para una fase posterior de ITER, donde se experimentará con la capacidad del isótopo minoritario Li6 del litio natural (que debe ser enriquecido para ello) para producirlo en la suficiente cantidad para mantener la reacción14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Nuevamente nos movemos en el campo de las expectativas, sólo la experimentación demostrará la viabilidad de la propuesta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Un poco de historia  Las radicales diferencias entre los procesos de fusión y fisión nuclear son la causa de que entre la primera reacción nuclear explosiva en cadena (Trinity ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' 16 de julio de 1945 en Alamogordo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Nuevo México,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' EEUU) y la primera producción comercial de energía eléctrica mediante la fisión controlada15 (18 de diciembre  1957,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' en Shippingport,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Pennsylvania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' EEUU) transcurriesen solamente 12 años,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' mientras que tras  la primera explosión termonuclear16 (Ivy Mike,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' 1 de noviembre de 1952 en el atolón Enewetak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' en las Islas Marshall) aún no se ha conseguido domesticar la fusión para mantenerla bajo control y producir energía aprovechable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  En el caso de la fisión se pretende (y consigue desde el año 1942) mantener controlada una reacción en cadena, donde los garantes de esa  continuidad son los neutrones producidos en cada reacción.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En el caso de la fusión los neutrones no juegan ningún papel en mantenimiento de la misma, sino que el  agente garante de la  reacción en cadena es el calor producido, que se debe traducir en temperatura (manteniendo la densidad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Cuando no se pretende el control de la misma sino todo lo contrario (bombas), ello se hace por fuerza bruta (nunca mejor dicho) recurriendo  a la extraordinaria presión de radiación originada por el fulminante de fisión17, sin que ésta escape antes de conseguir instantáneamente su objetivo (todo ocurre en unos pocos microsegundos18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En cambio, para mantener la reacción de fusión en un reactor no se puede, obviamente,  recurrir a ese mecanismo explosivo y se debe conseguir que el calor generado por las reacciones de fusión se recicle sin escapar para mantener la temperatura, al tiempo que la densidad se mantenga temporalmente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Una empresa formidable, que aún está por conseguirse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' paral  5  ¿Por qué se persigue conseguir tan elevadas densidades y temperaturas en un futuro reactor nuclear de fusión?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Porque es preciso conseguir que núcleos atómicos ligeros superen la repulsión debida a su carga eléctrica y se fundan en un núcleo mayor y  alguna otra partícula emergente;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' y además que lo hagan en la tasa (velocidad a la que se producen las reaccciones)  suficiente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' La clave radica en que la suma de las masas de los productos de la reacción es ligeramente inferior a la masa de los reaccionantes, convirtiéndose esa diferencia de masa m en energía E, según la archiconocida fórmula de Einstein E=m c2, donde c es la velocidad de la luz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Este mecanismo es el opuesto al de la fisión nuclear, aunque la consecuencia es la misma: conversión de masa en energía.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En la fisión un núcleo pesado captura un neutrón y se rompe en dos fragmentos de aproximadamente la mitad de su masa y  varios neutrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Aquí tenemos por tanto una gran diferencia cualitativa: a diferencia de la fusión, en la fisión el agente desencadenante (el neutrón, que como su nombre indica, carece de carga eléctrica) no tiene que superar en primer lugar la repulsión eléctrica por parte del núcleo (donde hay protones y neutrones que, en todo caso, lo atraen por la llamada interacción nuclear o fuerte, que es de corto alcance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Por ello en la fisión, si se dan las circunstancias adecuadas (características del núcleo progenitor y energía del neutrón incidente) el núcleo compuesto resultante, que se forma en un estado excitado, se rompe espontáneamente en  busca de una mayor estabilidad del sistema, es decir fisiona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Nos encontramos ante una situación radicalmente diferente a la de la fusión, donde los dos intervinientes han de superar su repulsión mutua (ambos están cargados positivamente), lo cual implica enormes temperaturas para conseguirlo19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Además la densidad ha de ser altísima para que la tasa de reacción sea la suficiente, como comentaré a continuación.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  La tasa de reacción es la clave  La tasa de una reacción nuclear (es decir, el número de reacciones por unidad de tiempo) es proporcional a la densidad de blancos20, al flujo de proyectiles que los bombardean y  a la probabilidad de que la reacción se produzca una vez que proyectil y blanco colisionan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Esta última cantidad es a su vez proporcional a una magnitud llamada sección eficaz21, que viene determinada por la estructura nuclear intrínseca de cada pareja proyectil-blanco y en la cual nuestro margen de maniobra está limitado a la velocidad relativa, es decir, a la  temperatura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Por lo tanto, para cada pareja de proyectil y blanco reaccionantes (fusionantes o fisionantes), conseguir una tasa de reacción suficiente exige unos valores adecuados de densidad y temperatura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  La tasa de reacción es la clave para producir energía aprovechable, porque las reacciones de fusión se producen  rutinariamente en laboratorio mediante el uso de aceleradores (controladas, pero no automantenidas ya que exigen aporte continuo de energía).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Una de las fuentes habituales de neutrones es la llamada DT (deuterio-tritio), la misma mezcla prevista en ITER,  en la cual mediante un acelerador se bombardea con deuterones un blanco de tritio gaseoso, en cada una de cuyas reacciones de fusión se liberan unos 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='6 MeV22 de energía, que se reparten entre una partícula alfa (núcleo de helio, que se lleva aproximadamente el 80% de diche energía) y un neutrón (de alta energía en la jerga especializada,  que se lleva el 20% restante).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Pero la producción energética en forma de calor (debido  mayoritariamente a la energía que transportan las partículas alfa23) es ínfima debido a los valores   de las tasas de reacción implicadas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Es decir,  esta fusión DT ( y lo mismo se puede decir de las fuentes de neutrones DD) no sirve para producir energía eléctrica aprovechable, su finalidad es producir  neutrones rápidos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  6  El balance energético  En principio, un argumento  en favor de la fusión nuclear frente a la fisión es la  energía específica (o energía por unidad de masa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Vamos a explicarlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Una característica de los núcleos atómicos, aunque no exclusiva, ya que lo siguiente es aplicable a cualquier sistema regido por las leyes de la Física Cuántica (es decir a todos), es que su masa es menor que  la suma de las masas de sus constituyentes por separado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Esa diferencia, traducida en energía por la fórmula de Einstein,  es lo que se conoce como energía de ligadura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Por lo tanto si en una reacción nuclear pasamos de una situación con menos ligadura (más masa) a otra de más ligadura (menor masa), la diferencia se transforma en energía cinética de los productos y radiación.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Y esa es la energía que, en forma de calor, se utiliza (en un reactor nuclear de fisión) o debería algún día poder  utilizarse (en un reactor nuclear de fusión)  para producir energía eléctrica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Si colocamos los isótopos conocidos (es decir tipos de núcleos atómicos) en orden creciente con sus masas, comenzando en el hidrógeno, encontramos que la ligadura por nucleón va aumentando en promedio  hasta el Fe56 (núcleo de hierro con 26 protones y 30 neutrones, donde alcanza casi 9 MeV por nucleón);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' a partir de ese punto comienza a disminuir suavemente hasta el final de la tabla , donde llega a unos 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='5  MeV por nucleón en la región de isótopos que nos interesa (la llamada zona de los actínidos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Ello quiere decir que, cuando dos isótopos ligeros (por debajo de la masa del Fe56) se fusionan, el resultado está  más ligado en general, es decir  tiene menos masa, y esa pérdida de masa se transforma en energía.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Por ejemplo en la típica reaccción DT la ligadura del deuterón son unos 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='2 MeV, es decir aproximadamente 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='1 MeV por nucleón ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' la ligadura del tritio son unos 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='5 MeV, es decir unos 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='8 MeV por nucleón;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' la ligadura de la partícula alfa son unos 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='3 MeV, es decir aproximadamente 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='1 MeV por nucleón.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Por lo tanto, se pasa de una ligadura incial en el sistema DT de aproximadamente 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='7 MeV a los 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='3 MeV de la partícula alfa, es decir hay una ganancia de energía ligadura de unos 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='5 MeV por nucleón inicial (~17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='6/5)   eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' La situación opuesta se presenta en el otro extremo de la tabla de isótopos cuando un núcleo pesado, por ejemplo el U235, captura un neutrón y fisiona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' La energía de ligadura del U235  son unos 1786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='7 MeV, es decir aproximadamente  unos 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='6 MeV por nucleón,  y la de dos típicos productos de fisión (recordemos que es un proceso probabilístico) está próxima a la máxima del Fe56, digamos que en torno a los 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='5 MeV por nucleón;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' por lo tanto se ha ganado aproximadamente 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='9 MeV por nucleón en energía de ligadura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Multiplicando esta cantidad por los 236 nucleones del núcleo compuesto inicial (el de U235  más el neutrón absorbido) resultan unos 212 MeV de energía liberada en una fisión típica, cantidad bastante próxima a los valores medidos experimentalmente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Desde este punto de vista,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' en principio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' la fusión DT es claramente más interesante,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' ya que la ganancia neta  de ligadura por nucleón del combustible  es casi 4 veces mayor (lo cual se traduce en una energía específica unas 4 veces mayor  del combustible DT respecto del U235),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' pero hay que considerar que la mayor parte de esa energía cinética corresponde a los neutrones (el ya mencionado 80% típicamente en el caso de la fusión DT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' frente al aproximadamente  5% en el caso de la fisión del U235),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' que es muy poco  aprovechable para producir calor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' es decir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' en última instancia energía eléctrica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En resumidas cuentas, un reactor de fusión es una magnífica fuente de neutrones muy energéticos, otro asunto muy diferente es cómo aprovechar la energía producida (de la que esos neutrones se llevan lel 80%), ya que sólo una pequeña parte de ella se podrá transformar en calor y, aún menos, en energía eléctrica a partir de él.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  ¿Por qué es tan difícil conseguir la fusión nuclear automantenida?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  En el caso de la fisión,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' en la que se basan las centrales nucleares actuales,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  la  densidad de núcleos está fijada por ser el combustible  un medio sólido (aunque puede ser líquido,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' que para el caso es lo mismo) y  la sección eficaz (es decir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' la probabilidad de que se produzca la reacción) se hace  7 enorme en los núcleos fisionables para una energía adecuada de los neutrones (recordemos que los neutrones no tienen carga eléctrica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' por lo que  se cuelan sin obstáculo en los núcleos blanco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' y  que la sección eficaz varía con su energía – es decir con la temperatura del medio que los  termaliza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' es decir que los frena- debido a los detalles de la estructura nuclear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Por este motivo, para  mantener bajo control la tasa de reacciones nucleares basta con controlar con precisión la población neutrónica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Con ello se consigue una tasa de reacción que  libera  la cantidad de calor suficiente para ser transformado comercialmente en energía eléctrica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Por el contrario, en el caso de la fusión (la que nos anuncian como fuente de energía limpia e inagotable del futuro) la sección eficaz  es extraordinariamente pequeña comparada con la de fisión.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Concretamente, en la fusión DT la sección eficaz  a la temperatura prevista de unos 150-200 millones de grados24  es  aproximadamente una cienmillonésima  de la sección eficaz de fisión de un núcleo de U235 bombardeado por neutrones termalizados (es decir con energía óptima para fisionar eficazmente este isótopo) en un reactor convencional  refrigerado por agua ligera a presión (LWR-PWR), que opera a  unos 350ºC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Conseguir la temperatura necesaria en el plasma de fusión (que es la sopa de núcleos y electrones en que se transforma la materia esas temperaturas), es ya en sí misma una empresa formidable, pero a eso hay que añadir la necesidad de alcanzar  una densidad suficiente de dicho plasma  y , además, que ambos parámetros se mantengan durante el tiempo suficiente para mantener una tasa de reacción que la haga utilizable para producir energía.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En las bombas  de fusión (empleadas con fines bélicos) se  utilizan una o varias bombas de fisión para conseguir  simultáneamente los objetivos anteriores (compresión y calentamiento), y en ellas, obviamente, ni el control ni el mantenimiento temporal son necesarios, sino todo lo contrario, desafortunadamente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Pero evidentemente  este mecanismo está excluido para aplicaciones pacíficas, de forma que el  objetivo de mantener el plasma en esas condiciones sólo se plantea  de forma pulsada, ya sea mediante compresión con láseres  o confinamiento magnético (con la que se pretende llegar a varios centenares de segundos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Ello explica el ya mencionado  hecho de que desde Ivy Mike en 1952 hayan transcurrido 70 años sin alcanzar la fusión controlada para la producción comercial de energía eléctrica, siendo las predicciones más optimistas de unos 30, 40, ¿50?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' años adicionales para conseguirlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Porque ITER , cuando funcione, está destinado ser la prueba de concepto  científica  de la fusión controlada y automantenida para producir energía eléctrica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Habrá que esperar a DEMO (como su nombre indica) para la prueba de concepto de ingeniería, que demuestre que es posible verter energía neta a la red eléctrica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Y  mientras tanto debe continuarse investigando exhaustivamente en los efectos que el extraordinario bombardeo con neutrones de alta energía  a semejantes temperaturas induce en las propiedades de los materiales estructurales del reactor , en particular la fragilización,  aparición de fallas y deformaciones25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Y tras todo ello,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' si se alcanza ese punto (cosa que en el mejor de los casos podrán ver nuestros nietos o bisnietos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' porque ninguno de nosotros tendrá la oportunidad de sonreir consultando la  hemeroteca) habrá de demostrarse la viabilidad económica de esta fuente de energía,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' que dados los enormes costes de desarrollo y su descomunal consumo energético previo acumulado en forma de consumo de combustibles fósiles y energía eléctrica de origen nuclear de fisión  (hay estudios al respecto) dista mucho de estar claro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  El tamaño importa  Considero también pertinente mencionar el previsible tamaño de una central de fusión para producción de energía eléctrica, si algún día llega a construirse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Una de las muchas críticas que  se han realizado en contra de las centrales nucleares de fisión es la gran concentración de infraestructuras y capital que implican y su tamaño,  que van radicalmente en contra de una producción distribuida y cercana a los puntos de consumo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Ello sin olvidar los riesgos inherentes a dicha concentración provenientes de posibles ataques terroristas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' A estas alturas del texto, creo que  8 resulta evidente que en una central de fusión estos aspectos criticados en una central de fisión aumentan  hasta dimensiones desconocidas hasta la fecha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' No hay más que comparar el tamaño de ITER con su precursor JET y, aún más, con el previsto para DEMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Los gabinetes de comunicación de los proyectos de fusión (NIF, ITER) nos inundan con informaciones grandilocuentes donde siempre aparece  lo más de lo más: los láseres más potentes del mundo (en el caso del NIF), los imanes superconductores mayores del mundo, la vasija de vacío mayor del mundo, la soldadura electrónica más sofisticada  del mundo, etc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='. (en el caso de ITER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Son innegables logros de ingeniería a gran escala (y puede que ya eso de por sí justifique el esfuerzo y la energía  invertidos), pero no deberían hacernos perder la visión de conjunto: de lo que se trata es de producir energía aprovechable en un futuro no demasiado lejano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Además, tampoco conviene olvidar que dicho desarrollo en busca de cuanto más grande mejor (porque esa es la única manera conocida de alcanzar las extremas condiciones descritas anteriormente) va en el sentido opuesto al seguido en los modernos prototipos de centrales de fisión modulares, destinados a su instalación a escala local, de los cual hay ya uno en fase operacional en Rusia26 y otro en China  en fase avanzada de construcción27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' hay muchos otros diseños  avanzados y prometedores en Japón, Europa y EEUU, que hasta ahora no se han podido llevar a la práctica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  El hecho de que hayan sido precisamente Rusia y China los países que primero hayan llevado a la práctica esta idea innovadora, dice mucho del panorama geopolítico actual, donde la segunda (a Rusia aún le quedan rescoldos científicos y tecnológicos de la época soviética) se ha convertido a pasos agigantados en un referente mundial en ciencia y tecnología en todas las áreas estratégicas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Igualmente, se continúa investigando exhaustivamente en el ciclo de fisión del torio28,29, desarrollando la tecnología para reactores más pequeños (llegando a la escala del MW), más seguros y con menos producción de residuos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' No olvidemos que ITER , cuando entre en funcionamiento, consumirá del orden de 300 MW sólo para mantener la temperatura  del plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Epílogo  El proyecto ITER,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' al igual que la Estación Espacial Internacional (ISS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' de sus siglas en inglés) surgieron en las mismas fechas (años 90) y con los mismos loables propósitos (fomentar la colaboración científico-técnica internacional),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' inmediatamente  tras el derrumbe del bloque soviético y el comienzo de una época de absoluto dominio del bloque llamado occidental  (aunque incluye también a Japón,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Corea del Sur y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' por supuesto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Australia y Nueva Zelanda) liderado por los EEUU de Norteamérica y  la postración absoluta de la otra antigua potencia hegemónica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Rusia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' China, aunque despegando, aún contaba poco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Era la época del famoso Final de la Historia de Francis Fukuyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' No creo necesario resaltar cómo ha cambiado el panorama internacional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' La ISS, con Rusia retirándose, además de la poca relevancia de los resultados científicos obtenidos, está abocada a convertirse pronto en un trozo más de chatarra orbital  destinada a desintegrarse en unos 10 años (si no antes, el silencio mediático es poco prometedor en ese sentido).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Opino que, aparte de los innegables avances tecnológicos asociados a su desarrollo,  ese es su principal ( y probablemente) único éxito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Los plazos han ido alargándose sin cesar: De la fecha inicialmente prevista de las primeras pruebas con plasma en ITER, 2016, se pasó a 2025 y hasta 2035 para las pruebas con la mezcla real DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Los rumores sugieren insistentemente un nuevo alargamiento y la situación geopolítica  mundial (al margen de los enormes problemas científico-técnicos asociados al proyecto) apunta a ello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Para DEMO ya ni siquiera se dan fechas concretas, sólo se habla de que será una realidad en la segunda mitad de la centuria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Vienen a la mente las palabras del Quijote: “Cuán largo me lo fiais, amigo Sancho”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  9 1 Un nanosegundo es una milésima de una millonésima de segundo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  2 El núcleo de hidrógeno, que consiste en un único protón, es el más simple de la Naturaleza y, obviamente es estable, es decir, no radiactivo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' El de deuterio o deuterón consiste en un protón más un neutrón y es también estable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' El de tritio consiste en un protón más dos neutrones y es radiactivo, con una semivida de 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='6 años (el tiempo en que tardan en desintegrarse la mitad de sus núcleos a partir de una población incial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  3 Inicialmente la idea del tokamak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' que es una  palabra rusa porque  fue un concepto propuesto en la Unión Soviética en los años 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' fue descartada por los investigadores norteamericanos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' que llevaban desde los primeros años 50 trabajando en la Universidad de Princeton y el Laboratorio Nacional de Los Alamos tratando de controlar la fusión nuclear para usos civiles mediante confinamiento magnético con la configuración llamada stellarator  (de aspecto exterior muy parecido al tokamak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' pero con importantes diferencias conceptuales entre ambos),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' en paralelo al desarrollo del programa militar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' de hecho ambos proyectos  compartían inicialmente gran parte de su nombre en clave: Matterhorn-B para el militar (la B de bomb) y Matterhorn-S para la civil (con la S de stellarator) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' A comienzos de los años 60 los científicos norteamericanos tuvieron que rendirse a la evidencia de que la idea del tokamak funcionaba mejor en muchos aspectos y comenzaron a investigar mayoritariamente  basándose en ella, aunque varios grupos continuaron   la investigación en la línea original (stellarator), que aún se mantiene hoy en día.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Como puede fácilmente deducirse, la íntima interconexión en el entramado científico-militar norteamericano era evidente desde los inicios en estas investigaciones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  Abundando en esta idea, cabe resaltar que el Livermore National Laboratory tiene una bien conocida vinculación íntima con la industria militar estadounidense en forma de contratos de investigación  orientada tanto a las armas de fisión nuclear como a las basadas en láseres de alta potencia (sobre todo a partir de la llamada Guerra de las Galaxias, que promovió su presidente Ronald Reagan en los años 80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  4 La explicación del mecanismo de producción de energía en el Sol fue propuesta por Hans Bethe en su genial y clarividente artículo (uno más entre tantos de los suyos)  “Energy Production in Stars”, Physical Review 55 (1939) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' 434 , considerado uno de  los diez mejores de la Historia de la Física moderna en una clasificación del Instituto Niels Bohr de Copenhague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Como curiosidad, este artículo fue inicialmente retirado por el autor para poder presentarlo a un concurso de ideas científicas inéditas (que obviamente ganó), con cuyo premio costeó la mudanza de su madre (judía perseguida en Alemania) a los Estados Unidos de América del Norte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En el mismo, entre otras muchas especulaciones basadas en las evidencias entonces disponibles, el autor propone una cadena que se inicia con la fusión de dos núcleos de hidrógeno, es decir dos protones, y que globalmente se traduce en  que a partir de 4 protones se forma una partícula alfa con gran liberación de energía.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  5 A no ser que se consiga crear algún día en la Tierra algo parecido a  una estrella de neutrones (en este caso de protones, vamos, un Sol .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='. pero  necesitaríamos su masa para tener la compresión gravitacional suficiente, es decir, su tamaño, con lo cual nos quedaríamos sin Tierra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='.).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Mi imaginación no da para tanto y esto entra dentro del campo de la ciencia ficción.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  6 Es la combinación que presenta mayor probabilidad  de fusión a las temperaturas que,  aunque enormes (del orden de los cien millones de grados), pueden alcanzarse  en una central de fusión.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  7 En los experimentos realizados hasta la fecha en JET, la única instalación operativa por confinamiento magnético capaz  de utilizarlo,  se ha evitado  en lo possible su utilización por las complicaciones que acarrea;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' de hecho, según la información de que dispongo, no se utiliza desde 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  8 Clery, Daniel (10 October 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' "Fusion "Breakthrough" at NIF?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Uh, Not Really …".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  9 Por  su ausencia de carga eléctrica, los neutrones son mucho menos eficientes que las partículas cargadas, como las alfas, para transformar su energía cinética en calor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  10 1 MW es un millón de vatios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Para hacernos una idea: uno de los últimos reactores nucleares instalados en España produce del orden de 1000 MW de potencia eléctrica  11 Steven B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Krivit, “The ITER Power Amplification Myth”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En New Energy Times, 6 Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' 2017  12 Ya que nos movemos en el ámbito de las estimaciones,  no encuentro  motivo a priori valorar unas más que otras: sólo la experiencia dará su veredicto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  10  13 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='org/mach/Blanket  14 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Giegerich et al, Development of a viable route for lithium-6 supply of DEMO and future fusion power plants, Fusion Engineering and Design, Volume 149, December 2019, 111339  15  Los primeros reactores nucleares de fisión estuvieron vinculados al programa militar norteamericano (proyecto Manhattan) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En concreto, la primera  reacción en cadena controlada fue  en el Pile-1 (“CP-1”, acónimo de Chicago Pile 1) situado bajo el graderío oeste del campo de fútbol americano de la Universidad de Chicago, bajo la dirección científica de Enrico Fermi, el 2 de diciembre de 1942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Por ello se puede afirmar que la fisión nuclear controlada y la explosiva se desarrollaron en paralelo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  16 El nombre que inicalmente se le dio fue el de bomba de hidrógeno, o simplemente bomba H porque utilizaba isótopos de hidrógeno para producir la fusión, aunque el detonante fuera una bomba atómica (varias en los diseños modernos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Como dato se aporta el hecho de que aproximadamente las tres cuartas partes de la energía liberada en una explosiión termonulear proviene de la fisión, mientras que sólo la cuarta parte restante proviene de la fusión.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Ello es debido a que un  detonante central (sparkplug, o bujía, en su denominación inicial)  de fisión convencional (Pu239 en el caso de Ivy Mike), que fisiona bajo el efecto de bombardeo con neutrones lentos,  comprime y calienta la mezcla de isótopos de hidrógeno deuterio y tritio, que fusionan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' En cada una de dichas fusiones se produce una partícula alfa  (núcleo de helio) y un neutrón  de alta energía que se utiliza para fisionar otro isótopo (U238 en Ivy Mike, dispuesto en la parte exterior del dispositivo), que precisamente fisiona muy eficientemente  bajo el bombardeo con neutrones muy energéticos (no lo hace con neutrones lentos, por lo cual no es útil en las bombas atómicas convencionales) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Por lo tanto puede afirmarse que una bomba termonuclear,de hidrógeno o de fusión  es en realidad un amplificador de la fisión mediante la fusión.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' De hecho, a pesar de su nombre,  aproximadamente el 75% de su energía liberada proviene de la fisión.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  17 La energía transportada por la radiación a temperaturas ordinarias es despreciable frente a la asociada a la agitación cinética a nivel microscópico, pero la primera aumenta con la cuarta ptencia de la temperatura, mientras que la segunada lo hace linealmente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' A las enormes temperaturas alcanzadas en una reacción en una bomba de fisión la presión de la radiación se comporta como un gigantesco mazo que se utiliza para comprimir y calentar el plasma de fusión.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Ello está descartado, obviamente, para aplicaciones pacíficas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  18 Un microsegundo es una millonésima de segundo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  19 La temperatura es una medida de la agitación a nivel microscópico, es decir de la energía cinética (aproximadamente proporcional a la velodidad al cuadrado en este contexto) de los constituyentes de la materia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  20 Número de partículas blanco por unidad de volumen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Obviamente en el caso de la fusión, proyectil y blanco son intercambiables, ya que ambos están en movimiento en el sistema del laboratorio, a diferencia de la fisión donde los blancos (usualmente  núcleos de U235) están en reposo y son bombardeados por los neutrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  21 El cálculo teórico y medida experimental de las secciones eficaces de las diferentes reacciones  es, en última instancia,  a lo que nos dedicamos los físicos nucleares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  22 El MeV es la unidad de energía típica de Física nuclear, siendo  la energíacinética que adquiere un electrón acelerado por una diferencia de potencial de un millón de voltios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  23 Las particulas cargadas (en este caso las partículas alfa) son las responsables de la generación de la mayor parte del calor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' El proceso es como sigue: al estar cargadas, interaccionan eléctricamente con los átomos del medio, arrancándoles electroles, es decir produciendo parejas iones positivos y electrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Estos posteriormente se recombinan para formar nuevamente átomos neutros, liberándose  la energía de ligadura correspondiente en forma de  energías de vibración y rotación de dichos átomos, lo cual  macroscópicamente se traduce en el aumento de temperatura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content=' Los neutrones, por el contrario, son muy poco eficientes para producir calor a partir de su energía cinética (es decir aumento de temperatura del medio) debido a su carencia de carga eléctrica, que obliga a producir la ionización sólo indirectamente, a través de partículas cargadas secundarias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  24 El máximo se alcanza a unos 600 milllones de grados y aumenta unos dos órdenes de magnitud, pero esa temperatura excede con mucho lo previsiblemente alcanzable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  11  25 La instalación IFMIF, asociada a la futura instalación  DEMO, para la cual la ciudad de Granada es una firme candidata, estará destinada a investigar los efectos de un  bombardeo masivo de neutrones (producidos mediante un intenso haz de deuterio sobre un blanco de litio)  en diferentes materiales que se pretende utilizar en el reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='  26 http://fnpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='info/  27 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='world nuclear news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='org/Articles/Linglong One reactor pit installed at Changjiang  28 http://ithec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='org  29 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='thmsr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}
+page_content='com/en/' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfSvvG/content/2301.01238v1.pdf'}

-dE1T4oBgHgl3EQf8QXB/content/tmp_files/2301.03544v1.pdf.txt

@@ -0,0 +1,1020 @@
+ 
+© 2023 Steven Fraser and Dennis Mancl 
+Report on the Future of Conferences 
+Steven Fraser, Innoxec, Santa Clara CA USA, [email protected]  
+Dennis Mancl, MSWX Software Experts, Bridgewater NJ USA, [email protected] 
+January 9, 2023 
+ABSTRACT 
+In 2020, virtual conferences became almost the only alternative to cancellation. Now that the pandemic is subsiding, 
+the pros and cons of virtual conferences need to be reevaluated. In this report, we scrutinize the dynamics and 
+economics of conferences and highlight the history of successful virtual meetings in industry. We also report on the 
+attitudes of conference attendees from an informal survey we ran in spring 2022. 
+1. Conferences must evolve 
+In March 2020, catalyzed by the need for “social distancing” due to the COVID-19 pandemic, conferences, trade 
+shows, symposia, workshops, and other “mass” meetings were canceled, postponed, or moved to virtual (online) 
+formats for the balance of the year. Government travel restrictions and individual health concerns made in-person 
+conferences difficult, if not impossible, to organize. 
+The authors wholeheartedly endorse the adoption and growth of virtual and hybrid conferences, even as the 
+pandemic subsides. We strongly believe that conference organizers must increase conference accessibility by 
+reducing the cost for attendees. Accessibility is improved and costs are reduced by the adoption of virtual and hybrid 
+conference strategies. Communities that sponsor conferences need to create new and more open conference models 
+to foster increased diversity, equity, inclusion, and accessibility while decreasing attendee cost and carbon footprint. 
+Pre-COVID, conferences were organized as face-to-face assemblies with participants congregating at convention 
+centers, hotel complexes, resorts, or on company/university campuses. Attendees would meet, talk, give 
+presentations, present papers, receive feedback, market and sell products/ideas, network, build community, and have 
+some fun together. 
+Based on experiences of the past two years, participants extol the benefits of no-travel conferences. Virtual events 
+eliminate conference-associated risks from the pandemic, reduce climate change impact, and increase accessibility 
+for those with limited travel budgets or government travel restrictions. Others yearn for a return to face-to-face 
+meetings, driven by a desire to return a pre-pandemic status quo with in-person networking and the attraction of 
+interesting destinations. The authors believe that each side of the debate has merits. However, we strongly believe 
+that virtual and hybrid public conferences will flourish, in spite of the nostalgia for pre-pandemic in-person 
+conferences. 
+Our report will explore aspects of in-person, virtual, and hybrid conferences. We will examine motivations, 
+logistics, technology, finances, and new ways of enabling interactions. 
+In the course of our study of conferences, the authors ran a community survey in spring 2022 to probe for opinions 
+about the value of in-person, virtual, and hybrid conferences [1]. 
+This report will not address the pandemic-era trend to “work from home” or the more recent debate over a “return to 
+the office.” We have already seen many recent changes in vision of the workplace of the future, including work from 
+open-plan offices, coworking spaces, work from home, work from anywhere, and hybrid models (a mix of 
+workplaces). A discussion of the workplace of the future is beyond the scope of this report. 
+This report is a general discussion of the structure of past and future conferences – and it is aimed at conference 
+attendees, organizers, and sponsors. The historical discussion is a recap of information that is already familiar to 
+frequent academic conference veterans. Even so, to better understand the pros and cons of virtual conferences, it is 
+useful to revisit the benefits of traditional conferences. 
+
+ 
+ 
+Page 2 
+1.1. A look back at history 
+Stepping back in time, many advances in technologies over the past 250 years have changed how civilization works, 
+learns, communicates, and plays. 
+Commercialization of inventions such as the steam engine (accelerating manufacturing and transportation), 
+electricity (working longer than daylight hours), telecommunications (telegraphy and telephony), aviation, and the 
+internet have each played a role in changing society. 
+Inter-personal communication has made enormous progress in 25 years. Emergent technologies have enabled the 
+virtualization of retail commerce, the evolution of print media from paper to websites, blogs, Twitter, and other 
+digital social media platforms, and the migration from POTS (Plain Old Telephone Service) to multi-media 
+platforms enabling text, images, voice, video, virtual reality, and augmented reality.  
+In the intervening years, internet technology has been adopted for general use in homes, offices, and schools. 
+Business has somewhat reluctantly embraced “work-from-home” virtual meeting technologies as a result of the 
+pandemic. Other virtual applications are emergent, from distance learning, tele-justice, tele-health, media live-
+streams, internet-gaming, to social networking. This brings us to the question of virtual conferences. Could 
+conferences be the next domain where “virtual” achieves a more significant level of adoption above the plateau 
+achieved during the pandemic? 
+Let us pose two questions before we delve into our prognostications on the future of conferences. 
+1.2. What is a conference? 
+For the purposes of this report, a conference is a meeting to share, discuss, and expand knowledge. Some exemplars 
+of conferences are: 
+• 
+Academic conferences are sponsored by professional societies, industry associations, or academic entities. 
+The core part of an academic conference often consists of peer-reviewed research paper presentations and 
+prepared talks; attendees come to learn, discuss, and network. Examples include the ACM/IEEE ICSE and 
+ACM SPLASH Conferences. Key characteristics are: Low registration fees, discounts for students, all 
+participants pay; size may range from small to large. 
+• 
+Commercial conferences are organized by media companies like InfoQ and O’Reilly and feature tutorials, 
+keynotes, and panels delivered by industry experts. Registration fees in comparison to academic 
+conferences are higher and speakers are paid to present. Conference size is generally large. 
+• 
+Developer conferences are focused on company product ecosystems, e.g., JavaOne, AWS Summit. 
+Conference size is generally large. 
+• 
+Trade association conferences are organized to showcase the latest products and innovations in an 
+industry. Participants market products to increase sales, e.g., CES, Interop, Mobile World Congress. 
+Conference size is generally large to ultra large with many tens of thousands of attendees. 
+• 
+Government/NGO conferences initiate and continue discussions on policies and innovations with societal 
+breadth – for example, the 2021 UN Climate Change Conference in Glasgow. Conference size varies from 
+small to ultra large. 
+1.3. What factors influence conferences? 
+Factor #1: Conference stakeholders. 
+It is important to understand the set of stakeholders (the people who are connected directly and indirectly to 
+conference registration and conference events) to ensure that future conferences deliver value. 
+Within each stakeholder category, the set of conference participants is constantly evolving. The conference 
+community is growing more global and diverse, especially in the domain of scientific and technical conferences. 
+Conferences have created a competitive ecosystem, but there is also a place for effective collaboration among the 
+stakeholders. 
+ Key stakeholders include: 
+• 
+Attendees – individuals who will benefit directly from participation through making presentations, 
+learning, and marketing ideas/products 
+
+ 
+ 
+Page 3 
+• 
+Attendee Proxies and Sponsors (companies, universities, or funding agencies that pay for attendees to 
+attend a conference) 
+• 
+Conference Organizers (individuals, companies, professional societies, NGOs, governments) 
+• 
+Conference Sponsors – organizations that support conferences through paid sponsorships or gifts 
+• 
+Venue Sponsors – tourism/hospitality businesses and regional development government interests 
+Factor #2: Motivations for attendance differentiated by “personal” or “organizational” benefits: 
+Personal 
+Self-improvement 
+Learning 
+Ideation 
+Problem Solving 
+Publishing 
+Networking 
+Fun 
+Company, University, or 
+Organization Benefit 
+ 
+Learning 
+Scouting Trends 
+Recruiting 
+Marketing/Selling Products 
+Enhancing Reputation 
+Motivations for Attending Conferences 
+In general, a conference is a great place to create and communicate, and there is almost always some reward in terms 
+of individual self-improvement. 
+“Learning” appears twice in the table above because Individuals and organizations both may benefit from the 
+conference “learning experience.” For example, a company may send a group of employees to a conference tutorial 
+or workshop. 
+Factor #3: Conference finances must be economically sound.  
+Revenues must balance costs. The COVID-19 pandemic has demonstrated that it is not simply a matter of “build it 
+and they will come.” Conference stakeholders (organizers, attendees, and sponsors), each have a different 
+perspective on economics. 
+Organizers orchestrate conference logistics by soliciting, curating, and marketing content to put on “the show” (i.e., 
+the conference). Organizers assume financial risks, and their “rewards” depend on the nature of the conference. In 
+some cases, conferences are commercial ventures where the organizers hope to turn a profit (with revenue greater 
+than the event’s costs). For academic conferences, success might be measured by “breaking even” after taking into 
+account government grants and sponsorships. Other conferences are organized by professional societies with some 
+costs offset by society membership fees. 
+Conference organizers need to balance costs and revenues. Some costs are fixed (independent of the number of 
+attendees): 
+Fixed conference costs 
+Insurance and legal 
+Registration services 
+Professional staff 
+Most IT services 
+Marketing/Advertising 
+Conference publications  
+(proceedings, Open Access fees) 
+Security  
+(physical and electronic) 
+Speaker costs 
+Other costs are variable (proportional to the number of conference attendees): 
+
+ 
+ 
+Page 4 
+Variable conference costs 
+Meeting room logistics 
+IT services  
+(wi-fi, streaming, attendee support) 
+Food and beverage 
+Support staff 
+Hotel logistics  
+Conference publications 
+Whether a conference is small or large, seed funding is required to cover initial planning, marketing, and deposits 
+when booking venues or reserving IT services. 
+As with any personal purchase, prospective attendees should assess the advertised value of a conference before 
+attempting to convince their “management” (corporate or academic) to “buy” a registration. Post-conference, 
+attendees (and their proxies and sponsors) will need to assess whether they received positive value for their time and 
+investment (registration cost and travel/living costs). This value can be demonstrated through conference reports, 
+inspiration, key learnings, and personal experience (fun, learning, and network growth). In times of economic 
+restraint, corporate employees might be fortunate enough to get time-off-with-pay while having to self-fund 
+conference travel and registration. Virtual conferences, with reduced costs, can prove attractive to budget-conscious 
+attendees. 
+To evaluate the delivered value of a conference, sponsors evaluate changes in sales, market opportunities, recruiting, 
+and other business goals. However, these are often impossible to directly quantify in the short term – and corporate 
+sponsorships frequently depend on a company’s desire to do “social good” or as part of a targeted marketing 
+campaign. Some conference sponsors are government agencies who have ongoing programs to provide funds for the 
+support of research, regional development, and information exchanges in selected fields. 
+Factor #4: Nostalgia for past conference experiences  
+Conferences of the “Future” will change as technology, social norms, and government policy evolve. Nostalgia 
+shouldn’t be ignored, but how many times should progress be sacrificed to satisfy a core set of repeat attendees? It is 
+useful to reflect on the following questions: 
+• 
+Which parts of traditional conference experiences are most attractive to attendees and organizers? 
+• 
+Will new conference formats be sufficiently engaging to attract “repeat attendees?” 
+Technology changes may make virtual meetings increasingly more effective. Over time, virtual meetings will feel 
+less awkward, especially as more people use video telephony for chatting with family and friends. Government 
+policies may constrain travel to react to worldwide crises: carbon offset requirements (global warming), quarantines 
+(pandemics), or diplomatic issues (sanctions, armed conflicts) that may make international travel impossible. Social 
+norms may also change – reducing the desire to travel or interact face-to-face. 
+Fifty years ago, if one had mentioned “games” – one would have imagined face-to-face participation on an outdoor 
+playing field, indoor gymnasium, or across a table. Today, over 3 billion people participate in “game” experiences 
+online – not in-person. Technology has also catalyzed everything from the evolution of retail sales from bricks-and-
+mortar to virtual retail shops on the web – to matchmaking. 
+The “Future of Conferences” will depend more than we can imagine on the evolution of technology, social norms, 
+and government policy. 
+2. Origins of virtual conferences 
+Virtual meetings weren’t “invented” as a result of the pandemic in the spring 2020. Virtual meetings emerged in the 
+late 1980s for companies to reduce costs [2]. In the 2010s, learned institutions experimented with virtual 
+conferencing to reduce their carbon footprint – for example, the Nearly Carbon-Neutral (NCN) conferences (UCSB) 
+[3]. In April 2020, an ACM task force published a report on best practices for virtual conferences [4].  
+Multinational companies were already embracing virtual meeting technology in the late 20th century, which they 
+used for business meetings and large company events. Virtual meetings helped cut travel costs and reduced “out-of-
+office” time. Internal company meetings made increasing use of virtual meeting technology in the late 20th century, 
+
+ 
+ 
+Page 5 
+even though public conferences remained in-person. Some large companies sponsor internal virtual forums: multi-
+site events to share best practices across business units [5]. 
+As technology developed, company meetings became hybrid with a mix of in-person and virtual participation. 
+Companies used the best communications technologies they could afford: teleconferencing in the 1980s, multi-site 
+video rooms (ISDN-based) in late 1980s to 2000s, telepresence systems beginning in the mid-1990s, and evolving 
+desktop video collaboration applications starting in the early 2000s. Early telepresence systems were costly to run 
+and required: specialized rooms, high performance equipment, and special low latency high bandwidth networking. 
+Companies welcomed the advent of desktop video collaboration applications – the earliest desktop applications 
+(such as WebEx and Skype) were primitive, but they were cheaper, easier to use, more accessible, and scalable 
+across enterprises. 
+There are social challenges associated with today’s virtual meeting technology. In a hybrid meeting, with a mix of 
+in-person and virtual, some virtual attendees feel they are “second-class” participants. Virtual attendees miss side 
+conversations and are limited in how they can influence the course of a meeting. Virtual attendees don’t always hear 
+what was said or miss attendee body language cues. In-person interactions have a much higher “social bandwidth” 
+than virtual interactions. 
+A recent ACM conference paper made this point: “[T]he most challenging asymmetry is the diverse experience 
+between co-located and remote meeting participants. Remote participants often feel isolated, while co-located 
+participants dominate the interaction. [6]” 
+For public conferences, virtual technology did not gain traction before 2020, even though there were some trials, 
+such as ACM and IEEE’s ICSE conference experiment with MBone (multicast backbone) in 1995 [7]. In the world 
+of “virtual meeting technology,” public conferences were late adopters. 
+Why wasn’t virtual technology adopted by public conferences, even though it was being used widely in company 
+business meetings? At the time, the authors believe, a transition to virtual public conferences would have been a 
+disruptive change in the participation and economic models. Most decision makers (conference organizers and long-
+time conference attendees) were likely reluctant to make changes to successful in-person conference models. In 
+contrast, internal company meetings are another matter: a company could easily realize significant travel savings 
+and time savings by going virtual. 
+Today, progressive conference organizers should consider the need to improve conference accessibility for students, 
+young professionals, women, and others with less influence in the conference hierarchy. A virtual conference 
+structure might serve to expand and diversify the conference community [8]. 
+3. Conferences are a business 
+Conferences and other in-person business meetings have been “big business.” The conference business exploits the 
+dimensions of entertainment, tourism, and wanderlust (the appeal of travel, especially to exotic locations). The 
+economic influence of a trillion dollar conference and event industry is difficult to resist [9]. The business meeting 
+industry advertises the charms of their meeting venues and cities to conference decision makers. In addition to the 
+business or academic attraction of conference content, a tourist destination will attract attendees wishing to mix 
+business and pleasure. The authors suspect that exotic conference locations raise the popularity of a conference – but 
+it has been difficult to obtain comparative statistics. 
+Virtual conferences are boring in comparison to “destination” conferences. 
+4. Logistics for virtual and hybrid conferences 
+During the pandemic, conferences borrowed many ideas from the classroom – adapting techniques for both “all-
+virtual” and “hybrid” learning. Schools chose to employ virtual and hybrid to keep students and teachers safe – 
+choosing “all-virtual” when the local infection rate was high, transitioning to a hybrid mix of in-person and virtual 
+instruction as infection rates declined. 
+Hybrid enabled students to be “socially distant” in a half-full classroom, and it also helped students feel less isolated 
+after months of virtual schooling. Teachers complained about the logistical challenges: it is very difficult to organize 
+classroom activities that can deliver an equivalent learning experience for in-person and virtual students. 
+
+ 
+ 
+Page 6 
+The motivation for choosing a virtual or hybrid structure is different for schools and conferences. For schools, the 
+main motivation has been local health concerns. On the other hand, conferences are a very different context from 
+schools. Differences for conferences (in contrast to schools) include: 
+• 
+No grades 
+• 
+No attendance requirements 
+• 
+Sharing new information on the leading edge 
+• 
+Audience a mix of experts and non-experts 
+• 
+Participants from multiple time zones 
+• 
+Global participants with expensive travel costs 
+• 
+Condensed time schedule (few days versus school year) 
+• 
+Participants attracted by well-known experts on the program 
+• 
+Participants attracted by the conference’s focus 
+4.1. Virtual presentations can be live or recorded 
+In all-virtual conferences, there are four major variations for making virtual talks and sessions available to virtual 
+attendees: 
+1. Live Program: Program is presented as a sequence of “live” presentations; attendees may ask questions in 
+real-time (Zoom, WebEx, …). 
+2. Recorded presentations with “live” Q&A: Each presenter pre-records their presentation which is 
+displayed in sequence; attendees may ask questions in real-time following recorded presentation. 
+3. Recorded “on-demand” presentations: Pre-recorded presentations may be viewed by attendees in any 
+order. 
+4. Mix of Live and Recorded Presentations: Presentations with Q&A are recorded in real time; conference 
+attendees have two additional choices for viewing: to watch a “mirror” replay at a designated time (e.g., 4, 
+6, 8, 12 hours) later the same day, or “on demand” (at any time after the “live” session). 
+“On demand” is ideal to avoid attendee “schedule conflicts” (two or more talks scheduled at the same time). 
+Attendees with large time zone offsets (more than 2 hours) appreciate options to view presentations at a more 
+convenient time. 
+However, “on demand” does not support interactions between the presenters and the audience. Live interactions are 
+possible only in options 1 and 2. In those options, organizers usually include a short question and answer session for 
+each talk – and this “feedback and interaction” can be the most interesting part of a conference session. Interactive 
+sessions need to be engaging and structured to support in-person and virtual participants equitably. Virtual/hybrid 
+conferences can be more staffing-intensive: a standard research paper session requires multiple facilitators per 
+session, including an in-person chair who works to keep all attendees engaged and several behind-the-scenes 
+production assistants. 
+Virtual conference panels enable global participation by experts who would not otherwise attend the conference in 
+person. For example, the authors recently organized ICSE and SPLASH virtual panels with diverse panelists from 
+four continents unlikely to attend ICSE or SPLASH. 
+4.2. Hybrid conference options 
+While in-person and virtual conferences are fairly straightforward to explain, hybrid conferences have different 
+options. A hybrid conference will include both in-person and virtual elements. 
+A hybrid conference could be “asynchronous,” consisting of a separate in-person conference and virtual conference 
+that are separated in time. 
+For example, ACM/IEEE ICSE 2022 had a virtual conference (May 10-13, 2022) and an in-person conference two 
+weeks later (May 25-27, 2022). At “asynchronous hybrid ICSE,” most of the in-person presenters were able to 
+present their talks for both conferences, but some presenters in the virtual conference were unable to travel and give 
+their presentations a second time at the in-person conference. 
+
+ 
+ 
+Page 7 
+Another hybrid conference option is “synchronous,” such as SPLASH 2022 (December 5-10, 2022). At “hybrid 
+SPLASH,” some presenters were in-person, others were virtual. In-person attendees and virtual attendees could view 
+any of the talks. Last but not least, a hybrid conference could be a blend of synchronous and asynchronous events. 
+Below are two tables summarizing key characteristics of in-person, virtual, and hybrid conferences. 
+ 
+In-Person Conference 
+Virtual Conference 
+“Live” presentations 
+Traditional in-person conference: 
+Live in-person presenters and 
+attendees 
+Virtual conference: all presenters and 
+attendees are virtual and presentations 
+occur in “real time” 
+“Recorded” presentations 
+In-person attendees view 
+prerecorded conference 
+presentations during conference  
+Virtual attendees view recorded 
+conference sessions/presentations at 
+any time during or following the 
+conference 
+In-Person and Virtual conference characteristics 
+ 
+ 
+Synchronous Hybrid 
+Conference 
+(overlap of in-person and 
+virtual sessions) 
+Blended Synchronous and 
+Asynchronous Hybrid 
+Conference Options 
+Asynchronous Hybrid Conference 
+(no overlap of in-person and virtual 
+sessions) 
+Attendees may be 
+in-person or virtual: Sessions 
+are synchonous 
+(SPLASH 2022) 
+Synchronous sessions for virtual 
+attendees plus presentations for 
+“local” attendees at conference hubs 
+In-person sessions are separated in time 
+from virtual sessions days or weeks apart 
+(ICSE 2022) 
+Hybrid conference options 
+A hybrid conference may have one or more “hubs” – a hub is a location where attendees can meet in-person, and 
+where some presenters may deliver in-person talks. In a hybrid conference with multiple hubs in different time 
+zones, it is preferable that virtual sessions be held at a time that is convenient for a majority of attendees. For 
+example, if there is a North America hub and a European hub, virtual sessions could be held in the afternoon for 
+Europe (in the morning for North America). Each hub could host a “local program” at a time most convenient to the 
+local in-person attendees. For corporate hybrid conferences – e.g., Cisco, Qualcomm, Nortel – conference “hubs” 
+were the regional R&D Labs plus the corporate headquarters. 
+5. Motivation for attending conferences: networking and learning 
+Conferences can be a place to share knowledge in a “formal” manner. In many scientific fields, publishing new 
+research work in conference papers can be preferable to (and faster than) publishing articles in scientific journals 
+[10]. 
+Conferences can also be a way to share ideas informally. Conferences offer an excellent opportunity to make new 
+connections, build networks, and to renew friendships. Conferences bring people with shared interests together. 
+Even if they are sometimes distracted by technology (e.g., cellphones, email, Facebook, Twitter, and TikTok), 
+attendees become more energized when they break out of their day-to-day universe of familiar faces. 
+The value of making new contacts is difficult to estimate. One approach might be to assess the value of increasing a 
+“personal network” by applying Metcalfe’s Law. Metcalfe’s Law proclaims that the value of a computer network is 
+proportional to the square of the number of connected users. 
+Extrapolating Metcalfe’s Law to social networks suggests that personal network growth is more impactful for 
+individuals with smaller personal networks. 
+
+ 
+ 
+Page 8 
+For an individual with a 10-person network, adding 10 more contacts increases the network’s value by 300%, while 
+those with a 50-person network, adding 10 more contacts increases the network’s value by only 44%. 
+Even if we choose a much more modest value model, such as “proportional to Nlog(N)” (as suggested by Briscoe, 
+et.al. [11]), the impact of expanding the network is still much more significant for people with small networks. 
+Adding 10 people to a 10-person network adds 160% to the network’s value, adding 10 people to a 50-person 
+network adds 26%. 
+Based on the premise of network value, the individuals who might benefit most from the networking opportunities at 
+an in-person conference are those least likely to be able to afford to travel: students, early-career professionals, and 
+individuals who have high travel costs – or visa related challenges. 
+A return to an in-person-only format is very short-sighted, in terms of attempts to foster greater diversity, equity, and 
+inclusiveness of conference participation. There is a large potential to help more people build more diverse 
+networks, if and only if we can organize our virtual conferences to support more effective interactions. 
+6. The dynamics of building personal networks 
+6.1. Conferences have both one-way and multi-way communication 
+Many conferences are centered around keynotes, tutorials, and paper presentations. These talks are similar to 
+university lectures: a one-way form of communication with limited audience interaction. Attendees also participate 
+in interactive (multi-way) activities, such as workshops, hands-on demos, and “shows” – be they artistic, musical, 
+multi-media, or product-oriented in nature. Other more casual settings for interaction and information sharing may 
+include social events and informal serendipitous conversations. 
+6.2. Knowledge transfer through personal contacts 
+Conference participation helps disseminate and incubate knowledge that is not yet widely available. One-on-one 
+networking is a key part of the knowledge transfer process, even in a world that has a wealth of information in 
+electronic media. 
+Today, the research community depends on the materials held in digital libraries, open-source repositories, open 
+access journals, and online forums. Static material is complemented by online education options which help the 
+global community of software professionals upskill and expand their knowledge. 
+But there are pitfalls relying exclusively on non-peer reviewed knowledge sources due to a low signal-to-noise ratio 
+(lots of noise). The internet serves up an amazing supply of scientific and technical information, practical YouTube 
+videos, and useful social media discussions; it also hosts misinformation and conspiracy theories. 
+Face-to-face conference discussions make it possible to ask questions directly. Tapping into personal networks via 
+an informal conversation or email exchange can help accelerate research. 
+Without personal interaction, “asking a quick question” or “having a conversation” is slowed by the constraints of 
+distance. The personal touch matters – interaction at a conference helps build long-term relationships with experts. 
+Conversing with an expert can be more helpful than a computer search engine inquiry. 
+A conference is an ideal setting for informal discussions. At home, our focus is on day-to-day job software 
+development, research experiments, meetings, writing and reading reports, and office bureaucracy. 
+6.3. Impromptu discussions and serendipitous interactions 
+Information exchanges are built on discussions and interactions, and in-person meetings improve the 
+communication. In contrast, interactions that use or apply technology (virtual collaboration tools) can be awkward. 
+The standard rules of interpersonal interaction have not caught up to the new wave of networking tools. In general, 
+the authors believe that face-to-face interactions still provide better support for ideation and incubation of 
+friendships and partnerships. 
+As humans, we gain insight from the tone of voice, body language, and eye contact. Interpersonal interactions at 
+conferences are not preprogrammed or prerecorded. The interactions are made much richer by: 
+• 
+impromptu discussions – without previous preparation 
+
+ 
+ 
+Page 9 
+• 
+serendipitous encounters – random meetings with individuals one is unlikely to meet elsewhere 
+• 
+serendipitous discussions – leading to valuable or interesting revelations  
+Conference attendees are uniquely positioned to learn from in-person impromptu discussions – ideas that are 
+difficult to acquire in any other fashion. For example, an in-person discussion may help to understand results or 
+spark new approaches. Serendipity often contributes to new connections and ideas. A dialog might start with 
+introductions followed by an exchange of recent experiences: “What did you try? Did it work? Was it a key 
+learning? A best practice? Or something to be avoided?” A short conversation can trigger an insight, inspire a 
+concept, or initiate a collaboration. 
+Serendipity happens “by chance” at a conference: at sessions, during breaks, or even on conference travel. For 
+example, Kent Beck tells the story of how he and Erich Gamma had a useful software design session on a flight 
+from Switzerland to the United States on their journey to attend ACM’s OOPSLA 1997. In three hours of 
+discussions, they collaborated to write the first version of the popular JUnit automated unit test framework [12]. 
+A serendipitous conversation is more than an opportunity to exchange social networking profiles or business cards – 
+it could inspire new ideas and collaborations. 
+6.4. Do virtual conferences support serendipity? 
+Most virtual conferences as currently designed have limited support for one-on-one serendipitous meetings. 
+There are two forces at work that limit serendipitous conversations in a virtual conference: technology obstacles and 
+social norms. The key obstacle for virtual conference technology is the inability to communicate “presence.” In day-
+to-day conversations, we often read body language, facial expressions, and tone of voice. With virtual meeting 
+technology, it isn’t easy to read non-verbal cues across meeting participants. The camera sees only speakers’ faces, 
+video resolution is poor, and audio can be masked by background noise. 
+Virtual conference attendees have low expectations for interactions with other attendees. They are resigned to being 
+a “viewer” – watching the set of “canned” presentations without interacting either with the speaker or other 
+attendees. Virtual conferees might be multi-tasking with non-conference activities – too busy for side conversations 
+with other conferees. 
+But even if current expectations are low for virtual technology, there is hope for both the present and the future. A 
+well-designed virtual conference program is not required to follow the same structure or timeline as an in-person 
+conference. There are many creative options for building an effective virtual conference program to catalyze more 
+active participation. 
+6.5. Engaging virtual conference attendees 
+The structure of conferences must evolve to better serve virtual attendees. In-person attendees benefit more from in-
+person contacts, impromptu discussions, and serendipity. Virtual attendees need similar benefits – conference 
+activities that give them a chance to be active participants, make connections with other attendees, and establish 
+opportunities for dialog with other conferees after the conference is over. 
+Conference organizers should leverage experiences from social media to get participants more engaged. Conference 
+organizers should increase participant engagement by borrowing community-building practices from social media. 
+A simple approach to get participants engaged is to use real-time polling throughout the conference. A session chair 
+could use a web-based tool like MentiMeter or Slido to run frequent audience polls to sustain audience engagement. 
+Social media can also support multiway discussions during a virtual conference. Today’s social media users have 
+opinionated exchanges with people they have never met in real life. Virtual conference participants could convene 
+an impromptu panel discussion with participants selected via real-time social media metrics. The panelists could be 
+the most frequent conference Twitter or Facebook posters – the posters who get the most likes or text responses to 
+their real-time blogging of conference talks. 
+In many conferences, virtual participants are totally anonymous, for example, when presentations are streamed via 
+YouTube. In other conferences, the virtual participants connect to an online conference platform, which displays a 
+list of session attendees – and no other valuable session context, e.g., organizational affiliation, contact information, 
+or chat links. 
+
+ 
+ 
+Page 10 
+While some platforms encourage attendees to add a “personal profile” associated with their conference login, there 
+is generally no incentive for participants to enter personal information, so most profiles are left blank. 
+To encourage participants to add to their profile, conference organizers can provide motivation through: 
+• 
+registration discounts 
+• 
+post-conference access to premium conference content 
+• 
+forums to enable post-conference conversations among attendees with similar interests  
+However, it is necessary to be mindful of GDPR [13] and other privacy concerns – and to be mindful of possible 
+abuses. Profiles are a useful way to connect attendees who have similar (or dissimilar) interests and backgrounds – 
+while offering privacy and diversity safeguards. Different conferences will likely require different templates. To 
+assist attendees profile information could be suggested from personal websites and public databases, with the 
+opportunity for participants to customize as they choose. 
+In order to support participant interactions during the conference (via chat or web video), it isn’t necessary to have 
+the conference platform be a “completely immersive environment.” Conference participants may have other ways to 
+chat. A conference platform ought to include some impromptu text chat or user-configurable small-group video 
+meeting capabilities. Conference attendees would then have the option to establish their own one-on-one 
+communications (using email, Twitter, LinkedIn, Slack, Skype, or whatever) during or after conference sessions. 
+Some of the experts in the conference community might also volunteer to host small-group informal chat sessions – 
+an opportunity for non-experts to meet some of the stars in the field. This is a practice that has started to become 
+commonplace. Some recent conferences have offered virtual sessions titled “Ask Me Anything” with a keynote 
+speaker or another notable person. 
+7. Virtual conferences – commitment to diversity, equity, and inclusion 
+Conference attendance costs include registration, travel, and time away from home and office. With virtual 
+conferences, conference costs are lower because physical logistics are unnecessary (food, meeting rooms, etc.), 
+participant travel costs are reduced (no need for transportation or hotels), and “away time” from home and office are 
+reduced (at least proportionally to travel time). That said, some would argue that getting away from “home and 
+office” is the attraction of attending a conference. Crista Lopes postulated that escaping to conference “destinations” 
+at the expense of your employer or grant agency is a key motivator for some to attend a conference [14]. 
+But in-person conferences might not be a “safe” environment for some attendees. Some people in the technical 
+community can be outright hostile to newcomers. Some of the hostility may include racism and sexism. But a subtle 
+hostility is elitism – rejecting academics from lower-rated universities, company participants who are not from 
+highly-ranked research programs, and “practitioners” who just come to learn. 
+As noted earlier, virtual conferences are much easier for students to attend, and many people who are unable to 
+travel appreciate the opportunity to attend conferences from home. At a town hall meeting discussion at ICSE 2022, 
+one attendee suggested that 50 students can attend a virtual ICSE for the cost of sending one person to the in-person 
+ICSE [15]. 
+Hybrid conferences can draw virtual attendees who live nearby. In metropolitan areas, traffic and parking can be an 
+enormous obstacle to attending a meeting – it may take more than an hour to navigate rush-hour traffic, especially in 
+congested metro regions. Hybrid conferences are a way to encourage more local participants. 
+8. Financial issues 
+The rise of virtual conferences has created many new revenue opportunities for conference organizers, but virtual 
+also adds new challenges. During the pandemic, organizers found it difficult to monetize virtual conferences since 
+attendees associated conference “cost” with in-person expenses (food & beverage plus physical meeting logistics). 
+In-person conferences fees are usually tiered by content elements, e.g., for an entire multi-day program – or by day, 
+by track, by workshop, or by tutorial. Different categories of attendees may be assessed different fees – for example: 
+presenters, industry participants, students, academics, members (ACM, IEEE), etc. 
+
+ 
+ 
+Page 11 
+For a virtual conference, the charging model can be more fine-grained. For example, if the conference presentations 
+are organized in one-quarter, one-third, or half day blocks, there could be a charge per block. It is one way to attract 
+attendees who are interested in a group of specialized talks, or just the keynotes. Fine-grained session charges are 
+likely more useful for conferences with large attendance. 
+With conference collateral such as session recordings, organizers have the choice between making these freely 
+available after the conference to registered conference attendees or to charge premium access fees to a wider 
+audience. However, organizers need to be mindful of digital rights issues. Without securing blanket permissions 
+from all attendees, only the presentations can be shared – but not the recordings of the Q&A sessions – assuming 
+that presenter permissions are a conference participation prerequisite. 
+Another question is how to set pricing for virtual attendance since attendees have expectations that virtual should be 
+cheaper than an in-person registration. This expectation is based on the assumption that there will be no cost 
+expenditures for physical rooms, coffee breaks, lunches, meals, or social events. However, the IT and production 
+costs may be higher for virtual conferences that go beyond simple video conferencing (Teams, WebEx, Zoom, etc.). 
+Advanced virtual conference services may require paid production staff, which can result in a higher per-attendee 
+cost than in-person food and beverage services. 
+Recruiting, training, and supporting volunteers is a significant burden on the conference organizers. There is a 
+tradeoff: Working with volunteers can help keep costs low. In contrast, paid staff may require less training and 
+support. 
+There are many more practical suggestions for virtual and hybrid conferences in the “High-Level Planning” section 
+of the report of the ACM Task Force on virtual conferences [4]. 
+9. What factors will drive the future of conferences? 
+The future of conferences depends on three key issues: 
+• 
+Increasing costs [financial and carbon footprint] and the inconvenience of travel. 
+• 
+Emerging technologies topics spawn new conferences. Research funding and commercial investments 
+incubate new communities that need to share and innovate. New virtual conferences have a low cost of 
+entry. This may stimulate fragmentation of existing conferences. 
+• 
+New collaboration technologies will enable new conference formats. Examples may include Virtual 
+Reality, Augmented Reality, and Gamification. 
+10. Attitudes of conference attendees: a community survey 
+To learn more about current attitudes about in-person and virtual conferences, the authors ran a community survey 
+in spring 2022 [1]. Although the survey population was not a random sample of the universe of all conference 
+attendees, it did include a range of geographies (70% of survey respondents were from North America, 23% from 
+Europe, 7% from the rest of the world) and there were respondents from industry (77%), academia (19%), and other 
+(4%). 
+The primary result of the survey: “hybrid” was the preferred conference mode. 
+• 
+54% said they preferred hybrid 
+• 
+36% preferred in-person 
+• 
+9% preferred virtual 
+It is possible that hybrid was preferred by many because they wanted to have the option to attend an in-person event 
+after two years of pandemic-imposed isolation. On the other hand, hybrid might have been the top option because 
+many respondents are still nervous about traveling – but they didn’t want to bar others from being able to attend a 
+face-to-face conference. 
+In the survey’s text comments, respondents shared a range of opinions. Some had serious issues with virtual 
+attendance and made a good case for choosing in-person conferences. Comments included: 
+• 
+Nothing today can replace the human networking and high-intensity one-on-one networking that happens at 
+an in-person conference 
+
+ 
+ 
+Page 12 
+• 
+Virtual is too much one-way broadcast 
+• 
+Virtual conferences are absolutely abysmal experiences 
+• 
+Virtual – I find it really difficult to pay attention 
+Other respondents found virtual or hybrid conferences valuable: 
+• 
+I especially value the global participation of virtual conferences, this democratizes technology development 
+and sharing 
+• 
+Hybrid offers flexibility that can meet the needs and constraints of diverse potential attendees 
+• 
+Hybrid is the way of the future 
+• 
+Hybrid allows me to make the choice of how to attend 
+We asked the survey participants to list the conferences they attended. Respondents replied with an amazingly 
+diverse set of conferences. The 331 survey respondents reported attending over 500 different conferences, 
+everything from AAAI to ICSE to JavaOne to SPLASH to Zoomtopia. The most frequent conferences attended were 
+software engineering conferences (such as ICSE and SPLASH), consumer products gatherings (CES), agile 
+development conferences (sponsored by Agile Alliance or Scrum Alliance). 
+The survey asked respondents to rate potential obstacles for virtual and in-person conferences. The purpose of these 
+questions was to solicit improvements for conference program design and logistics. Responses identified aspects of 
+virtual and in-person conferences that might limit participation. 
+Identified challenges for virtual conferences were: 
+• 
+Ineffective support for casual discussions 
+• 
+Ineffective tools for interactive discussions 
+• 
+Time zone issues 
+• 
+Fatigue due to long virtual meetings 
+Identified challenges for in-person conferences were: 
+• 
+Registration and travel costs are too high 
+• 
+Ongoing pandemic related health risks 
+• 
+Time away from family or work 
+The challenges for virtual conferences focused on communications and interactions – while the obstacles for in-
+person conferences related to economic and health issues (cost, travel, and time). 
+We asked how many conferences participants attended in 2021 and how many conferences they planned to attend in 
+2022. 
+• 
+86% of respondents reported attending at least one virtual or hybrid conference in 2021 
+• 
+79% said they would attend at least one virtual or hybrid conference in 2022 
+Also, the average number of “conferences attended” rose significantly per survey respondent. 
+• 
+The mean number of 2021 conferences attended = 2.1 
+• 
+The mean number of planned 2022 conferences = 3.5 
+This ratio held firm across several job roles: managers, university faculty, and industry software developers. 
+To increase the viability of virtual conferences, enabling casual discussions and interactive discussions requires 
+creativity and effort by organizers and attendees. Conference organizers need to incorporate conference activities 
+that will help the online conference community to get to know one another. Online participants will get more out of 
+their conference experience if they would be willing to do more than just “view” talks. Questions and dialog 
+between conference attendees help increase understanding. 
+Our survey suggests that in order to increase the value of in-person conferences, conference organizers need to 
+convert at least part of their meetings to a virtual event – to attract attendees who would not normally participate 
+
+ 
+ 
+Page 13 
+because of cost and travel barriers. Organizers need to keep in mind that the effectiveness of virtual and hybrid 
+events will vary depending on the conference content and structure. 
+11. Keep conferences simple, understand motivations of the participants 
+11.1. Practices for virtual and hybrid conferences 
+To reduce the complexity and increase the appeal of virtual and/or hybrid conferences, here are some practices the 
+authors have found useful: 
+• 
+Smaller conferences (fewer talks, fewer tracks, fewer days) that simplify logistics  
+• 
+Shorter conference days (4 hours of conference program per day instead of 6 or 8 hours)  
+• 
+Keynote talks in “the middle of day” to anchor the program 
+• 
+Moderated Q&A: host-curated questions received via chat 
+• 
+Easy-to-navigate conference program: with hyperlinks to program elements 
+• 
+Web-based conference programs: attendee sees the schedule with automatic time zone localization for their 
+time zone 
+• 
+Support: a “help line” for technical assistance (either via chat or phone) 
+• 
+Be kind to presenters: avoid scheduling 3:00 a.m. presentations 
+• 
+Registration options: sell registrations “by program element” for broad spectrum conferences; also sell an 
+“all-access” registration 
+Less helpful strategies (“antipatterns”) include: 
+• 
+“Live” anonymous chat feeds – with inappropriate, vitriolic, or profane comments 
+• 
+Conference program that is difficult to navigate 
+• 
+Incomplete presenter and attendee profiles 
+11.2. Motivations for attendees and organizers 
+Many conference activities are linked to the “motivations for conference attendance.” Virtual conferences can 
+adequately address some of them – but there are some activities that work much better at in-person conferences. 
+[Note that ratings in these tables are the subjective opinions of the authors.] 
+Attendee 
+Motivation 
+Conference Program 
+Element 
+In-person 
+Virtual 
+Learning 
+All program elements 
++++ 
+++ 
+Ideation & 
+Problem Solving 
+Collaborative workshops 
++++ 
++ 
+Publishing 
+Conference papers 
++++ 
++++ 
+Networking 
+Social networking 
++++ 
++ 
+Fun 
+Social activities 
++++ 
++ 
+Relative Value of Conference Program Elements (for attendee self-development) 
+
+ 
+ 
+Page 14 
+Attendee/Organization Goal 
+Conference Program Element 
+In-person 
+Virtual 
+Cost-effective Learning 
+Presentations, Tutorials, Workshops 
++ 
++++ 
+Scout Trends 
+Expert chats, Demos, Exhibits, Posters 
++++ 
++ 
+Social Networking 
+Snacks/Lunch/Dinner/Hallway chats 
++++ 
++ 
+Recruiting 
+Presentations, 
+Social networking 
+++ 
+++ 
+Marketing Products 
+Special events, 
+Sponsor receptions, Tradeshows 
++++ 
+++ 
+Enhancing Reputation 
+Peer reviewed papers, 
+Organization success stories, 
+Sponsor keynotes 
++++ 
+++ 
+Relative Effectiveness of In-person vs Virtual Conference Program Elements  
+In the opinion of the authors, the list of “goals” in the left column are the principal benefits to organizations when 
+their employees attend conferences. Recruiting and marketing are much more effective when done in person. On the 
+other hand, cost-effective learning is a key motivation for companies to have their staff attend virtual conferences. 
+Conference organizers need to monitor the primary motivations of their attendees to ensure that activities will meet 
+the needs of both repeat attendees and conference newbies. 
+Conference Organizer Objective 
+How 
+In-person 
+Virtual 
+Education / Training 
+Feature “hot topics” 
+attractive to attendees 
+Tutorials and workshops that 
+support “collaborative and 
+experiential learning” 
+++ 
+++ 
+Maximize Revenue 
+Hot topics 
+Feature “experts” 
+Targeted marketing/discounts 
++++ 
++ 
+Community Building 
+Targeted marketing 
+Feature community experts 
+Community sponsors 
+++ 
++ 
+Increased Accessibility  
+Global marketing 
+Sponsor attendees 
++ 
++++ 
+Showcase University or Company 
+Promote location benefits 
++++ 
++ 
+Relative Value of In-Person vs Virtual Conferences for Organizers 
+Again, these relative assessments are the opinions of the authors. Our framework is a starting point for the reader to 
+evaluate the most relevant tradeoffs for conference attendees, sponsors, and organizers. 
+A virtual conference may have a different mission than an in-person conference – and it may be judged as 
+“successful” even if it doesn’t achieve all of the goals listed above. 
+11.3. Motivations for conference presenters 
+Conference presenters have a wide range of motivations. Most of their goals are similar to conference attendees – 
+especially for presenters who are members of the core community. Presenters may also attend the full conference to 
+learn, scout tech trends, recruit, and market products. One of the most important collateral benefits of being a 
+conference presenter is the potential for an increase in reputation as a subject matter expert. 
+
+ 
+ 
+Page 15 
+Some non-community members (i.e., individuals who have not attended previous conferences associated with the 
+community) may be invited to participate in a conference program as a keynote speaker or panelist. Each conference 
+has its own guidelines for keynotes and speaker compensation. Featured speakers might include: 
+• 
+Famous researchers, authors, and experts 
+• 
+Celebrities: executives, entertainers, athletes, writers, and inspirational individuals 
+Compensation can be an issue for speakers. An invited speaker may have a mercenary interest. Speaker bureaus 
+have a reputation for negotiating high appearance fees for famous individuals. Other invited speakers may be willing 
+to forego direct compensation, because they view their appearance as a publicity and marketing opportunity for their 
+company’s products and services. 
+“Virtual speakers” – speakers who aren’t required to travel – can sometimes be less expensive. Most speakers are 
+more willing to deliver a virtual talk, because they can avoid the time and inconvenience of travel. “Virtual” can 
+simplify scheduling, and it is especially useful for organizing panel sessions – where panelists might participate 
+from any continent. 
+On the other hand, when a virtual talk is broadcast online to a large conference audience, it raises the question of 
+“digital rights management” which if not addressed might lead to the illegal bootlegging of screen-capture 
+recordings by conference attendees – however this can now be a challenge for in-person conferences too! 
+Alternatively, speakers might desire larger fees for a wider distribution of their presentations – although in the age of 
+YouTube videos and TedTalks – the world is moving slowly towards “Open Access.” 
+12. Reflecting on the past, trying new things in the future 
+12.1. Conference surveys, retrospectives, and experiments 
+It is essential that conference organizers keep asking questions of their stakeholders – to sustain a conference’s 
+relevance. Every conference should run a post-conference survey to identify trends (year over year) and run a 
+retrospective to learn and improve. Also, because the best practices for virtual and hybrid conferences are evolving, 
+organizers should experiment with new approaches. 
+Conference organizers need to track the changing attitudes of their own community of conference attendees and 
+presenters, just as the authors’ (Fraser/Mancl) community survey in the spring of 2022 gathered opinions from 
+people who attend a spectrum of conferences. There are many things for conference organizers to assess: 
+• 
+Are attendees satisfied with the conference’s virtual platform? If not – why not? 
+• 
+Is the conference is serving the needs its community? For example, if the conference is intended to serve an 
+international audience, how successful is its marketing? How effective is the conference at delivering 
+value? 
+• 
+What changes might improve accessibility and attract a diverse community? 
+A “retrospective” is an essential management practice for conference stakeholders to drive ongoing improvements to 
+a conference. A retrospective is an informal meeting of conference organizers after the event to reflect on which 
+strategies worked well or what to consider for the next event (assuming a conference series). 
+A survey is an essential part of the feedback process. Why collect opinions from conference attendees? The views of 
+community members evolve over time. Organizers might believe a virtual or hybrid conference cannot be successful 
+based on a dated pre-pandemic survey. Attitudes change. Expanding access, reaching out to an international 
+community, and improving diversity are also increasingly important goals for conferences in the third decade of the 
+21st century. 
+12.2. Attendee self-assessment after a conference 
+The authors have attended many conferences, and we are still learning. We have personally assessed our likes and 
+dislikes about in-person, virtual, and hybrid conferences – because we perform our own “self-assessment” after each 
+conference experience. 
+For example, we have both worked for many years in a corporate culture where technical staff members would write 
+and present short “trip reports” following any outside activities. A good conference report focuses on two things. 
+
+ 
+ 
+Page 16 
+• 
+What did the conference attendee learn at the conference? (short summaries of interesting presentations, 
+ideas collected from hallway conversations) 
+• 
+How well did the attendee’s conference activities meet personal and corporate goals? (learning specific 
+technology trends, recruiting new staff, or doing targeted marketing) 
+With a series of self-assessment reports for conferences, the value of participation becomes more evident. In the mix 
+of in-person, virtual, and hybrid conferences, reports help us to assess the effectiveness of each type of conference 
+participation. Although it requires daily effort during the conference, a report doesn’t need to be a long narrative. A 
+report might consist of one or two paragraphs summarizing each day’s program – an outline of conference topics, 
+good questions from the conference sessions, and a short list of “new contacts.” 
+The authors look forward to learning more about the evolution of conferences from new surveys shared by 
+conference organizers and informal conference reports shared by our colleagues. 
+12.2. Attendee multi-tasking at a conference 
+Two situations to consider unrelated to the conference focus: (1) an in-person conference where attendees are 
+distracted for reasons directly unrelated to the conference (work or personal); or (2) a virtual conference where 
+attendees are multi-tasking on non-conference related matters. The degree of “distraction” is likely due to an 
+attendee not being fully engaged by the conference or not having any conference related “deliverables.” For 
+example, will they be evaluated post-conference via a trip report/presentation which requires them to remain focused 
+– or is their personal value self-assessed? 
+13. The future? 
+Although “virtual” changed the conference experience in the early stages of the pandemic, the flexibility of virtual 
+participation has had important side benefits. Virtual has led to increased conference accessibility through lower 
+attendee travel costs (money, time, carbon, government visas), and reduced expenditures by companies, universities, 
+and governments. One conference category where “virtual” meeting technologies are not making a significant 
+difference currently – are trade shows. For example, the annual Consumer Technology Association’s CES 2022 
+reported a 75% drop in attendance (40,000 attendees instead of the pre-pandemic 170,000 attendees) [16]. 
+Areas ripe for “virtual” improvement include:  
+• 
+Increased support for casual serendipitous interactions 
+• 
+Support for interactive discussions for ideation 
+• 
+Convenient post-event access to video and presentation content 
+• 
+Sustainable revenue models for virtual events 
+While “online fatigue” is an issue for virtual conferences, it is not obvious whether this is more than the fatigue 
+experienced with travel to an in-person conference. While some might argue that the lure of a conference destination 
+mitigates travel fatigue – the authors suggest that more data and research is required to assess the comparative 
+impact of online versus travel fatigue on conference attendees. In-person conferences appear to catalyze increased 
+attendee interaction. In comparison, virtual conference environments, as currently implemented, seem to foster less 
+engagement between attendees. 
+The questions of in-person, virtual, or hybrid conferences – and the appropriate ways to apply advances in virtual 
+technology – need to be answered in the context of global and personal issues. The world is facing a climate crisis. 
+Society is becoming more aware of diversity and equity issues. All of us are facing individual challenges: keeping 
+our knowledge and skills up to date, expanding our set of personal contacts, and protecting our health by reducing 
+unnecessary travel. Our employers and sponsors are trying to get maximum value at the lowest cost. There is no 
+single simple answer for conference organizers, but we should all work together to try new ideas. 
+The authors believe that it is short-sighted and reduces accessibility if all conferences return to an in-person format. 
+Our informal survey suggests that individuals prefer conference attendance options. While there is more work to be 
+done to make virtual conferences truly effective and collaborative – we should all recognize that reducing travel and 
+carbon footprints is a good thing. In our view, we all need to foster the adoption of virtual conferences – beyond the 
+plateau achieved during the pandemic. 
+
+ 
+ 
+Page 17 
+“The only thing we know about the future is that it will be different.” – Peter Drucker 
+“Alone we can do so little; together we can do so much.” – Helen Keller 
+14. About the authors 
+This report reflects the opinions and the combined 50+ years’ experience with in-person, virtual, and hybrid 
+conferences of the authors. Fraser and Mancl have participated and presented at over one hundred ACM, IEEE, 
+Agile Alliance, and university hosted conferences and workshops – including the 2000, 2021, and 2022 virtual and 
+hybrid ACM SPLASH, ACM/IEEE ICSE, and Agile Alliance’s XP conferences. 
+Fraser pioneered virtual hybrid corporate forums at Nortel, Qualcomm, and Cisco Systems starting in the 1990s with 
+ISDN based videoconferencing (25+ sites worldwide). The Nortel Design Forum (a global internal hybrid technical 
+conference with 30+ ISDN video meeting room hubs and audio/web desktop participation) attracted up to 2,000 
+attendees per forum and ran through more than a dozen editions featuring peer-reviewed paper presentations, 
+keynotes, and interactive workshops. The hybrid QTech and CTech forums at Qualcomm and Cisco used a 
+combination of desktop video applications (e.g., WebEx) and TelePresence – with the program anchored by in-
+person presentations at corporate headquarters. 
+Mancl has been a presenter at internal technical conferences on software tools and technology at Lucent and Alcatel-
+Lucent beginning in 1990. He also has worldwide experience in corporate education and training, developing a wide 
+range of software technology courses and delivering them in person and virtually using multiple generations of 
+collaboration technology. 
+ACKNOWLEDGMENTS 
+Thanks to our anonymous reviewers and a special thanks to Moshe Vardi, Crista Lopes, and Dave Parnas for their 
+perspectives on conferences. We would also like to thank Robert Crawhall and Steve McConnell for feedback on a 
+draft of this report. Lastly, we would like to thank Teresa Foster and Ellen Grove from the Agile Alliance for their 
+support of our community survey on conference preferences that the authors ran in the spring 2022. 
+REFERENCES 
+[1] Steven Fraser and Dennis Mancl. 2022. The Future of Conferences Research Survey, 
+https://manclswx.com/survey2022.html, Accessed 5 Jan 2023. 
+[2] Clarence A. Ellis, Simon J. Gibbs, and Gail Rein. 1991. Groupware: some issues and experiences. 
+Communications of the ACM 34, 1 (Jan. 1991), 39–58. https://doi.org/10.1145/99977.99987  
+[3] Ken Hiltner. A Nearly Carbon-Neutral (NCN) Conference Model, 2016, 
+https://hiltner.english.ucsb.edu/index.php/ncnc-guide/, Accessed 5 Jan 2023. 
+[4] ACM Presidential Task Force on What Conferences Can Do to Replace Face to Face Meetings. Virtual 
+Conferences, A Guide to Best Practices, 2020, https://www.acm.org/virtual-conferences, Accessed 5 Jan 2023. 
+[5] Steven Fraser. 2021. Five Strategies for the Future of Work: Accelerating Innovation through Tech Transfer. 
+Experience report from XP 2021 Conference. https://www.agilealliance.org/wp-
+content/uploads/2021/06/S.Fraser.Five-Strategies-for-the-Future-of-Work-Accelerating-Innovation-through-
+Tech-Transfer.pdf. Accessed 5 Jan 2023. 
+[6] Marios Constantinides and Daniele Quercia. 2022. The Future of Hybrid Meetings. In 2022 Symposium on 
+Human-Computer Interaction for Work (CHIWORK 2022). Association for Computing Machinery, New York, 
+NY, USA, Article 6, 1–6. https://doi.org/10.1145/3533406.3533415 
+[7] Kevin Sullivan, Interview with Michael Gorlick: On the Mbone. In Will Tracz (editor). 1995. 17th International 
+Conference on Software Engineering: Window on the World. SIGSOFT Software Engineering Notes 20, 3 (July 
+1995), 18-19. https://doi.org/10.1145/219308.773575  
+[8] Alexandra Ridgway. 2022. “Conference water-cooler moments are not accessible to everyone,” Times Higher 
+Education (THE), https://www.timeshighereducation.com/opinion/conference-water-cooler-moments-are-not-
+accessible-everyone, Accessed 5 Jan 2023. 
+
+ 
+ 
+Page 18 
+[9] Liang Zhao. 2020. How the Trillion Dollar Global Business Events Industry Is Adjusting to a Rapid Shift 
+Online, Grit Daily, June 11, 2020, https://gritdaily.com/how-the-trillion-dollar-global-business-events-industry-
+is-adjusting-to-a-rapid-shift-online/, Accessed 5 Jan 2023. 
+[10] Moshe Y. Vardi. 2020. Reboot the computing-research publication systems. Communications of the ACM 64, 1 
+(January 2021), 7. https://doi.org/10.1145/3437991  
+[11] Bob Briscoe, Andrew Odlyzko and Benjamin Tilly. 2016. Metcalfe’s law is wrong - communications networks 
+increase in value as they add members-but by how much? IEEE Spectrum, vol. 43, no. 7, pp. 34-39, July 2006, 
+https://doi.org/10.1109/MSPEC.2006.1653003  
+[12] Kent Beck. 2005. Video interview with Kent Beck at the Agile 2005 Conference, July 22, 2005. 
+https://www.youtube.com/watch?v=1zaCvLVU70o starting at 10:13], Accessed 5 Jan 2023. 
+[13] European Union GDPR website, https://gdpr.eu/what-is-gdpr/, Accessed 5 Jan 2023. 
+[14] Crista Lopes. The Future of Conferences, Strange Loop 2022 conference, 
+https://www.youtube.com/watch?v=LkJNA88R_5w, Accessed 5 Jan 2023. 
+[15] Dennis Mancl. 2022. Personal notes from ICSE 2022 conference, 
+https://manclswx.com/notes/icse2022_report.html#icse_town_meeting, Accessed 5 Jan 2023. 
+[16] Richard N. Velotta. 2022. “CES attendance down more than 75%, organizers say,” Las Vegas Review-Journal, 
+January 7, 2022. https://www.reviewjournal.com/business/conventions/ces/ces-attendance-down-more-than-75-
+organizers-say-2509439/, Accessed 5 Jan 2023. 
+ 
+© 2023 Steven Fraser and Dennis Mancl 
+ 
+This report is licensed under the terms of the Creative Commons Attribution 4.0 
+International License (https://creativecommons.org/licenses/by/4.0/), which 
+permits use, sharing, adaptation, distribution and reproduction in any medium or 
+format, as long as you give appropriate credit to the original author(s) and the 
+source, provide a link to the Creative Commons license and indicate if changes 
+were made. 
+
+BY

.gitattributes

@@ -9879,3 +9879,99 @@ qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf filter=lfs diff=lfs merge=lfs -tex
 A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf filter=lfs diff=lfs merge=lfs -text
 3tFAT4oBgHgl3EQflR3K/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
 29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf filter=lfs diff=lfs merge=lfs -text
+YdAyT4oBgHgl3EQfiPiV/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+UtE3T4oBgHgl3EQf0Au1/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+X9E3T4oBgHgl3EQfGAla/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+R9E2T4oBgHgl3EQfBwbf/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+JtFQT4oBgHgl3EQfSTYo/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf filter=lfs diff=lfs merge=lfs -text
+fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf filter=lfs diff=lfs merge=lfs -text
+a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf filter=lfs diff=lfs merge=lfs -text
+s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf filter=lfs diff=lfs merge=lfs -text
+jNE5T4oBgHgl3EQfGA7Z/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf filter=lfs diff=lfs merge=lfs -text
+z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf filter=lfs diff=lfs merge=lfs -text
+A9AyT4oBgHgl3EQfd_iI/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf filter=lfs diff=lfs merge=lfs -text
+3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf filter=lfs diff=lfs merge=lfs -text
+aNE0T4oBgHgl3EQfnQFP/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf filter=lfs diff=lfs merge=lfs -text
+zNE5T4oBgHgl3EQfNw6V/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+cdE0T4oBgHgl3EQfWQDo/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf filter=lfs diff=lfs merge=lfs -text
+OtFJT4oBgHgl3EQfICxT/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+DdAzT4oBgHgl3EQfT_w7/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+V9FPT4oBgHgl3EQfqjWj/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+c9E0T4oBgHgl3EQfoAGA/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+dtAzT4oBgHgl3EQfn_1r/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf filter=lfs diff=lfs merge=lfs -text
+P9FJT4oBgHgl3EQf2S1v/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf filter=lfs diff=lfs merge=lfs -text
+gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf filter=lfs diff=lfs merge=lfs -text
+ItE3T4oBgHgl3EQfugv_/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ONE3T4oBgHgl3EQfxAvE/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf filter=lfs diff=lfs merge=lfs -text
+XdE3T4oBgHgl3EQf1Qur/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+C9AzT4oBgHgl3EQfiP2U/content/2301.01497v1.pdf filter=lfs diff=lfs merge=lfs -text
+Q9E0T4oBgHgl3EQfkQH6/content/2301.02472v1.pdf filter=lfs diff=lfs merge=lfs -text
+3tAyT4oBgHgl3EQf1_ko/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+69AyT4oBgHgl3EQfpvg-/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf filter=lfs diff=lfs merge=lfs -text
+hdE0T4oBgHgl3EQfXwBq/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf filter=lfs diff=lfs merge=lfs -text
+XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf filter=lfs diff=lfs merge=lfs -text
+qdE4T4oBgHgl3EQfvw0Q/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+8NAyT4oBgHgl3EQfp_h6/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf filter=lfs diff=lfs merge=lfs -text
+zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf filter=lfs diff=lfs merge=lfs -text
+etE3T4oBgHgl3EQfHQmJ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf filter=lfs diff=lfs merge=lfs -text
+etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf filter=lfs diff=lfs merge=lfs -text
+Q9E0T4oBgHgl3EQfkQH6/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+pdE4T4oBgHgl3EQfvQ2I/content/2301.05240v1.pdf filter=lfs diff=lfs merge=lfs -text
+w9E2T4oBgHgl3EQfLwYb/content/2301.03716v1.pdf filter=lfs diff=lfs merge=lfs -text
+BNFRT4oBgHgl3EQfuzgt/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+PdAzT4oBgHgl3EQfWvxw/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf filter=lfs diff=lfs merge=lfs -text
+xdFQT4oBgHgl3EQfATUq/content/2301.13222v1.pdf filter=lfs diff=lfs merge=lfs -text
+z9AzT4oBgHgl3EQf8f40/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf filter=lfs diff=lfs merge=lfs -text
+UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf filter=lfs diff=lfs merge=lfs -text
+WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf filter=lfs diff=lfs merge=lfs -text
+ztFLT4oBgHgl3EQfoC-E/content/2301.12130v1.pdf filter=lfs diff=lfs merge=lfs -text
+ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf filter=lfs diff=lfs merge=lfs -text
+cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf filter=lfs diff=lfs merge=lfs -text
+zdFQT4oBgHgl3EQfCTXT/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+49FKT4oBgHgl3EQf9y5B/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+s9A0T4oBgHgl3EQfLP_n/content/2301.02117v1.pdf filter=lfs diff=lfs merge=lfs -text
+8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf filter=lfs diff=lfs merge=lfs -text
+OdFRT4oBgHgl3EQf4zhC/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+UNE2T4oBgHgl3EQfCwYa/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+8tAyT4oBgHgl3EQfc_f6/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+8NE4T4oBgHgl3EQfdAy1/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf filter=lfs diff=lfs merge=lfs -text
+1NE2T4oBgHgl3EQf5Ahd/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+RtAzT4oBgHgl3EQfXPx9/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+itE3T4oBgHgl3EQf4wuZ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+V9E1T4oBgHgl3EQfJAMR/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+zdAzT4oBgHgl3EQfQ_uv/content/2301.01209v1.pdf filter=lfs diff=lfs merge=lfs -text
+1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf filter=lfs diff=lfs merge=lfs -text
+ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf filter=lfs diff=lfs merge=lfs -text
+ctAzT4oBgHgl3EQfLvsV/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf filter=lfs diff=lfs merge=lfs -text
+zdAzT4oBgHgl3EQfQ_uv/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+k9E1T4oBgHgl3EQfNwMc/content/2301.03005v1.pdf filter=lfs diff=lfs merge=lfs -text
+j9A0T4oBgHgl3EQfIv_e/content/2301.02081v1.pdf filter=lfs diff=lfs merge=lfs -text
+fdE0T4oBgHgl3EQf6AIY/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+1NE4T4oBgHgl3EQfzQ1E/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+u9AzT4oBgHgl3EQf6_6o/content/2301.01884v1.pdf filter=lfs diff=lfs merge=lfs -text
+INE4T4oBgHgl3EQfhA00/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+sdFKT4oBgHgl3EQfJy1E/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+29FLT4oBgHgl3EQfrS9R/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+WtE0T4oBgHgl3EQf3QI4/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+k9E1T4oBgHgl3EQfggTF/content/2301.03231v1.pdf filter=lfs diff=lfs merge=lfs -text
+49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf filter=lfs diff=lfs merge=lfs -text
+ztFLT4oBgHgl3EQfoC-E/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+3NE1T4oBgHgl3EQfAQKK/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf filter=lfs diff=lfs merge=lfs -text
+u9AzT4oBgHgl3EQf6_6o/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text

1NE2T4oBgHgl3EQf5Ahd/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:8cd96bf473aa8aa559b56fae80230cd08b7f7e0e04ee8a3f4ce3b450e6a9dedb
+size 1114157

1NE2T4oBgHgl3EQf5Ahd/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:66cefe25893598aedf004bbdee9d2875ae9994563d8f1dc268421bc1be334728
+size 39949

1NE4T4oBgHgl3EQfzQ1E/content/2301.05272v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:2a9b53dad5f1a141ecbf63d356c2575b5dad2f1045505091129208fc8fa551f1
+size 152015

1NE4T4oBgHgl3EQfzQ1E/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:610746013cb2b397b45dd76d374829a5ef01f4b92418a94eaa818806c79a4471
+size 1638445

1NE4T4oBgHgl3EQfzQ1E/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d76f0cdf4950a5f5bfa6f47aa0727738ba60007206923ff18c0d1be8ce4ae71f
+size 62380

29FLT4oBgHgl3EQfrS9R/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ea019431ae87b00ef9a2e93f2722f540a0af9ed231f4ae7096c4807a86f02335
+size 13565997

39E1T4oBgHgl3EQfSgNA/content/tmp_files/2301.03066v1.pdf.txt

@@ -0,0 +1,1416 @@
+Impacts of momentum dependent interaction, symmetry energy and near-threshold
+NN → N∆ cross sections on isospin sensitive flow and pion observables
+Yangyang Liu,1, ∗ Yingxun Zhang,1, 2, † Junping Yang,1 Yongjia Wang,3, ‡ Qingfeng Li,3, 4, § and Zhuxia Li1
+1China Institute of Atomic Energy, Beijing 102413, China
+2Guangxi Key Laboratory of Nuclear Physics and Technology,
+Guangxi Normal University, Guilin, 541004, China
+3School of Science, Huzhou University, Huzhou 313000, China
+4Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
+(Dated: January 10, 2023)
+Based on the ultra-relativistic quantum molecular dynamics (UrQMD) model, the impacts of
+momentum dependent interaction, symmetry energy and near-threshold NN → N∆ cross sections
+on isospin sensitive collective flow and pion observables are investigated. Our results confirm that
+the elliptic flow of neutrons and charged particles, i.e. vn
+2 and vch
+2 , are sensitive to the strength
+of momentum dependence interaction and the elliptic flow ratio, i.e., vn
+2 /vch
+2 , is sensitive to the
+stiffness of symmetry energy. For describing the pion multiplicity near the threshold energy, accurate
+NN → N∆ cross sections are crucial. With the updated momentum dependent interaction and
+NN → N∆ cross sections in UrQMD model, seven observables, such as directed flow and elliptic
+flow of neutrons and charged particles, the elliptic flow ratio of neutrons to charged particles, charged
+pion multiplicity and its ratio π−/π+, can be well described by the parameter sets with the slope
+of symmetry energy from 5 MeV to 70 MeV. To describe the constraints of symmetry energy at
+the densities probed by the collective flow and pion observables, the named characteristic density
+is investigated and used. Our analysis found that the flow characteristic density is around 1.2ρ0
+and pion characteristic density is around 1.5ρ0, and we got the constrains of symmetry energy at
+characteristic densities are S(1.2ρ0) = 34 ± 4 MeV and S(1.5ρ0) = 36 ± 8 MeV. These results are
+consistent with previous analysis by using pion and flow observable with different transport models,
+and demonstrate a reasonable description of symmetry energy constraint should be presented at the
+characteristic density of isospin sensitive observables.
+I.
+INTRODUCTION
+The isospin asymmetric nuclear equation of state is
+crucial for understanding the isospin asymmetric objects,
+such as the structure of neutron-rich nuclei, mechanism
+of neutron-rich heavy ion collisions, the properties of neu-
+tron stars including neutron star mergers and core col-
+lapse supernovae[1–4].
+The symmetric part of isospin
+asymmetric equation of state has been well constrained
+by using the flow and Kaon condensation[5]. However,
+the symmetry energy away from the normal density still
+have large uncertainty, and it leads that the constraint
+of symmetry energy becomes one of the important goal
+in nuclear physics[6, 7].
+The ultimate goal of symmetry energy constraint is to
+obtain the density dependence of symmetry energy over
+a wide range, and many efforts have been devoted to con-
+strain the symmetry energy from subsaturation density
+to suprasaturation density. For probing the symmetry
+energy at suprasaturation density, the isospin sensitive
+observables in heavy ion collisions (HICs), such as the
+ratio of elliptic flow of neutrons to charged particles, hy-
+drogen isotopes or protons (vn
+2 /vch
+2 , vn
+2 /vH
+2 or vn
+2 /vp
+2)[8–
+12] and the multiplicity ratio of charged pions (i.e.,
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+§ [email protected]
+M(π−)/M(π+) or named as π−/π+)[13–23], were mainly
+used.
+By comparing the calculations to transverse-
+momentum-dependent or integrated FOPI/LAND and
+ASY-EOS elliptic flow data of nucleons and hydrogen
+isotopes, a moderately soft to linear symmetry energy
+is obtained with UrQMD[8, 10, 11] and T¨ubingen quan-
+tum molecular dynamics (T¨uQMD) models[9]. The lower
+limit of the slope of symmetry energy L obtained with the
+flow ratio data is L > 60 MeV[24], which overlaps with
+the upper limits of the constraints from nuclear structure
+and isospin diffusion, i.e., L ≈ 60±20 MeV[25–27]. How-
+ever, the constraints of symmetry energy from π−/π+
+show strong model dependence[15–21, 28], and the ex-
+tracted L values ranges from 5 MeV to 144 MeV. It may
+be caused by the different treatments on the nucleonic
+potential, ∆ potential, threshold effects, pion potential,
+Pauli blocking, in-medium cross sections and so on, and
+also by the different numerical technical for solving the
+transport equations.
+To reduce the model dependence and enhance the reli-
+ability of the constraints of symmetry energy, especially
+at suprasaturation density, the transport model evalu-
+ations are required.
+The transport model evaluation
+project has made important progress on benchmarking
+the treatment of particle-particle collision[29, 30] and
+nucleonic mean field potential[31] in both Boltzmann-
+Uehling-Uhlenbeck (BUU) type and Quantum molecu-
+lar dynamics (QMD) type models. For simulating the
+collisions or decay of resonance particles, the time-step-
+arXiv:2301.03066v1  [nucl-th]  8 Jan 2023
+
+2
+free method is suggested[29, 30] since this method au-
+tomatically determine whether the resonance will col-
+lide or decay according to their collision time or decay
+time. In the UrQMD model, the time-step-free method
+is adopted in the collision part[29, 30], and the nucleonic
+potential is also involved for extending its applications
+in low-intermediate energy HICs[10, 21].
+This model
+has been successfully used to study the HICs from low-
+intermediate energy to high energies[10, 21, 24, 32, 33].
+Another method to reduce the model uncertainties is si-
+multaneously describing the observables data (or named
+doing combination analysis), such as isospin sensitive col-
+lective flow and pion observables. For the combination
+analysis on the isospin sensitive nucleonic and pion ob-
+servables, there were few works to simultaneously inves-
+tigate them except the T¨uQMD model[20] as far as we
+know.
+Thus, it will be interesting to do combination
+analysis on nucleonic and pion observables back-to-back
+by the UrQMD model for increasing the reliability of the
+constraints of symmetry energy in the community.
+In previous analysis on the neutrons to protons or
+to hydrogen isotopes elliptic flow ratios[10] or π−/π+
+ratios[21] by UrQMD model, the momentum dependent
+interaction (MDI) form, i.e., t4 ln2(1+t5(p1−p2)2)δ(r1−
+r2), was used. This form was extracted from the Arnold’s
+optical potential data [34, 35]. In 1990’s, the real part
+of the global Dirac optical potential (Schr¨odinger equiv-
+alent potential) was published by Hama et al. [36], in
+which angular distribution and polarization quantities
+in proton-nucleus elastic scattering were analyzed in the
+range of 10 MeV to 1 GeV. The Hama’s data generated
+Lorentzian-type momentum-dependent interaction [35],
+which give a stronger momentum dependent potential
+than the Arnold’s form at high momentum, have been
+used in many version of transport models[37–42] for
+studying high energy HICs. In another, the cross sections
+of NN → N∆ channel, i.e.,σNN→N∆, used in UrQMD
+model are obtained by fitting CERN data [43], and the
+fitting formula underestimate σNN→N∆ near the thresh-
+old energy which will be shown in Figure 2. Thus, the
+refinements of MDI and formula of NN → N∆ cross
+section σNN→N∆ near the threshold are necessary for si-
+multaneously describing the flow and pion observables.
+In this work, we will address these issues with the
+UrQMD model and investigate their influence on nucle-
+onic flow and pion observables. Further, the constraints
+of symmetry energy at suprasaturation density are dis-
+cussed with the updated version of UrQMD model. The
+paper is organized as follows: in Sect.II, we briefly intro-
+duce the nucleonic potential, momentum dependent in-
+teraction and refined cross sections of NN → N∆ chan-
+nel. In Set.III, the impacts of momentum dependent in-
+teraction, symmetry energy and refined NN → N∆ cross
+sections on flow and pion observables are presented and
+discussed. By comparing the calculations with the ASY-
+EOS flow data and FOPI pion data, the constraints of
+symmetry energy at characteristic density are discussed.
+Sec.IV is the summary of this work.
+II.
+URQMD MODEL
+The version of UrQMD model we used is the same as
+that in Ref.[21], in which the cross sections of N∆ →
+NN channel are replaced with a more delicate form by
+considering the ∆-mass dependence of the M-matrix in
+the calculation of N∆ → NN cross section[44].
+This
+version has been successfully used to describe the FOPI
+experimental data of multiplicity and ratio of charged
+pion[21], but did not use to simultaneously describe the
+pion and flow observables.
+Since we focus on the effects of different forms of MDI,
+symmetry energy, and cross sections of NN → N∆, we
+briefly introduce them in the following. The nucleonic
+potential energy U is calculated from the potential energy
+density, i.e., U =
+�
+ud3r. The u reads as
+u = α
+2
+ρ2
+ρ0
++
+β
+η + 1
+ρη+1
+ρη
+0
+(1)
++gsur
+2ρ0
+(∇ρ)2 + gsur,iso
+ρ0
+[∇(ρn − ρp)]2
++umd + usym.
+The parameters α, β, and η are related to the two, three-
+body interaction term. The third and fourth terms are
+isospin independent and isospin dependent surface term,
+respectively. The umd is from the MDI term, and we will
+adopt two forms in this work. The usym is the symmetry
+energy term.
+The energy density associated with the MDI, i.e., umd,
+is calculated according to the following relationship,
+umd =
+�
+ij
+�
+d3p1d3p2fi (⃗r, ⃗p1) fj (⃗r, ⃗p2) vmd(∆p12).
+(2)
+The form of vmd(∆p12) is assumed as,
+vmd(∆p12) = t4 ln2(1 + t5∆p2
+12) + C,
+(3)
+where ∆p12 = |p1 − p2|, and the parameters t4, t5 and
+C are obtained by fitting the data of the real part of
+optical potential. In details, we fit the data of real part
+of nucleon-nucleus optical potential Vmd(p) according to
+the following ansatz,
+Vmd(p1) =
+�
+p2<pF
+vmd(p1 − p2)d3p2/
+�
+p2<pF
+d3p2.
+(4)
+This method is as the same as that in Ref.[35].
+Two
+sets of data of the real part of optical potential are used.
+One is from Arnold et al. [34] which were used in pre-
+vious version of UrQMD[10, 21]. Another is from Hama
+et al. [36]. They are presented as green squares and red
+circles in Fig. 1 (a), respectively. The lines are momen-
+tum dependence interaction vmd(∆p12) at normal den-
+sity obtained by fitting Arnold’s or Hama’s data by using
+Eq.(3) and Eq.(4) within the kinetic energy Ekin ≈ 750
+MeV. The values of t4, t5 and C obtained from Arnold’s
+
+3
+data and Hama’s data are listed in Table I. The momen-
+tum dependence of vHama
+md
+(∆p12) is stronger than that of
+vArnold
+md
+(∆p12), and the value of vHama
+md
+(∆p12) is higher
+than vArnold
+md
+(∆p12) at high momentum region.
+To keep the incompressibility of symmetric nuclear
+matter K0 = 231 MeV for two different MDIs, the pa-
+rameter α, β, and η are readjusted and the values of
+parameters and corresponding effective mass m∗/m are
+listed in Table I.
+TABLE I. Parameters used in the present work.
+t4, C, α,
+β and K0 are in MeV. t5 is in MeV−2, η and m∗/m are
+dimensionless. The width of Gaussian wave packet is taken
+as 1.414 fm for Au+Au collision.
+Para.
+t4
+t5
+C
+α
+β
+η
+K0 m∗/m
+vArnold
+md
+1.57 5×10−4 -54 -221 153 1.31 231
+0.77
+vHama
+md
+3.058 5×10−4 -86 -335 253 1.16 231 0.635
+For the potential energy density of symmetry en-
+ergy part, i.e., usym, we take two forms in the calcu-
+lations. One is the Skyrme-type polynomial form ( (a) in
+Eq. (5)) and another is the density power law form ((b)
+in Eq. (5)). It reads,
+usym = Spot
+sym(ρ)ρδ2
+(5)
+=
+�
+(A( ρ
+ρ0 ) + B( ρ
+ρ0 )γs + C( ρ
+ρ0 )5/3)ρδ2, (a)
+Cs
+2 ( ρ
+ρ0 )γiρδ2.
+(b)
+The symmetry energy coefficient is S0 = S(ρ0) and the
+slope of symmetry energy is L = 3ρ0∂S(ρ)/∂ρ|ρ0. Based
+on the values of S0, L and parameters in Table I, one
+can also obtain the parameters of Eq.(5) based on the
+relationship described in Ref. [27, 45]. In following cal-
+culations, we taken S(ρ0) = 30−34 MeV and L = 5−144
+MeV, as shown in Table.II.
+For L < 35 MeV, we use the Skyrme polynomial form
+of Spot
+sym(ρ) because the simple power law form of symme-
+try energy can not give reasonable values at subnormal
+density. Further, the L < 5 MeV sets are not adopted be-
+cause the corresponding symmetry energy becomes nega-
+tive at the densities above 2.7ρ0 and the EOS will not be
+favored by the properties of the neutron stars. Thus, the
+lower limit of L in our calculations is 5 MeV. For L > 35
+MeV, we use the simple power law form of symmetry en-
+ergy. As an example, we present the density dependence
+of symmetry energy in Fig.1 (b) for L = 20, 144 MeV at
+S0=30 and 34 MeV.
+TABLE II. Parameters of symmetry energy and effective mass
+used in the calculations.
+Para. Name
+Values
+Description
+S0
+[30, 34]
+symmetry energy coefficient
+L
+[5,144]
+slope of symmetry energy
+m∗/m
+0.635,0.77
+isoscalar effective mass
+0
+300
+600
+900
+-50
+0
+50
+100
+0.00
+0.16
+0.320
+40
+80
+ 
+ 
+Re Uopt (MeV)
+Ekin (MeV)
+Arnold
+Hama 
+ v
+Arnold
+md
+ v
+Hama
+md
+(a)
+L=20 MeV
+ S0=30 MeV
+ S0=34 MeV
+ S(r) (MeV)
+L=144 MeV
+r (fm-3)
+(b)
+FIG. 1. (a) Real part of the optical potential Vmd and momen-
+tum dependent interaction vmd. The symbols are the optical
+potential data obtained from Arnold et al. [34] and Hama et
+al. [36]. Lines are the vArnold
+md
+and vHama
+md
+obtained through
+Eq.(4). (b) density dependence of the symmetry energy with
+different S0 and L values.
+In the collision term, the medium modified nucleon-
+nucleon elastic cross sections are used as the same as that
+in our previous works[24]. For the NN → N∆ cross sec-
+tions, we found that the default formula used in UrQMD
+model in Ref. [32] underestimates the data [43] near the
+threshold energy. The discrepancy is shown in Fig.2 (a),
+where the blue line is the fitting formula in Ref. [32] and
+solid symbols are the data taken from Ref. [43].
+Thus, one can expect that we have to use an accurate
+form of NN → N∆ cross section near the threshold en-
+ergy for describing the pion production at 0.4A GeV. To
+refine the fitting of NN → N∆ cross section near the
+threshold energy, we adopt a Hubbert function form to
+describe the NN → N∆ cross sections at √s < 2.21
+GeV. That is,
+σNN→N∆(√s) = A1 + 4A2 ∗ e−(√s−A3)/A4
+(1 + e−(√s−A3)/A4)2 ,
+(6)
+√s < 2.21GeV.
+In which, A1=-1.11 mb, A2=26.30 mb, A3=2.24 GeV,
+and A4=0.05 GeV. We named it as σHub
+NN→N∆ to distin-
+guish the default form in Ref.[32]. The fitting results are
+represented as the red line in Fig.2 (a). Above 2.21 GeV,
+the original fitting function is used.
+As shown in Fig.2 (a), the σHub
+NN→N∆ is closer to the
+experimental data than the original formula. The right
+panels show that the ratio of R = σHub/σDefault, and one
+can see that the cross sections σHub
+NN→N∆ are increased
+by a factor of 8.56 at the beam energy of 0.4A GeV.
+Consequently, one can expect a higher pion multiplicity
+with σHub
+NN→N∆ than the one with σDefault
+NN→N∆. The N∆ →
+NN cross sections are obtained based on the detailed
+balance, in which a ∆ mass dependent N∆ → NN cross
+sections was also considered as in Refs. [21, 44].
+
+4
+2.0
+2.1
+2.2
+0
+20
+40
+2.0
+2.1
+2.2
+0
+5
+10
+2
+3
+4
+0
+20
+ 
+s
+Default
+NN→ND
+ s
+Hub
+NN→ND
+s1/2 (GeV)
+sNN→ND (mb)
+(a)
+0.4A GeV
+(b)
+8.56
+0.4A GeV
+R=s
+Hub/s
+Default
+s1/2 (GeV)
+ DATA
+(c)
+FIG. 2. (a) The cross section of NN → N∆ channel used in
+the default UrQMD model σDefault
+NN→N∆ and obtained by refitting
+the experimental data with Hubbert function σHub
+NN→N∆ near
+threshold energy. (b) The ratio of σHub
+NN→N∆ over σDefault
+NN→N∆
+as a function of √s.
+III.
+RESULTS AND DISCUSSIONS
+The collective flow reflects the directional features of
+the transverse collective motion, and it can be quantified
+in terms of the moments of the azimuthal angle relative
+to the reaction plane, i.e., vn = ⟨cos(nφ)⟩, n = 1, 2, 3, · · · .
+Among the vn, the elliptic flow v2 has been used to de-
+termine the MDI [46], and the ratio between v2 of neu-
+trons and protons, i.e., vn
+2 /vp
+2, or ratio between v2 of neu-
+trons and charged particles, i.e., vn
+2 /vch
+2 , are proposed to
+determine the symmetry energy at suprasaturation den-
+sity [10–12]. It is known that pions are mainly produced
+through ∆ resonance decay in suprasaturation density re-
+gion at early stage; and the multiplicity ratio of charged
+pions, i.e., π−/π+, was also supposed as a probe to con-
+strain the symmetry energy at suprasaturation density
+and widely studied[13–17, 20, 21]. In this work, we first
+investigate the nucleonic flow observables to determine
+the form of MDI and pion production to determine the
+form of NN → N∆ cross sections near the threshold en-
+ergy. Then, the symmetry energy at suprasaturation den-
+sity will be extracted by comparing the UrQMD calcula-
+tions of vn
+2 /vch
+2 to ASY-EOS data and comparing π−/π+
+results to FOPI data.
+A.
+collective flow and pion observable
+In this work, we perform the calculations of Au+Au
+collision at 0.4A GeV witsubsectionh 200,000 events at
+each impact parameter.
+The final observables are ob-
+tained by integrating over b from 0 ot bmax with a certain
+weight. The weight of b is reconstructed by the central-
+ity selection used in the experiments where the Zbound
+or Zrat and the detected charge particle multiplicity or
+the ratio of total transverse to longitudinal kinetic en-
+ergies in the center-of-mass (c.m.) system are used as
+in Refs. [11, 47]. The corresponding impact parameter
+distributes in a wide range and the weight of b is a Gaus-
+sian shape rather than a triangular shape [11], which also
+have been discussed in Refs.[48–50]. The seven observ-
+ables are investigated in the following analysis, as listed
+in Table.III.
+TABLE III. Seven experimental observables used in this work.
+obsevable
+rapidity y0 cut
+θlab cut
+< b >
+vn
+1 (pt/A)
+−0.5 − 0.5
+37◦ − 53◦ 5.69 fm [11]
+vch
+1 (pt/A)
+−0.5 − 0.5
+37◦ − 53◦ 5.69 fm[11]
+vn
+2 (pt/A)
+−0.5 − 0.5
+37◦ − 53◦ 5.69 fm[11]
+vch
+2 (pt/A)
+−0.5 − 0.5
+37◦ − 53◦ 5.69 fm[11]
+vn
+2 /vch
+2 (pt/A)
+−0.5 − 0.5
+37◦ − 53◦ 5.69 fm [11]
+M(π)
+−
+−
+<2a[47]
+π−/π+
+−
+−
+<2a[47]
+a We did not put the average b value here since experimental
+paper only provides b/bmax < 0.15, which is obtained by
+estimating the impact parameter b from the measured
+differential cross sections for the ERAT under a geometrical
+sharp-cut approximation.
+-0.2
+0.0
+0.2
+0.2
+0.4
+0.6
+-0.1
+0.0
+0.2
+0.4
+0.6
+0.8
+ 
+v1
+neutrons
+(c)
+L=20 MeV
+L=144 MeV
+(a)
+ 
+ 
+v2
+pt/A (GeV/c)
+ASY-EOS
+ 
+(d)
+Au+Au Ebeam=0.4A GeV
+Charged particles
+V
+Arnold s
+Default
+V
+Hama  s
+Default
+V
+Hama  s
+Hub
+(b)
+ 
+pt/A (GeV/c)
+FIG. 3.
+Panel (a) v1(pt/A) for neutrons; (b) v1(pt/A) for
+charged particles; (c) v2(pt/A) for neutrons, and (d) v2(pt/A)
+for charged particles.
+The green lines are for V Arnold
+md
+and
+σDefault
+NN→N∆, blue lines for V Hama
+md
+and σDefault
+NN→N∆, and red lines
+for V Hama
+md
+and σHub
+NN→N∆. The dash and solid lines represent
+the results with L = 20 MeV and L = 144 MeV. The ASY-
+EOS data of collective flow for neutrons and charged particles
+are shown as circle and triangle symbols[11].
+Fig.3 (a) and (b) show directed flow as a function
+of pt/A for neutrons vn
+1 (pt/A) and for charged parti-
+cles vch
+1 (pt/A) at given rapidity region and angle cut.
+The symbols are the ASY-EOS data from Ref.[11]. The
+lines represent the results of UrQMD calculations with
+different forms of MDI, symmetry energy and different
+
+5
+NN → N∆ cross sections. The green lines are the re-
+sults with vArnold
+md
+and σDefault
+NN→N∆, blue lines are the results
+with vHama
+md
+and σDefault
+NN→N∆. By comparing the green and
+blue lines, one can understand the effects of MDI. The red
+lines are the results for vHama
+md
+and σHub
+NN→N∆. By com-
+paring the blue lines and red lines, the effects of σNN→N∆
+can be understood. The dashed lines and solid lines rep-
+resent the results with L = 20 MeV and L = 144 MeV at
+S0 = 32.5 MeV, respectively. The calculations show that
+the vn
+1 (pt/A) and vch
+1 (pt/A) increase from negative val-
+ues to positive values with the increasing of pt/A, and the
+sign of v1 changes around pt/A ≈ 0.5 GeV/c. Further-
+more, the calculations show that there is no sensitivities
+of v1 to L, MDI and σNN→N∆ at the selected rapidity
+region, due to the spectator matter blocking effect. In
+addition, the calculations with different combination of
+L, MDI and σNN→N∆ falls in the data region.
+Fig.3 (c) and (d) show the elliptic flow for neutrons
+vn
+2 (pt/A) and for charged particles vch
+2 (pt/A), with differ-
+ent L, MDI and σNN→N∆. The symbols and lines have
+the same meaning as in panels (a) and (b). Both the vn
+2
+and vch
+2
+have negative values and decrease with pt/A in-
+creasing, which means a preference for particle emission
+out of the reaction plane, towards 90◦ and 270◦. The im-
+portant point is that both vn
+2 and vch
+2
+at high pt region
+are strongly sensitive to the strength of MDI and L, but
+hardly influenced by the forms of σNN→N∆. The reason
+is that only 6% of NN collisions belong to NN → N∆
+collision in the present studied beam energy [21]. The
+values of v2 obtained with the vHama
+md
+are always lower
+than that with vArnold
+md
+due to the stronger momentum
+dependence of vHama
+md
+than that of vArnold
+md
+. The calcula-
+tions of vn
+2 and vch
+2
+with vHama
+md
+are more closed to the
+ASY-EOS experiment data than the one obtained with
+vArnold
+md
+, which means that the vHama
+md
+is favored. Thus,
+the following analyzing on the symmetry energy effects
+are based on the MDI of vHama
+md
+.
+In addition, both the vn
+2 and vch
+2 exhibit some sensitiv-
+ity to the stiffness of the symmetry energy. As shown in
+Fig.3 (c), the values of vn
+2 obtained with L = 144 MeV
+(stiff) are lower than that with L = 20 MeV (soft) case.
+The reason is that the stiff symmetry energy provides
+the stronger repulsive force on neutrons at suprasatu-
+ration density than that for soft symmetry energy cases.
+For charged particles, as shown in panel (d), vch
+2 obtained
+with stiff symmetry energy case are higher than that with
+soft symmetry energy case. This is because the emitted
+charged particles are mainly composed of free protons,
+which feel stronger attractive interaction for stiff symme-
+try energy case than that for soft symmetry energy case
+at suprasaturation density. However, vn
+2 or vch
+2 cannot be
+used individually to constrain the symmetry energy, be-
+cause both vn
+2 and vch
+2
+not only depend on the symmetry
+energy but also on the MDI and incompressibility. For
+example, the calculations with different incompressibility
+can lead to different results of the elliptic flow[51].
+To isolate the contributions from the isocalar poten-
+tial, vn
+2 /vch
+2
+ratio was proposed to probe symmetry en-
+ergy and several analysis have been performed by using
+the UrQMD model or T¨uQMD model[11, 20]. Fig.4 (a)
+shows the calculations for vn
+2 /vch
+2
+as a function of pt/A
+obtained with vHama
+md
+. The symbols are the data points.
+The upper two lines are the calculations with L = 144
+MeV, and the lower two lines are for L = 20 MeV. The vi-
+olet lines are for S0=30 MeV and red lines are for S0=34
+MeV. The calculations show that vn
+2 /vch
+2
+is sensitive to
+L, especially at the low pt region in which the mean-field
+play more important role. The values of vn
+2 /vch
+2 obtained
+with stiff symmetry energy cases are larger than that
+with soft symmetry energy case. This behavior can be
+understood from Fig.3 (c) and (d). By comparing the
+calculations of vn
+2 /vch
+2
+to ASY-EOS experimental data
+and doing a χ2 analysis, one can find the data favored
+parameter sets. In our work, the parameter sets are dis-
+tinguished by the values of S0 and L. Our conclusion
+is that the parameter sets with L = 5 − 70 MeV and
+S0 = 30 − 34 MeV can describe the data.
+0.30
+0.45
+0.60
+0
+1
+2
+0
+50
+100
+1500
+10
+20
+30
+L=144 MeV
+L=20 MeV
+ ASY-EOS
+   197Au+
+197Au Ebeam=0.4A GeV
+vn2 / vch
+2
+ 
+ 
+pt/A (GeV/c) 
+(a)
+S0=34 MeV
+(b)
+c
+2
+ 
+L(MeV)
+S0=30 MeV
+FIG. 4. Panel (a) vn
+2 /vch
+2
+as a function of pt/A for L = 20
+MeV and 144 MeV at S0=30 MeV (violet line) and S0=34
+MeV (red line). The black symbols represent the ASY-EOS
+experimental data[11]; (b) χ2 as a function of L with different
+S0.
+Fig.5 (a) shows the calculated Mπ/Apart as a func-
+tion of L with vHama
+md
+, under different forms of σNN→N∆.
+Apart is the nucleon number of the participant, which is
+90% of the number of system. The blue lines represent
+the calculations obtained with σDefault
+NN→N∆ in the UrQMD
+model.
+By using the σDefault
+NN→N∆, Mπ/Apart is underes-
+timated by about 30% relative to the data.
+This dis-
+crepancy can be understood from the underestimation of
+NN → N∆ cross sections by using the default formula
+σDefault
+NN→N∆ in UrQMD model, as shown in Fig. 2. The vi-
+olet and red lines represent the results obtained with the
+σHub
+NN→N∆ at S0 varying from 30 to 34 MeV. The calcu-
+lated results of Mπ/Apart fall into the data region since
+the σHub
+NN→N∆ enhance the cross sections by a factor of
+8.56 at 0.4A GeV relative to the default formula. But
+Mπ/Apart can not be used to probe L, since Mπ/Apart is
+insensitive to L based on the calculations.
+
+6
+0
+50
+100
+0.00
+0.01
+0.02
+0
+50
+100
+1502.0
+2.5
+3.0
+3.5
+ 
+FOPI
+L(MeV)
+Mp/Apart
+(a)
+S0=34 MeV
+p
+-/p
++
+(b)
+ sHub
+ sDefault
+L(MeV)
+S0=30 MeV
+FIG. 5. Mπ/Apart and π−/π+ as a function of L with two
+forms of σNN→N∆.
+The blue shaded region is the FOPI
+data[47].
+The blue dashed lines represent the calculations
+obtained with σDefault
+NN→N∆, and the violet and red lines are the
+calculations with σHub
+NN→N∆ for S0 = 30 and 34 MeV.
+In Fig.5 (b), we present the calculated ratios π−/π+ as
+a function of L with different forms of σNN→N∆. Calcu-
+lations show that π−/π+ is sensitive to L for both forms
+of σNN→N∆. Even the calculations with σDefault
+NN→N∆ can
+reproduce the π−/π+ (blue line), one can not believe the
+conclusion since the pion multiplicity is underestimated
+relative to the data. For the calculations with σHub
+NN→N∆,
+both the multiplicity of charged pion and its ratio π−/π+
+can be reproduced. By comparing the calculations to the
+FOPI data, the parameter sets with L = 5 − 70 MeV are
+also favored at S0 = 30 − 34 MeV.
+B.
+Characteristic density of nucleonic flow
+observable and symmetry energy constraints
+Before extracting the constraints of symmetry en-
+ergy at suprasaturation density with collective flow and
+charged pion production, it is interesting to check the
+characteristic density probed by charged pion produc-
+tion and nucleonic flow observable. For pion observable,
+the characteristic density is obtained by averaging the
+compressed density with pion production rate and force
+acting on ∆s in spatio-temporal domain in our previous
+work[21], and the calculations show that the character-
+istic density of pion observable is around 1.5±0.5 times
+normal density.
+For the collective flow of neutrons and charged par-
+ticles, the idea of calculating characteristic density is
+as same as pion characteristic density in our previ-
+ous work[21], but the weight is replaced by momentum
+change of nucleons. The momentum changes of nucleons
+during the time interval reflect the strength of the driven
+force for the collective motion of emitted particles, and
+can be used to understand the origins of v1 and v2. In the
+following calculations, two kinds of momentum change of
+nucleons are used. One is the momentum change in x-
+direction,
+⟨ρc, flow ⟩|∆px| =
+� t1
+t0 Σi
+��∆pi
+x(t)/∆t
+�� ρc(t)dt
+� t1
+t0 Σi |∆pix(t)/∆t| dt
+(7)
+and another is the momentum change in tranverse direc-
+tion,
+⟨ρc, flow ⟩|∆pt| =
+� t1
+t0 Σi
+��∆pi
+t(t)/∆t
+�� ρc(t)dt
+� t1
+t0 Σi
+��∆pi
+t(t)/∆t
+�� dt
+.
+(8)
+The summation over i runs over the nucleons belong-
+ing to the emitted nucleons and particles. More details,
+|∆pi
+x/t(t)/∆t| = |(pi
+x/t(t) − pi
+x/t(t − ∆t))/∆t|, i.e., the
+momentum changes of nucleon during the time interval.
+ρc(t) is obtained in a spherical region centered at c.m.
+of the system and with a radius of 3.35 fm. The region
+is used to represent the overlap region in semi-peripheral
+collisions of Au+Au.
+In Figure.6 (a), we plot the time evolution of the aver-
+aged central density ρc(t) for a semi-peripheral collision
+of Au+Au.
+The averaged central density beyond nor-
+mal density from 8 fm/c to 28 fm/c and reaches maxi-
+mum values of 1.8ρ0 at 16 fm/c with the interactions we
+adopted. For convenience, we use ∆p/∆t to represent the
+momentum change per nucleon for nucleons and emitted
+particles, i.e.,
+∆p
+∆t = Σi
+��∆pi(t)/∆t
+��
+N(t)
+,
+(9)
+N(t) is the total number of nucleons in the emitted nucle-
+ons and particles. Panel (b) shows the average momen-
+tum changes of emitted particles ∆p
+∆t as a function of time
+in x-direction and transverse direction. It illustrates that
+the drastic momentum changes of nucleons occur around
+16 fm/c when the participant region reaches the maxi-
+mum density. It confirms that nucleonic flow observables
+mainly carry the EOS information at high density. Two
+forms of symmetry energy are tested, and they did not
+change the results dramatically.
+By using Eq.(7) and Eq.(8), the characteristic density
+for the collective flow are obtained, and they are around
+1.2 ± 0.6ρ0. It is consistent with the characteristic den-
+sity obtained in the Ref.[52] and Ref.[53], but is smaller
+than the characteristic density obtained with pion ob-
+servable. Thus, by comparing the calculations of vn
+2 /vch
+2
+and π−/π+ to data, one can give the constraints of sym-
+metry energy at two densities, i.e., 1.2 ρ0 and 1.5 ρ0. The
+values of them we got are S(1.2ρ0) = 34 ± 4 MeV and
+S(1.5ρ0) = 36 ± 8 MeV, and we present them as black
+symbols in Fig.7.
+The important point is that the constraints of S(ρ) at
+flow characteristic density, i.e., at 1.2ρ0, are consistent
+with the analysis of elliptic flow ratios or elliptic flow
+difference by UrQMD[10] or T¨uQMD calculations[9, 12]
+which are presented by blue symbols. The constraints of
+
+7
+0
+20
+40
+0
+1
+2
+0
+20
+40
+600.00
+0.02
+time (fm/c)
+ 
+ 
+rc/r0
+197Au+
+197Au
+   0.4A GeV
+(a)
+ L=20 MeV
+ L=144 MeV
+(b)
+ Dp
+ 
+x/Dt
+ Dp
+ 
+t/Dt
+time (fm/c)
+ 
+Dp/Dt (GeV/fm)
+FIG. 6.
+(a) Time evolution of the averaged density in the
+center of reaction system, (b) time evolution of momen-
+tum changes in x direction ∆px/∆t and transverse direction
+∆pt/∆t .
+S(ρ) at pion characteristic density, i.e., at 1.5ρ0, is consis-
+tent with our previous analysis[21] and constraints from
+the analysis of SπRIT by using dcQMD[23] and isospin-
+dependent Boltzmann-Uehling-Uhlenbeck (IBUU) [22],
+analysis of FOPI data by using T¨uQMD[20], IBUU [15]
+and isospin-dependent Boltzmann-Langevian (IBL) [17]
+within statistical uncertainties,
+except for the con-
+straints obtained by Lanzhou quantum molecular dy-
+namics (LQMD) model[16]. Furthermore, if we extrap-
+olate our constraints to subsaturation density, it also
+consists with the one at its characteristic density from
+the neutrons to protons yield ratios in HIC (n/p)[54],
+isospin diffusion in HIC (isodiff)[55], mass calculated
+by the Skyrme[56] and density functional theory (DFT)
+theory[57], Isobaric analog state (IAS)[58], electric dipole
+polarization αD[59], at their sensitive density, which are
+decoded by Lynch and Betty in Ref.[60]. Further, the ex-
+trapolated region is also consistent with the results from
+theoretical calculation with chiral effective field theory
+(χEFT)[61].
+IV.
+SUMMARY AND OUTLOOK
+In summary, we have investigated the influence of dif-
+ferent momentum dependent interactions, symmetry en-
+ergy and NN → N∆ cross sections on nucleonic ob-
+servables and pion observables, such as vn
+1 , vch
+1 , vn
+2 ,
+vch
+2 , vn
+2 /vch
+2 , M(π) and π−/π+, with UrQMD model for
+Au+Au at the beam energy of 0.4A GeV. Our results
+confirm that the elliptic flow of neutrons and charged
+particles, i.e. vn
+2 and vch
+2 , are sensitive to the momentum
+dependence potential. The ASY-EOS flow data favors
+the calculations with a strong momentum dependent in-
+teraction, i.e., vHama
+md
+.
+However, the calculations with
+vHama
+md
+underestimate the pion multiplicity by about 30%
+relative to FOPI data if the σDefault
+NN→N∆ is adopted. Our
+calculations illustrate that the underestimation can be
+fixed by considering an accurate NN → N∆ cross sec-
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+20
+40
+60
+80
+30
+40
+50
+vn
+2/vH
+2  , Russotto 
+vn
+2/vH
+2  , Wang 
+vn
+2/vp
+2 , Cozma
+vn
+2/vp
+2 , Wang
+vn
+2-vp
+2 , Cozma
+vn
+2-vp
+2 , Wang
+vn
+2/vch
+2  , Russotto
+vn
+2-vH
+2  , Wang
+vn
+2/vH,p,ch
+2
+, Cozma
+ p-/p+, Xiao
+ p-/p+, Feng
+ p-/p+, Xie
+ p-/p+, Cozma
+ p-/p+, Liu
+ p-/p+, Yong
+ p-/p+, Estee
+ HIC (n/p)       
+ HIC (isodiff)
+ Mass (skyrme
+ IAS
+ Mass (DFT)   
+ aD
+ PREX-II        
+ Lynch
+ 
+ cEFT
+S(r) (MeV)
+r/r0
+ 
+ 
+ 
+ 
+ 
+FIG. 7.
+The constrains of the density dependence of symme-
+try energy at the collective flow characteristic density 1.2ρ0
+and the pion characteristic density 1.5ρ0.
+tions σHub
+NN→N∆ in UrQMD model.
+Further, the constraints on the symmetry energy at
+flow and pion characteristic densities are investigated
+with the updated UrQMD model.
+The characteristic
+density probed by flow is around 1.2ρ0, which is smaller
+than the pion characteristic density 1.5ρ0[21]. By simul-
+taneously describing the data of vn
+2 /vch
+2
+and π−/π+ with
+UrQMD calculations, the favored effective interaction pa-
+rameter sets are obtained and we got the S(1.2ρ0) =
+34±4 MeV and S(1.5ρ0) = 36±8 MeV. These results are
+consistent with previous analysis by using pion and flow
+observable with different transport models, and the con-
+sistency suggests that the reliable description of the con-
+straints on symmetry energy should be presented at the
+characteristic density of isospin sensitive observables. By
+using more than one isospin sensitive observables which
+have different characteristic densities, the reliable of the
+extrapolation of symmetry energy at normal density can
+be enhanced. The extrapolated values of L in this work
+are in 5−70 MeV within 2σ uncertainty for S0 = 30−34
+MeV, which is below the analysis of PREX-II results with
+a specific class of relativistic energy density functional,
+but is consistent with the constrains from charged ra-
+dius of 54Ni, from the combining astrophysical data with
+PREX-II and chiral effective field theory, and the SπRIT
+pion data for Sn+Sn at 0.27A GeV.
+ACKNOWLEDGEMENTS
+The authors thank the discussions on the transport
+model and symmetry energy constraints at TMEP weekly
+meeting. This work was supported by the National Natu-
+ral Science Foundation of China Nos.11875323, 12275359,
+
+8
+12205377, 11875125, U2032145, 11790320, 11790323,
+11790325, and 11961141003, the National Key R&D Pro-
+gram of China under Grant No.
+2018 YFA0404404,
+the Continuous Basic Scientific Research Project (No.
+WDJC-2019-13), and the funding of China Institute of
+Atomic Energy (No. YZ222407001301), and the Lead-
+ing Innovation Project of the CNNC under Grant No.
+LC192209000701, No.
+LC202309000201.
+We acknowl-
+edge support by the computing server C3S2 in Huzhou
+University.
+[1] B. A. Li, L. W. Chen, and C. M. Ko, Phys. Rept. 464,
+113 (2008).
+[2] C. J. Horowitz, E. F. Brown, Y. Kim, W. G. Lynch,
+R. Michaels, A. Ono, J. Piekarewicz, M. B. Tsang, and
+H. H. Wolter, J. Phys. G 41, 093001 (2014).
+[3] J. M. Lattimer and M. Prakash, Science 304, 536 (2004),
+https://www.science.org/doi/pdf/10.1126/science.1090720,
+URL
+https://www.science.org/doi/abs/10.1126/
+science.1090720.
+[4] A. W. Steiner, J. M. Lattimer, and E. F. Brown, The
+Astrophysical Journal 722, 33 (2010), URL https://dx.
+doi.org/10.1088/0004-637X/722/1/33.
+[5] P. Danielewicz, R. Lacey, and W. G. Lynch, Science 298,
+1592 (2002).
+[6] J. Carlson, M. P. Carpenter, R. Casten, C. Elster,
+P. Fallon,
+A. Gade,
+C. Gross,
+G. Hagen,
+A. C.
+Hayes, D. W. Higinbotham, et al., Progress in Par-
+ticle and Nuclear Physics 94, 68 (2017), ISSN 0146-
+6410, URL https://www.sciencedirect.com/science/
+article/pii/S0146641016300722.
+[7] A. Bracco, Europhysics News 48, 21 (2017).
+[8] P. Russotto, P. Wu, M. Zoric, M. Chartier, Y. Leifels,
+R. Lemmon, Q. Li, J. �Lukasik, A. Pagano, P. Paw�lowski,
+et al., Physics Letters B 697, 471 (2011), ISSN 0370-
+2693, URL https://www.sciencedirect.com/science/
+article/pii/S037026931100178X.
+[9] M. D. Cozma, Y. Leifels, W. Trautmann, Q. Li, and
+P. Russotto, Phys. Rev. C 88, 044912 (2013).
+[10] Y. J. Wang, C. C. Guo, Q. F. Li, H. F. Zhang, Y. Leifels,
+and W. Trautmann, Phys. Rev. C 89, 044603 (2014).
+[11] P. Russotto, S. Gannon, S. Kupny, P. Lasko, L. Acosta,
+M. Adamczyk,
+A. Al-Ajlan,
+M. Al-Garawi,
+S. Al-
+Homaidhi,
+F.
+Amorini,
+et
+al.,
+Phys.
+Rev.
+C 94,
+034608 (2016), URL https://link.aps.org/doi/10.
+1103/PhysRevC.94.034608.
+[12] M. D. Cozma, Eur. Phys. J. A 54, 40 (2018).
+[13] B. A. Li, Phys. Rev. Lett. 88, 192701 (2002).
+[14] B. A. Li, Nucl. Phys. A 708, 365 (2002).
+[15] Z. G. Xiao, B. A. Li, L. W. Chen, G. C. Yong, and
+M. Zhang, Phys. Rev. Lett. 102, 062502 (2009).
+[16] Z. Q. Feng and G. M. Jin, Phys. Lett. B 683, 140 (2010).
+[17] W. J. Xie, J. Su, L. Zhu, and F. S. Zhang, Phys. Lett. B
+718, 1510 (2013).
+[18] J. Hong and P. Danielewicz, Phys. Rev. C 90, 024605
+(2014).
+[19] T. Song and C. M. Ko, Phys. Rev. C 91, 014901 (2015).
+[20] M. D. Cozma, Phys. Lett. B 753, 166 (2016).
+[21] Y. Y. Liu, Y. J. Wang, Y. Cui, C. J. Xia, Z. X. Li, Y. J.
+Chen, Q. F. Li, and Y. X. Zhang, Phys. Rev. C 103,
+014616 (2021).
+[22] G. C. Yong, Phys. Rev. C 104, 014613 (2021).
+[23] J. Estee, W. G. Lynch, C. Y. Tsang, J. Barney, G. Jhang,
+M. B. Tsang, R. Wang, M. Kaneko, J. W. Lee, T. Isobe,
+et al. (SπRIT Collaboration), Phys. Rev. Lett. 126,
+162701 (2021), URL https://link.aps.org/doi/10.
+1103/PhysRevLett.126.162701.
+[24] Y. J. Wang and Q. F. Li, Front. Phys. (Beijing) 15, 44302
+(2020).
+[25] B. A. Li and X. Han, Phys. Lett. B 727, 276 (2013).
+[26] M. Oertel, M. Hempel, T. Kl¨ahn, and S. Typel, Rev.
+Mod. Phys. 89, 015007 (2017).
+[27] Y. X. Zhang, M. Liu, C. J. Xia, Z. X. Li, and S. K.
+Biswal, Phys. Rev. C 101, 034303 (2020).
+[28] G.
+Jhang,
+J.
+Estee,
+J.
+Barney,
+G.
+Cerizza,
+M. Kaneko, J. Lee, W. Lynch, T. Isobe, M. Kurata-
+Nishimura,
+T.
+Murakami,
+et
+al.,
+Physics
+Letters
+B
+813,
+136016
+(2021),
+ISSN
+0370-2693,
+URL
+https://www.sciencedirect.com/science/article/
+pii/S0370269320308194.
+[29] Y.-X. Zhang, Y.-J. Wang, M. Colonna, P. Danielewicz,
+A.
+Ono,
+M.
+B.
+Tsang,
+H.
+Wolter,
+J.
+Xu,
+L.-
+W.
+Chen,
+D.
+Cozma,
+et
+al.,
+Phys.
+Rev.
+C
+97,
+034625 (2018), URL https://link.aps.org/doi/10.
+1103/PhysRevC.97.034625.
+[30] A. Ono, J. Xu, M. Colonna, P. Danielewicz, C. M.
+Ko,
+M.
+B.
+Tsang,
+Y.-J.
+Wang,
+H.
+Wolter,
+Y.-
+X. Zhang, L.-W. Chen, et al., Phys. Rev. C 100,
+044617 (2019), URL https://link.aps.org/doi/10.
+1103/PhysRevC.100.044617.
+[31] M. Colonna, Y.-X. Zhang, Y.-J. Wang, D. Cozma,
+P. Danielewicz, C. M. Ko, A. Ono, M. B. Tsang,
+R. Wang,
+H. Wolter,
+et al.,
+Phys. Rev. C 104,
+024603 (2021), URL https://link.aps.org/doi/10.
+1103/PhysRevC.104.024603.
+[32] S. Bass, M. Belkacem, M. Bleicher, M. Brandstet-
+ter, L. Bravina, C. Ernst, L. Gerland, M. Hofmann,
+S. Hofmann, J. Konopka, et al., Progress in Parti-
+cle and Nuclear Physics 41, 255 (1998), ISSN 0146-
+6410, URL https://www.sciencedirect.com/science/
+article/pii/S0146641098000581.
+[33] M. Bleicher, E. Zabrodin, C. Spieles, S. A. Bass, C. Ernst,
+S. Soff, L. Bravina, M. Belkacem, H. Weber, H. St¨ocker,
+et al., Journal of Physics G: Nuclear and Particle Physics
+25, 1859 (1999), URL https://dx.doi.org/10.1088/
+0954-3899/25/9/308.
+[34] L. G. Arnold, B. C. Clark, E. D. Cooper, H. S. Sherif,
+D. A. Hutcheon, P. Kitching, J. M. Cameron, R. P. Liljes-
+trand, R. N. MacDonald, W. J. McDonald, et al., Phys.
+Rev. C 25, 936 (1982), URL https://link.aps.org/
+doi/10.1103/PhysRevC.25.936.
+[35] C. Hartnack and J. Aichelin, Phys. Rev. C 49, 2801
+(1994).
+[36] S. Hama, B. C. Clark, E. D. Cooper, H. S. Sherif, and
+R. L. Mercer, Phys. Rev. C 41, 2737 (1990).
+[37] M. Isse, A. Ohnishi, N. Otuka, P. K. Sahu, and Y. Nara,
+Phys. Rev. C 72, 064908 (2005).
+[38] L.-W. Chen, C. M. Ko, B.-A. Li, C. Xu, and J. Xu, Eur.
+Phys. J. A 50, 29 (2014).
+
+9
+[39] Y. Nara, T. Maruyama, and H. Stoecker, Phys. Rev. C
+102, 024913 (2020).
+[40] M. D. Cozma and M. B. Tsang, Eur. Phys. J. A 57, 309
+(2021).
+[41] F. Zhang and G.-C. Yong, Phys. Rev. C 106, 054603
+(2022).
+[42] Q. Li and M. Bleicher, J. Phys. G 36, 015111 (2009).
+[43] A. Baldini, V. Flaminio, W. G. Moorhead, and D. R. O.
+Morrison, Total Cross-Sections for Reactions of High En-
+ergy Particles (Including Elastic, Topological, Inclusive
+and Exclusive Reactions), vol. 12a of Landolt-Boernstein
+- Group I Elementary Particles, Nuclei and Atoms
+(Springer, 1988), ISBN 978-3-540-18386-0, 978-3-540-
+47940-6.
+[44] Y. Cui, Y. X. Zhang, and Z. X. Li, Chin. Phys. C 44,
+024106 (2020).
+[45] Y. Zhang, N. Wang, Q.-F. Li, L. Ou, J.-L. Tian, M. Liu,
+K. Zhao, X.-Z. Wu, and Z.-X. Li, Front. Phys. (Beijing)
+15, 54301 (2020).
+[46] P. Danielewicz, Nucl. Phys. A 673, 375 (2000).
+[47] W. Reisdorf et al. (FOPI), Nucl. Phys. A 848, 366 (2010).
+[48] J. D. Frankland, D. Gruyer, E. Bonnet, B. Borderie,
+R. Bougault, A. Chbihi, J. E. Ducret, D. Durand, Q. Fa-
+ble, M. Henri, et al. (INDRA Collaboration), Phys. Rev.
+C 104, 034609 (2021), URL https://link.aps.org/
+doi/10.1103/PhysRevC.104.034609.
+[49] L. Li, Y. Zhang, Z. Li, N. Wang, Y. Cui, and J. Winkel-
+bauer, Phys. Rev. C 97, 044606 (2018).
+[50] L. Li, X. Chen, Y. Cui, Z. Li, and Y. Zhang (2022),
+2201.12586.
+[51] Y. Wang, C. Guo, Q. Li, A. Le F`evre, Y. Leifels, and
+W. Trautmann, Phys. Lett. B 778, 207 (2018).
+[52] A. Le F`evre, Y. Leifels, W. Reisdorf, J. Aichelin, and
+C. Hartnack, Nucl. Phys. A 945, 112 (2016).
+[53] B. Gao, Y. Wang, Z. Gao, and Q. Li (2022), 2210.08213.
+[54] P. Morfouace, C. Tsang, Y. Zhang, W. Lynch, M. Tsang,
+D. Coupland, M. Youngs, Z. Chajecki, M. Famiano,
+T. Ghosh, et al., Physics Letters B 799, 135045 (2019),
+ISSN 0370-2693,
+URL https://www.sciencedirect.
+com/science/article/pii/S0370269319307671.
+[55] M. B. Tsang, Y. Zhang, P. Danielewicz, M. Famiano,
+Z. Li, W. G. Lynch, and A. W. Steiner, Phys. Rev. Lett.
+102, 122701 (2009).
+[56] B. A. Brown, Phys. Rev. Lett. 111, 232502 (2013).
+[57] M. Kortelainen, J. McDonnell, W. Nazarewicz, P. G.
+Reinhard, J. Sarich, N. Schunck, M. V. Stoitsov, and
+S. M. Wild, Phys. Rev. C 85, 024304 (2012).
+[58] P. Danielewicz, P. Singh, and J. Lee, Nucl. Phys. A 958,
+147 (2017).
+[59] Z. Zhang and L.-W. Chen, Phys. Rev. C 92, 031301
+(2015).
+[60] W. G. Lynch and M. B. Tsang, Phys. Lett. B 830, 137098
+(2022).
+[61] C. Drischler, R. J. Furnstahl, J. A. Melendez, and D. R.
+Phillips, Phys. Rev. Lett. 125, 202702 (2020).
+

3NE1T4oBgHgl3EQfAQKK/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7f310e3b9e94dc71310429ec565b92929bc863be4081f02a958220013dd74b6d
+size 2752557

3NE1T4oBgHgl3EQfAQKK/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:8c83cdac37896734d32c754ae3976103e4813870f6af319638381abbc4d2ffe2
+size 109158

3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:aceb683bbfa164e5bd7a75bd54924ae32eb2f247f5638a885cf310c8f88d7e69
+size 328107

3tAyT4oBgHgl3EQf1_ko/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f7b6c9d26dc5f4e7a0af159f71f0310daec2377292d9326ff63e69a2f41e4a81
+size 2818093

3tAyT4oBgHgl3EQf1_ko/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5d0ab39aa386f1e08fd0057729268d036517e2f66aa9fda22fe77856b8424d98
+size 98284

49FKT4oBgHgl3EQf9y5B/content/2301.11955v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9b9dac2e812769bddd448577c75d26ca9cea3e7862ec33dbb0f41ff1891f383f
+size 4092998

49FKT4oBgHgl3EQf9y5B/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b96169ffad74c9954590475f5d0a22d4048ee70cda76826ad7c8320854696ba2
+size 3735597

49FKT4oBgHgl3EQf9y5B/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f18719168c3918b7d210abd3e117acd30fdf5726b13cf2346564fad7027de6ce
+size 141607

5tE0T4oBgHgl3EQfewA6/content/tmp_files/2301.02394v1.pdf.txt

@@ -0,0 +1,352 @@
+arXiv:2301.02394v1  [cond-mat.stat-mech]  6 Jan 2023
+Increase in Rod Diffusivity Emerges even in Markovian Nature
+Fumiaki Nakai,1, ∗ Martin Kr¨oger,2, † Takato Ishida,1
+Takashi Uneyama,1 Yuya Doi,1 and Yuichi Masubuchi1
+1Department of Materials Physics, Graduate School of Engineering,
+Nagoya University, Furo-cho, Chikusa, Nagoya 464-8603, Japan
+2Polymer Physics, Department of Materials,
+ETH Zurich, CH-8093 Zurich, Switzerland
+1
+
+Abstract
+Rod-shaped particles embedded in certain matrices have been reported to exhibit an increase in their
+center of mass diffusivity upon increasing the matrix density. This increase has been considered to be
+caused by a kinetic constraint in analogy with tube models. Here, we investigate a mobile rod-like particle
+in a three-dimensional sea of immobile point obstacles using a kinetic Monte Carlo scheme equipped with a
+Markovian process, that generates gas-like collision times and positions stochastically, so that such kinetic
+constraints do essentially not exist. We find that even in such a system, the unusual increase in diffusivity
+emerges. This result implies that the kinetic constraint is not a necessary condition for the increase in the
+diffusivity. More generally, this work will provide fresh insight into the kinetics of non-spherical particles.
+The translational diffusion coefficient Dc of a particle is generally known to decrease with in-
+creasing matrix density or increasing amount of obstacles. It is understood as a consequence of
+the ballistic particle motion being disturbed during collisions with the surrounding matrix. How-
+ever, if the particle is rod-shaped, a counter-intuitive motion can occur; the Dc of a rod may
+increase as the matrix concentration increases, provided the concentration is sufficiently high.
+Frenkel and Maguire [1, 2] first observed such behavior for fluids consisting of infinitely thin
+hard rods, whose static properties are exactly the same as those of an ideal gas. This finding was
+later confirmed with higher accuracy [3, 4]. Their systems do not have any hidden particles or
+thermostats; the constituent particle moves ballistically between elastic collisions. Following the
+previous studies[1, 2], an increase in Dc has been observed in various systems: (i) an infinitely
+thin rod in a two-dimensional (2D) sea of fixed point obstacles [5], (ii) a thick rod in a 2D matrix
+of circular obstacles [6], and (iii) an active matter fluid consisting of a rod swimming in direction
+of its major axis [7]. In these systems, the increase in Dc is not triggered by a phase transition.
+Still, some rod systems exhibit an increase in Dc accompanied by the isotropic-nematic transition
+[8]. Such multi-particle effects remain beyond the scope of the present work.
+Various loosely defined concepts have been considered previously to explain the increase in Dc:
+so-called dynamic correlation, steric hindrance, geometrical constraints, confinement, or tube [2,
+5]. We refer to these concepts as the ”kinetic constraint” in what follows. In this work, we define
+the kinetic constraint as the constraint that prevents the rod from crossing an obstacle until the
+rod moves about the rod length. Using the kinetic constraint, the increase in Dc can be explained.
+∗ [email protected]
+† [email protected]
+2
+
+Namely, the rotational motion of the rod is kinetically constrained via the surrounding matrix in
+the concentrated matrix regime. Even in such a regime, the ballistic motion along the major axis
+of an infinitely thin rod is not hindered, while the relevance of collisions in direction of the major
+axis increases with increasing width of the rod or size of the obstacles. Consequently, the ballistic
+motion with the major axis may persist for a relatively long time. This duration may increase with
+matrix density or the degree of confinement and ultimately leads to an increase in Dc. In the so-
+called active rod fluid [7], a similar behavior is caused by swimming along the axial direction of
+the rod instead of ballistic motion. In light of these studies one question may arise; Is the kinetic
+constraint a necessary condition for the emergence of the increase in diffusivity?
+On our way towards an answer, we have been guided by our naive belief that such an increase
+can be caused by the reduction of the rotational diffusivity alone, without the hindrance of the
+axially directed motion. To test our hypothesis rigorously, we consider a simple model system
+where the rotational diffusivity reduces with increasing matrix density, whereas the ballistic mo-
+tion along the major axis of the rod-like particle remains largely undisturbed. One possible such
+system is a single mobile rod embedded in a 3D arrangement of spatially fixed point obstacles. It
+can be regarded as the extension of the Lorentz gas systems [9–11]; the single spherical particle in
+fixed obstacles.
+In this work, we report that the upturn of Dc emerges even in the presence of a Markovian
+process where the kinetic constraint does essentially not exist. We investigate the trajectories of a
+sphero-cylinder in a 3D matrix of stochastically homogeneously distributed point obstacles using a
+Markovian kinetic Monte Carlo (KMC) scheme [12, 13]. The Dc of this rod-like particle increases
+in an intermediate matrix density regime if the rod is sufficiently long. In our system, Dc reaches a
+peak value and subsequently decreases with increasing obstacle density due to the thickness of the
+rod. On the basis of the Markovian nature, we give scaling relations between Dc and the obstacle
+density for dilute, intermediate, and concentrated density regimes. This work will generate fresh
+insight into the kinetics of the non-spherical shaped particles [14].
+Model and methods. — The model consists of a rod-like sphero-cylinder (also termed capsule
+or stadium of revolution) with radius σ, mass M, and length L of its major axis. The effective ”rod”
+length is Le = L + 2σ due to the half-spherical end-caps, and the inertia tensor I is determined
+by assuming that the mass is homogeneously distributed over the volume of the rod [15]. The
+point obstacles are statistically homogeneously distributed in the unbounded 3D space at number
+density ρ. The interaction between the rod and obstacles is modeled by a hard-core potential; the
+3
+
+time
+ballistic motion
+collision
+FIG. 1. Schematic representation of the KMC method. r(t), e(t), v(t), and ω(t) are the position, direction
+unit vector, velocity, and angular velocity, respectively, of the rod at time t. z and n characterize the
+coordinate of the collision point; z ∈ [−L/2, L/2] is the axial coordinate and n is the surface normal at the
+collision point. τ is the collision time interval between successive collisions. τ, z, and n are stochastically
+sampled based on the collision statistics corresponding to Eq. (1). From the sampled variables, r, e, v and
+ω at time t + τ are obtained.
+obstacles do not penetrate the rod, and they do not move during a collision. The rod ballistically
+moves except when it elastically collides with an obstacle. The center of mass velocity v and
+angular velocity ω are changed during a collision, conserving the rod particle’s translational and
+rotational kinetic energy. The total energy of this system is 5kBT/2, where kB and T are the
+Boltzmann constant and temperature. Due to the assumed elastic collisions, the total energy is a
+conserved quantity and does not change during the course of time. We choose M, σ, and kBT
+to define dimensionless units. All physical quantities are therefore presented without physical
+dimensions, as they follow by dimensional arguments from the three units. Within these settings,
+the remaining parameters are the effective rod length Le and the number density of the obstacles,
+ρ. Here, we avoid the density ρLe > 1, which physically corresponds to the trapping transition
+regime.
+To calculate the dynamics of a rod subject to a Markovian collision process, we extend the
+kinetic Monte Carlo (KMC) simulation method [12, 13]. It requires two inputs; (i) statistics of
+collisions and (ii) the change of dynamical variables by a collision. For (i), we here extend the
+calculations for a sphere [14, 16] to the collision statistics of a sphero-cylinder and obtain the
+collision frequency with the coordinates of collision for a given v(t), ω(t), and the direction vector
+of the rod e(t). In the following, we denote Γ(t) as the 8-dimensional time-dependent phase space
+variable (v(t), ω(t), e(t)) characterizing the state of the rod (Fig. 1). The explicit expression for
+4
+
+-20
+0
+20
+-80
+-60
+-40
+-20
+0
+20
+40
+60
+80
+FIG. 2. Trajectories of the rod’s (Le = 2502) center of mass for various scaled obstacle densities ρL2
+e from
+the KMC simulation. 3D motions are projected onto the XY -plane and scaled by Le.
+the collision frequency for a given Γ(t), F(Γ(t)), arises from a surface integral of the collision
+frequency density
+f(z, n; Γ(t)) = ρve(z; Γ(t)) · nΘ[ve(z; Γ(t)) · n]
+× {δ(e(t) · n) + δ (z − L/2) Θ[e(t) · n]
++δ (z + L/2) Θ[−e(t) · n]}
+(1)
+where z is the axial coordinate along the rod direction and n an unit vector normal to the rod’s
+surface (Fig. 1). These two variables characterize the coordinate ze + n of the collision point
+between the rod and an obstacle, while ve(z; Γ(t)) = v(t) + zω(t) × e(t) is the rod’s velocity at
+the collision point. In Eq. (1), the first, second, and third terms in the curly bracket are relevant to
+the collision on the side (∥) and two opposing (±) edges of the rod. Based on f(z, n; Γ(t)) and
+F(Γ), the coordinate of the collision point and the collision time interval τ between successive
+collisions are sampled using stochastic techniques [17]. (ii) From these sampled variables, r,
+e, v, and ω are updated based on the rules of classical mechanics for a rigid body. Repeating
+these samplings and updates, we calculate the dynamics of the mobile rod. The details of the
+derivation of the collision statistics, sampling method, and the update scheme are described in the
+supplementary material.
+Results. — Qualitatively different behaviors occur during a change of ρ at fixed Le = 2502, as
+visually captured by representative trajectories in Fig. 2. The observed time duration is 2.0 × 106.
+For ρL2
+e = 1 and 10, the mobile rod seems to move randomly. At higher number densities ρL2
+e =
+100, the straight motion persists over longer distances compared with those for lower densities
+ρL2
+e = 1 and 10. For ρL2
+e = 1000, we observe straight and bouncing motions.
+To quantify these motions (Fig. 2), we calculate Dc of the mobile rod from its center-of-mass
+5
+
+101
+102
+103
+104
+105
+106
+10-11 10-10 10-9
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+101
+100
+101
+102
+103
+FIG. 3.
+Translational diffusion coefficient of the mobile rod (various rod lengths Le) from the KMC
+simulations. Data are shown as (a) Dc versus ρ and (b) in scaled form DcL−1
+e
+versus ρL2
+e. Error bars and
+asymptotic exponents are also displayed.
+mean square displacement (MSD) in the linear time domain. Dc versus the obstacle number
+density ρ are displayed in Fig. 3(a) for various mobile rod lengths Le (error bars arise from the
+linear fitting). In this figure, Dc shows non-monotonic behaviors with increasing ρ for the highly
+elongated rods Le ≳ 66; Dc at large Le exhibits both a local minimum and maximum. When
+the same data are represented in scaled forms, DcL−1
+e
+and ρL2
+e, as shown in Fig. 3(b), the curves
+collapse except for the larger density regime. From Figs. 3(a,b), the asymptotic forms are observed
+for small, intermediate, and large density regimes as Dc ∝ (ρL3
+e)−1, Dc ∝ ρL3
+e, and Dc ∝
+ρ−1, respectively. We emphasize that the non-monotonic ρ dependency for Dc arises even under
+the Markovian process. In contrast to Dc, the rotational diffusion coefficient Dr in the current
+system exhibits monotonic behavior with increasing obstacle density, Dr ∼ (ρL3
+e)−1 as shown in
+supplementary Fig. S4.
+The scaling relations between Dc and ρ can be simply explained based on the Markovian na-
+ture. Here, the Dc is also calculated from the integration of the velocity auto-correlation function
+over time lag, instead of the mean square displacement. Thus, the diffusion coefficient would be
+roughly approximated as the relaxation time of the center of mass velocity in the dimensionless
+units. The collision frequency can be decomposed into two contributions: collision frequencies
+from the side F∥ and edges F±. These contributions scale as F∥ ∼ ρLe and F± ∼ ρ. These
+estimates are confirmed by the rigorous calculations for the collision frequencies as shown in sup-
+plementary Eqs. S7 and S11. The average angular velocity scales as ¯ω ∼ L−1
+e . In the dilute regime
+ρL2
+e ≲ 1, the relation ¯ω > F∥ is satisfied. In this low density regime, the rod mainly rotates and
+occasionally collides with an obstacle on its side. By a few collisions, the motion of the rod largely
+6
+
+changes since the rod experiences the impulsive forces from various directions. Then, Dc scales
+linearly as the collision time interval as Dc ∼ F −1
+∥
+∼ ρ−1L−1
+e . This description is consistent
+with the observed random motions for the lower density regimes ρL2
+e = 1 and 10 in Fig. 2. In
+the higher density regime ρL2
+e ≳ 1, where the relation ¯ω > F∥ is fulfilled, the rotational motion
+of the rod is diffusive, and thus the direction of the rod slowly changes. In this density regime,
+the velocity with the orthogonal direction rapidly relaxes, whereas that with the axial direction is
+not largely disturbed. In such a case, there are possible relaxation mechanisms for the velocity
+with axial direction: the change of rod direction or the collision on the edge. Here, the change of
+rod direction between collisions is approximately ∆θ ∼ ¯ω/F∥, and the rotational relaxation time
+scales as τrot ∼ ∆θ−2/F∥ ∼ ρL3
+e. This estimate also predicts the rotational diffusion coefficient
+Dr = (2τr) ∼ (ρL3
+e)−1, in full agreement with our measurements, c.f., supplementary Fig. S4.
+The collision time interval on the edge is about F −1
+± . In the intermediate density regime where
+Dc increases, the rotational relaxation time is smaller than the collision time interval on the edge.
+Thus, the velocity relaxes by the rotation of the direction, and consequently the diffusion coeffi-
+cient is approximately Dc ∼ ρL3
+e. Within the high density regime where Dc decreases again, the
+collision on the edge is the main mechanism causing velocity relaxation with the axial direction,
+and we obtain Dc ∼ ρ−1. These mechanisms explained above seem to be consistent with the
+persistence of the straight motion with ρL2
+e = 100 and the straight and bouncing motions with
+ρL2
+e = 1000 displayed in Fig. 2, and the estimated exponents also successfully agree with the
+simulation results in Fig. 3.
+One may suspect that the increase in Dc is an artifact since we assume a Markovian process
+even in the high density regime. However, we next show that this assumption is indeed a good
+approximation to calculate Dc for a rod embedded in a 3D sea of point obstacles. To this end, we
+calculate the dynamics of a rod using conventional molecular dynamics (MD) simulations [18].
+Here, instead of a hard-core potential, the repulsive Weeks-Chandler-Andersen potential [19] is
+employed for the elastic interaction between rod and point obstacles. The details of the simulation
+method are described in the supplementary material. Fig. 4 displays Dc (symbols) for various rod
+lengths Le obtained via MD. Error bars are again calculated from linear fitting for the MSDs. Due
+to the computational cost, data for large rod lengths Le = 16002 and 100002 could not be sampled.
+For comparison, the KMC data from Fig. 3 are shown in Fig. 4 (solid curves). The MD results
+quantitatively agree with those obtained via KMC. This indicates that multi-body correlations are
+negligible in the estimation of Dc within the explored wide regime of obstacle densities.
+7
+
+10-1
+100
+100
+101
+102
+103
+FIG. 4. Translational diffusion coefficient Dc versus obstacle density with the error bars from MD simula-
+tions (Symbols). Data for three rod lengths Le are displayed. For comparison, the KMC simulation results
+(Fig. 3) are shown by solid curves.
+Discussion.— This work shows that Dc increases even in a Markovian process and that the
+observed exponents are easily rationalized. This result does not imply that the exponents in prior
+studied systems can be simply understood. Frenkel and Maguire [1, 2] investigated Dc of a con-
+stituent particle in a system of infinitely thin hard rods, where Dc was found to be proportional
+to the root of the rod density. For a 2D rod in the presence of point obstacles studied by H¨ofling,
+Frey, and Franosch [5], the power exponent of Dc versus obstacle density is 0.8 in the concentrated
+regime. Mandal et al [7] investigated the dynamics of a rod-shaped active swimmer (along the ax-
+ial direction) and showed that Dc depends on the square of the density of the constituent. In these
+prior systems, the kinetic constraints are not negligible, and they should be taken into account to
+explain the observed exponents.
+Some works investigated similar systems to ours. Tucker and Hernandez [6, 20] numerically
+studied the dynamics of a 5 ˚A long mobile rod in the presence of spatially fixed spherical obstacles
+with radius 0.5 ˚A for rod thickness values 0, 0.1, and 0.5 ˚A. They argued that the increase in Dc
+does not occur in their 3D system, while it can occur in the corresponding 2D setup. One may
+8
+
+think that these findings are inconsistent with our results. However, if one identifies the lengths in
+their system with ours, the effective aspect ratio of the rod becomes about 10 since the interaction
+distance between the rod and the obstacle is the rod thickness plus obstacle size. For the rod with
+such an aspect ratio 10, an increase in Dc does not occur. Conversely, in Tucker and Hernandez’s
+system, the increase in Dc will occur for a much smaller obstacle radius or much larger rod length.
+Otto, Aspelmeier, and Zippelius [21] theoretically analyzed the dynamics of a constituent particle
+of infinitely thin rods under the assumption of the Markovian process. They argued that an increase
+in Dc should not occur under such circumstances. This result obviously contradicts our findings.
+However, they did not consider the long-time persistence of the ballistic motion with the axial
+direction. Thus, the increase in the Dc could not be captured.
+It should be emphasized that a rise of Dc can occur for a ballistic system [1–5] or some active
+matter systems [7] due to the persistence of the motion with the axial direction. One may think
+that an increase in Dc can occur for passive rod-shaped particles in some solvents or some porous
+media. However, it can not exhibit the increase in diffusivity by the same mechanism as our system
+since the persistence of the motion with axial direction rapidly relaxes by the Brownian motion.
+Recently, the increase in diffusivity with increasing aspect ratio is observed for rod in a gel [22],
+although the mechanism would be different to our system.
+The current system consists of a rod colliding with immobile, or infinitely heavy point obsta-
+cles. Let us consider the situation where obstacles move in an equilibrium state. As long as the
+obstacle mass is sufficiently larger than M, the obstacle motion is slow because of the Maxwell-
+Boltzmann velocity distribution. In this case, the situation would not be largely different from
+the current system since the moving particles can be approximated as the fixed obstacles for the
+rod particle, and the increase in the diffusivity will emerge in this case. In contrast to this, if the
+obstacle mass is comparable to M, the situation can be different from the current system since
+the translational and rotational relaxation times vary largely with the obstacle mass. Even in this
+case, the increase in the diffusivity can emerge since it simply originates from the reduction of the
+rotational motion and the persistence of the axial motion. The analyses for the effects of obstacle
+mass on the increase in diffusivity will be future interesting work.
+In conclusion, this study demonstrated that a Dc upturn can emerge even in Markovian na-
+ture, where the kinetic constraint does not exist. As a simple model system, we investigated the
+single mobile rod-shaped particle in immobile fixed obstacles in three-dimensions using highly
+efficient Markovian kinetic Monte Carlo simulations. The translational diffusion coefficient of the
+9
+
+rod decreases, increases, and decreases again as the obstacle density increases. These non-trivial
+behaviors could be explained based on the Markovian process. This work sheds light on the ki-
+netics of non-spherical particles where the elementary dynamic processes are ballistic motion and
+collisions.
+FN and MK were supported by the “Young Researchers Exchange Programme between Japan
+and Switzerland” under the “Japanese-Swiss Science and Technology Programme”. FN was also
+supported by a Grant-in-Aid (KAKENHI) for JSPS Fellows (Grant No. JP21J21725 from the
+Ministry of Education, Culture, Sports, Science and Technology, MEXT).
+[1] D. Frenkel and J. F. Maguire, Phys. Rev. Lett. 47, 1025 (1981).
+[2] D. Frenkel and J. Maguire, Mol. Phys. 49, 503 (1983).
+[3] J. Magda, H. Davis, and M. Tirrell, J. Chem. Phys. 85, 6674 (1986).
+[4] J. J. Magda, M. Tirrell, and H. T. Davis, J. Chem. Phys. 88, 1207 (1988).
+[5] F. H¨ofling, E. Frey, and T. Franosch, Phys. Rev. Lett. 101, 120605 (2008).
+[6] A. K. Tucker and R. Hernandez, J. Phys. Chem. A 114, 9628 (2010).
+[7] S. Mandal, C. Kurzthaler, T. Franosch, and H. L¨owen, Phys. Rev. Lett. 125, 138002 (2020).
+[8] M. P. Allen, Phys. Rev. Lett. 65, 2881 (1990).
+[9] H. A. Lorentz, Proc. K. Ned. Akad. Wet. 7, 438 (1905).
+[10] B. Alder and W. Alley, Physica A 121, 523 (1983).
+[11] F. H¨ofling and T. Franosch, Phys. Rev. Lett. 98, 140601 (2007).
+[12] D. T. Gillespie, J. Comput. Phys. 22, 403 (1976).
+[13] A. B. Bortz, M. H. Kalos, and J. L. Lebowitz, J. Comput. Phys. 17, 10 (1975).
+[14] J. R. Dorfman, H. van Beijeren, and T. R. Kirkpatrick, Contemporary Kinetic Theory of Matter (Cam-
+bridge University Press, Cambridge, U.K., 2021).
+[15] L. Pournin, M. Weber, M. Tsukahara, J.-A. Ferrez, M. Ramaioli, and T. M. Liebling, Granul. Matter
+7, 119 (2005).
+[16] G. F. Mazenko, Nonequilibrium statistical mechanics (John Wiley & Sons, Hoboken, NJ, United
+States, 2008).
+[17] L. Devroye, Non-Uniform Random Variate Generation (Springer, New York, 1986).
+[18] M. P. Allen and D. J. Tildesley, Computer simulation of liquids (Oxford University Press, Oxford,
+10
+
+U.K., 1989).
+[19] J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem. Phys 54, 5237 (1971).
+[20] A. K. Tucker and R. Hernandez, J. Phys. Chem. B 115, 4412 (2011).
+[21] M. Otto, T. Aspelmeier, and A. Zippelius, J. Chem. Phys. 124, 154907 (2006).
+[22] K. A. Rose, N. Gogotsi, J. H. Galarraga, J. A. Burdick, C. B. Murray, D. Lee, and R. J. Composto,
+Macromolecules (2022).
+11
+

5tE0T4oBgHgl3EQfewA6/content/tmp_files/load_file.txt

@@ -0,0 +1,388 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf,len=387
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='02394v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='stat-mech]  6 Jan 2023  Increase in Rod Diffusivity Emerges even in Markovian Nature  Fumiaki Nakai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' ∗ Martin Kr¨oger,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' † Takato Ishida,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='1  Takashi Uneyama,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='1 Yuya Doi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='1 and Yuichi Masubuchi1  1Department of Materials Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Graduate School of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  Nagoya University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Furo-cho,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Chikusa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Nagoya 464-8603,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Japan  2Polymer Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Department of Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  ETH Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' CH-8093 Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Switzerland  1  Abstract  Rod-shaped particles embedded in certain matrices have been reported to exhibit an increase in their  center of mass diffusivity upon increasing the matrix density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' This increase has been considered to be  caused by a kinetic constraint in analogy with tube models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Here, we investigate a mobile rod-like particle  in a three-dimensional sea of immobile point obstacles using a kinetic Monte Carlo scheme equipped with a  Markovian process, that generates gas-like collision times and positions stochastically, so that such kinetic  constraints do essentially not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' We find that even in such a system, the unusual increase in diffusivity  emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' This result implies that the kinetic constraint is not a necessary condition for the increase in the  diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' More generally, this work will provide fresh insight into the kinetics of non-spherical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  The translational diffusion coefficient Dc of a particle is generally known to decrease with in-  creasing matrix density or increasing amount of obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' It is understood as a consequence of  the ballistic particle motion being disturbed during collisions with the surrounding matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' How-  ever, if the particle is rod-shaped, a counter-intuitive motion can occur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' the Dc of a rod may  increase as the matrix concentration increases, provided the concentration is sufficiently high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  Frenkel and Maguire [1, 2] first observed such behavior for fluids consisting of infinitely thin  hard rods, whose static properties are exactly the same as those of an ideal gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' This finding was  later confirmed with higher accuracy [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Their systems do not have any hidden particles or  thermostats;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' the constituent particle moves ballistically between elastic collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Following the  previous studies[1, 2], an increase in Dc has been observed in various systems: (i) an infinitely  thin rod in a two-dimensional (2D) sea of fixed point obstacles [5], (ii) a thick rod in a 2D matrix  of circular obstacles [6], and (iii) an active matter fluid consisting of a rod swimming in direction  of its major axis [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In these systems, the increase in Dc is not triggered by a phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  Still, some rod systems exhibit an increase in Dc accompanied by the isotropic-nematic transition  [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Such multi-particle effects remain beyond the scope of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  Various loosely defined concepts have been considered previously to explain the increase in Dc:  so-called dynamic correlation, steric hindrance, geometrical constraints, confinement, or tube [2,  5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' We refer to these concepts as the ”kinetic constraint” in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In this work, we define  the kinetic constraint as the constraint that prevents the rod from crossing an obstacle until the  rod moves about the rod length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Using the kinetic constraint, the increase in Dc can be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  ∗ nakai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='fumiaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='c7@s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='nagoya-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='jp  † mk@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='ch  2  Namely, the rotational motion of the rod is kinetically constrained via the surrounding matrix in  the concentrated matrix regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Even in such a regime, the ballistic motion along the major axis  of an infinitely thin rod is not hindered, while the relevance of collisions in direction of the major  axis increases with increasing width of the rod or size of the obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Consequently, the ballistic  motion with the major axis may persist for a relatively long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' This duration may increase with  matrix density or the degree of confinement and ultimately leads to an increase in Dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In the so-  called active rod fluid [7], a similar behavior is caused by swimming along the axial direction of  the rod instead of ballistic motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In light of these studies one question may arise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Is the kinetic  constraint a necessary condition for the emergence of the increase in diffusivity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  On our way towards an answer, we have been guided by our naive belief that such an increase  can be caused by the reduction of the rotational diffusivity alone, without the hindrance of the  axially directed motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' To test our hypothesis rigorously, we consider a simple model system  where the rotational diffusivity reduces with increasing matrix density, whereas the ballistic mo-  tion along the major axis of the rod-like particle remains largely undisturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' One possible such  system is a single mobile rod embedded in a 3D arrangement of spatially fixed point obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' It  can be regarded as the extension of the Lorentz gas systems [9–11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' the single spherical particle in  fixed obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  In this work, we report that the upturn of Dc emerges even in the presence of a Markovian  process where the kinetic constraint does essentially not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' We investigate the trajectories of a  sphero-cylinder in a 3D matrix of stochastically homogeneously distributed point obstacles using a  Markovian kinetic Monte Carlo (KMC) scheme [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The Dc of this rod-like particle increases  in an intermediate matrix density regime if the rod is sufficiently long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In our system, Dc reaches a  peak value and subsequently decreases with increasing obstacle density due to the thickness of the  rod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' On the basis of the Markovian nature, we give scaling relations between Dc and the obstacle  density for dilute, intermediate, and concentrated density regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' This work will generate fresh  insight into the kinetics of the non-spherical shaped particles [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  Model and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' — The model consists of a rod-like sphero-cylinder (also termed capsule  or stadium of revolution) with radius σ, mass M, and length L of its major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The effective ”rod”  length is Le = L + 2σ due to the half-spherical end-caps, and the inertia tensor I is determined  by assuming that the mass is homogeneously distributed over the volume of the rod [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The  point obstacles are statistically homogeneously distributed in the unbounded 3D space at number  density ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The interaction between the rod and obstacles is modeled by a hard-core potential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' the  3  time  ballistic motion  collision  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Schematic representation of the KMC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' r(t), e(t), v(t), and ω(t) are the position, direction  unit vector, velocity, and angular velocity, respectively, of the rod at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' z and n characterize the  coordinate of the collision point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' z ∈ [−L/2, L/2] is the axial coordinate and n is the surface normal at the  collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' τ is the collision time interval between successive collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' τ, z, and n are stochastically  sampled based on the collision statistics corresponding to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' From the sampled variables, r, e, v and  ω at time t + τ are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  obstacles do not penetrate the rod, and they do not move during a collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The rod ballistically  moves except when it elastically collides with an obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The center of mass velocity v and  angular velocity ω are changed during a collision, conserving the rod particle’s translational and  rotational kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The total energy of this system is 5kBT/2, where kB and T are the  Boltzmann constant and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Due to the assumed elastic collisions, the total energy is a  conserved quantity and does not change during the course of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' We choose M, σ, and kBT  to define dimensionless units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' All physical quantities are therefore presented without physical  dimensions, as they follow by dimensional arguments from the three units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Within these settings,  the remaining parameters are the effective rod length Le and the number density of the obstacles,  ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Here, we avoid the density ρLe > 1, which physically corresponds to the trapping transition  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  To calculate the dynamics of a rod subject to a Markovian collision process, we extend the  kinetic Monte Carlo (KMC) simulation method [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' It requires two inputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' (i) statistics of  collisions and (ii) the change of dynamical variables by a collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' For (i), we here extend the  calculations for a sphere [14, 16] to the collision statistics of a sphero-cylinder and obtain the  collision frequency with the coordinates of collision for a given v(t), ω(t), and the direction vector  of the rod e(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In the following, we denote Γ(t) as the 8-dimensional time-dependent phase space  variable (v(t), ω(t), e(t)) characterizing the state of the rod (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The explicit expression for  4  20  0  20  80  60  40  20  0  20  40  60  80  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Trajectories of the rod’s (Le = 2502) center of mass for various scaled obstacle densities ρL2  e from  the KMC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 3D motions are projected onto the XY -plane and scaled by Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  the collision frequency for a given Γ(t), F(Γ(t)), arises from a surface integral of the collision  frequency density  f(z, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Γ(t)) = ρve(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Γ(t)) · nΘ[ve(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Γ(t)) · n]  × {δ(e(t) · n) + δ (z − L/2) Θ[e(t) · n]  +δ (z + L/2) Θ[−e(t) · n]}  (1)  where z is the axial coordinate along the rod direction and n an unit vector normal to the rod’s  surface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' These two variables characterize the coordinate ze + n of the collision point  between the rod and an obstacle, while ve(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Γ(t)) = v(t) + zω(t) × e(t) is the rod’s velocity at  the collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' (1), the first, second, and third terms in the curly bracket are relevant to  the collision on the side (∥) and two opposing (±) edges of the rod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Based on f(z, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Γ(t)) and  F(Γ), the coordinate of the collision point and the collision time interval τ between successive  collisions are sampled using stochastic techniques [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' (ii) From these sampled variables, r,  e, v, and ω are updated based on the rules of classical mechanics for a rigid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Repeating  these samplings and updates, we calculate the dynamics of the mobile rod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The details of the  derivation of the collision statistics, sampling method, and the update scheme are described in the  supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' — Qualitatively different behaviors occur during a change of ρ at fixed Le = 2502, as  visually captured by representative trajectories in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The observed time duration is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='0 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  For ρL2  e = 1 and 10, the mobile rod seems to move randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' At higher number densities ρL2  e =  100, the straight motion persists over longer distances compared with those for lower densities  ρL2  e = 1 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' For ρL2  e = 1000, we observe straight and bouncing motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  To quantify these motions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 2), we calculate Dc of the mobile rod from its center-of-mass  5  101  102  103  104  105  106  10-11 10-10 10-9  10-8  10-7  10-6  10-5  10-4  10-3  10-2  10-1  100  101  100  101  102  103  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  Translational diffusion coefficient of the mobile rod (various rod lengths Le) from the KMC  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Data are shown as (a) Dc versus ρ and (b) in scaled form DcL−1  e  versus ρL2  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Error bars and  asymptotic exponents are also displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  mean square displacement (MSD) in the linear time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Dc versus the obstacle number  density ρ are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 3(a) for various mobile rod lengths Le (error bars arise from the  linear fitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In this figure, Dc shows non-monotonic behaviors with increasing ρ for the highly  elongated rods Le ≳ 66;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Dc at large Le exhibits both a local minimum and maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' When  the same data are represented in scaled forms, DcL−1  e  and ρL2  e, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 3(b), the curves  collapse except for the larger density regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' From Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 3(a,b), the asymptotic forms are observed  for small, intermediate, and large density regimes as Dc ∝ (ρL3  e)−1, Dc ∝ ρL3  e, and Dc ∝  ρ−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' We emphasize that the non-monotonic ρ dependency for Dc arises even under  the Markovian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In contrast to Dc, the rotational diffusion coefficient Dr in the current  system exhibits monotonic behavior with increasing obstacle density, Dr ∼ (ρL3  e)−1 as shown in  supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  The scaling relations between Dc and ρ can be simply explained based on the Markovian na-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Here, the Dc is also calculated from the integration of the velocity auto-correlation function  over time lag, instead of the mean square displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Thus, the diffusion coefficient would be  roughly approximated as the relaxation time of the center of mass velocity in the dimensionless  units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The collision frequency can be decomposed into two contributions: collision frequencies  from the side F∥ and edges F±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' These contributions scale as F∥ ∼ ρLe and F± ∼ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' These  estimates are confirmed by the rigorous calculations for the collision frequencies as shown in sup-  plementary Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' S7 and S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The average angular velocity scales as ¯ω ∼ L−1  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In the dilute regime  ρL2  e ≲ 1, the relation ¯ω > F∥ is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In this low density regime, the rod mainly rotates and  occasionally collides with an obstacle on its side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' By a few collisions, the motion of the rod largely  6  changes since the rod experiences the impulsive forces from various directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Then, Dc scales  linearly as the collision time interval as Dc ∼ F −1  ∥  ∼ ρ−1L−1  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' This description is consistent  with the observed random motions for the lower density regimes ρL2  e = 1 and 10 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In  the higher density regime ρL2  e ≳ 1, where the relation ¯ω > F∥ is fulfilled, the rotational motion  of the rod is diffusive, and thus the direction of the rod slowly changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In this density regime,  the velocity with the orthogonal direction rapidly relaxes, whereas that with the axial direction is  not largely disturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In such a case, there are possible relaxation mechanisms for the velocity  with axial direction: the change of rod direction or the collision on the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Here, the change of  rod direction between collisions is approximately ∆θ ∼ ¯ω/F∥, and the rotational relaxation time  scales as τrot ∼ ∆θ−2/F∥ ∼ ρL3  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' This estimate also predicts the rotational diffusion coefficient  Dr = (2τr) ∼ (ρL3  e)−1, in full agreement with our measurements, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=', supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  The collision time interval on the edge is about F −1  ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In the intermediate density regime where  Dc increases, the rotational relaxation time is smaller than the collision time interval on the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  Thus, the velocity relaxes by the rotation of the direction, and consequently the diffusion coeffi-  cient is approximately Dc ∼ ρL3  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Within the high density regime where Dc decreases again, the  collision on the edge is the main mechanism causing velocity relaxation with the axial direction,  and we obtain Dc ∼ ρ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' These mechanisms explained above seem to be consistent with the  persistence of the straight motion with ρL2  e = 100 and the straight and bouncing motions with  ρL2  e = 1000 displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 2, and the estimated exponents also successfully agree with the  simulation results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  One may suspect that the increase in Dc is an artifact since we assume a Markovian process  even in the high density regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' However, we next show that this assumption is indeed a good  approximation to calculate Dc for a rod embedded in a 3D sea of point obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' To this end, we  calculate the dynamics of a rod using conventional molecular dynamics (MD) simulations [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  Here, instead of a hard-core potential, the repulsive Weeks-Chandler-Andersen potential [19] is  employed for the elastic interaction between rod and point obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The details of the simulation  method are described in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 4 displays Dc (symbols) for various rod  lengths Le obtained via MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Error bars are again calculated from linear fitting for the MSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Due  to the computational cost, data for large rod lengths Le = 16002 and 100002 could not be sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  For comparison, the KMC data from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 3 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 4 (solid curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The MD results  quantitatively agree with those obtained via KMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' This indicates that multi-body correlations are  negligible in the estimation of Dc within the explored wide regime of obstacle densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  7  10-1  100  100  101  102  103  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Translational diffusion coefficient Dc versus obstacle density with the error bars from MD simula-  tions (Symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Data for three rod lengths Le are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' For comparison, the KMC simulation results  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 3) are shown by solid curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='— This work shows that Dc increases even in a Markovian process and that the  observed exponents are easily rationalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' This result does not imply that the exponents in prior  studied systems can be simply understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Frenkel and Maguire [1, 2] investigated Dc of a con-  stituent particle in a system of infinitely thin hard rods, where Dc was found to be proportional  to the root of the rod density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' For a 2D rod in the presence of point obstacles studied by H¨ofling,  Frey, and Franosch [5], the power exponent of Dc versus obstacle density is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='8 in the concentrated  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Mandal et al [7] investigated the dynamics of a rod-shaped active swimmer (along the ax-  ial direction) and showed that Dc depends on the square of the density of the constituent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In these  prior systems, the kinetic constraints are not negligible, and they should be taken into account to  explain the observed exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  Some works investigated similar systems to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Tucker and Hernandez [6, 20] numerically  studied the dynamics of a 5 ˚A long mobile rod in the presence of spatially fixed spherical obstacles  with radius 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='5 ˚A for rod thickness values 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='1, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='5 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' They argued that the increase in Dc  does not occur in their 3D system, while it can occur in the corresponding 2D setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' One may  8  think that these findings are inconsistent with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' However, if one identifies the lengths in  their system with ours, the effective aspect ratio of the rod becomes about 10 since the interaction  distance between the rod and the obstacle is the rod thickness plus obstacle size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' For the rod with  such an aspect ratio 10, an increase in Dc does not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Conversely, in Tucker and Hernandez’s  system, the increase in Dc will occur for a much smaller obstacle radius or much larger rod length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  Otto, Aspelmeier, and Zippelius [21] theoretically analyzed the dynamics of a constituent particle  of infinitely thin rods under the assumption of the Markovian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' They argued that an increase  in Dc should not occur under such circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' This result obviously contradicts our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  However, they did not consider the long-time persistence of the ballistic motion with the axial  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Thus, the increase in the Dc could not be captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  It should be emphasized that a rise of Dc can occur for a ballistic system [1–5] or some active  matter systems [7] due to the persistence of the motion with the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' One may think  that an increase in Dc can occur for passive rod-shaped particles in some solvents or some porous  media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' However, it can not exhibit the increase in diffusivity by the same mechanism as our system  since the persistence of the motion with axial direction rapidly relaxes by the Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  Recently, the increase in diffusivity with increasing aspect ratio is observed for rod in a gel [22],  although the mechanism would be different to our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  The current system consists of a rod colliding with immobile, or infinitely heavy point obsta-  cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Let us consider the situation where obstacles move in an equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' As long as the  obstacle mass is sufficiently larger than M, the obstacle motion is slow because of the Maxwell-  Boltzmann velocity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In this case, the situation would not be largely different from  the current system since the moving particles can be approximated as the fixed obstacles for the  rod particle, and the increase in the diffusivity will emerge in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' In contrast to this, if the  obstacle mass is comparable to M, the situation can be different from the current system since  the translational and rotational relaxation times vary largely with the obstacle mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Even in this  case, the increase in the diffusivity can emerge since it simply originates from the reduction of the  rotational motion and the persistence of the axial motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The analyses for the effects of obstacle  mass on the increase in diffusivity will be future interesting work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  In conclusion, this study demonstrated that a Dc upturn can emerge even in Markovian na-  ture, where the kinetic constraint does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' As a simple model system, we investigated the  single mobile rod-shaped particle in immobile fixed obstacles in three-dimensions using highly  efficient Markovian kinetic Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' The translational diffusion coefficient of the  9  rod decreases, increases, and decreases again as the obstacle density increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' These non-trivial  behaviors could be explained based on the Markovian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' This work sheds light on the ki-  netics of non-spherical particles where the elementary dynamic processes are ballistic motion and  collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  FN and MK were supported by the “Young Researchers Exchange Programme between Japan  and Switzerland” under the “Japanese-Swiss Science and Technology Programme”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' FN was also  supported by a Grant-in-Aid (KAKENHI) for JSPS Fellows (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' JP21J21725 from the  Ministry of Education, Culture, Sports, Science and Technology, MEXT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Frenkel and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Maguire, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 47, 1025 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Frenkel and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Maguire, Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 49, 503 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Magda, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Davis, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Tirrell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 85, 6674 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Magda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Tirrell, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Davis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 88, 1207 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' H¨ofling, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Frey, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Franosch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 101, 120605 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Tucker and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Hernandez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' A 114, 9628 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Mandal, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Kurzthaler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Franosch, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' L¨owen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 125, 138002 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Allen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 65, 2881 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Lorentz, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Ned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Wet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 7, 438 (1905).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [10] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Alder and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Alley, Physica A 121, 523 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [11] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' H¨ofling and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Franosch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 98, 140601 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [12] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Gillespie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 22, 403 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Bortz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Kalos, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Lebowitz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 17, 10 (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Dorfman, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' van Beijeren, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Kirkpatrick, Contemporary Kinetic Theory of Matter (Cam-  bridge University Press, Cambridge, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [15] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Pournin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Weber, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Tsukahara, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Ferrez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Ramaioli, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Liebling, Granul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Matter  7, 119 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [16] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Mazenko, Nonequilibrium statistical mechanics (John Wiley & Sons, Hoboken, NJ, United  States, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Devroye, Non-Uniform Random Variate Generation (Springer, New York, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Allen and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Tildesley, Computer simulation of liquids (Oxford University Press, Oxford,  10  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=', 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Weeks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Chandler, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Andersen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Phys 54, 5237 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Tucker and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Hernandez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' B 115, 4412 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Otto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Aspelmeier, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Zippelius, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' 124, 154907 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  [22] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Rose, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Gogotsi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Galarraga, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Burdick, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Murray, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Lee, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content=' Composto,  Macromolecules (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}
+page_content='  11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE0T4oBgHgl3EQfewA6/content/2301.02394v1.pdf'}

69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5829c0b7fe085c85489e6e2c14e12472199a31bf390aee6c7946d2f3927bc24c
+size 683406

69AyT4oBgHgl3EQfpvg-/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c3e45abbee112ee7d23873137d38abdb7412890c37908b5c4011a5e3d609bf77
+size 2359341

69AyT4oBgHgl3EQfpvg-/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:bb5ed9b2d5ebdc4aae21530895519154ca8449d79f165a816308d45377523d9c
+size 87672

6NFIT4oBgHgl3EQf8isw/content/tmp_files/2301.11402v1.pdf.txt

@@ -0,0 +1,1730 @@
+A Hybrid Deep Neural Operator/Finite Element Method for Ice-Sheet Modeling
+QiZhi Hea, Mauro Peregob, Amanda A. Howardc, George Em Karniadakisc,d, Panos Stinisc
+aDepartment of Civil, Environmental, and Geo- Engineering, University of Minnesota, 500 Pillsbury Drive S.E., Minneapolis, MN 55455
+bCenter for Computing Research, Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185
+cAdvanced Computing, Mathematics and Data Division, Pacific Northwest National Laboratory, Richland, WA 99352
+dDivision of Applied Mathematics and School of Engineering, Brown University, 182 George Street, Providence, RI 02912
+Abstract
+One of the most challenging and consequential problems in climate modeling is to provide probabilistic projections
+of sea level rise. A large part of the uncertainty of sea level projections is due to uncertainty in ice sheet dynamics.
+At the moment, accurate quantification of the uncertainty is hindered by the cost of ice sheet computational models.
+In this work, we develop a hybrid approach to approximate existing ice sheet computational models at a fraction
+of their cost. Our approach consists of replacing the finite element model for the momentum equations for the ice
+velocity, the most expensive part of an ice sheet model, with a Deep Operator Network, while retaining a classic finite
+element discretization for the evolution of the ice thickness. We show that the resulting hybrid model is very accurate
+and it is an order of magnitude faster than the traditional finite element model. Further, a distinctive feature of the
+proposed model compared to other neural network approaches, is that it can handle high-dimensional parameter spaces
+(parameter fields) such as the basal friction at the bed of the glacier, and can therefore be used for generating samples
+for uncertainty quantification. We study the impact of hyper-parameters, number of unknowns and correlation length of
+the parameter distribution on the training and accuracy of the Deep Operator Network on a synthetic ice sheet model.
+We then target the evolution of the Humboldt glacier in Greenland and show that our hybrid model can provide accurate
+statistics of the glacier mass loss and can be effectively used to accelerate the quantification of uncertainty.
+Keywords: hybrid model, finite element, neural operator, ice-sheet dynamics, deep learning surrogate
+1. Introduction
+Ice sheet models are important components of climate models and are crucial for providing projections of sea-level
+rise. In fact, sea-level rise is due in large part to added water to the ocean originating from mass loss of Greenland and
+Antarctic ice sheets [1, 2, 3].
+Quantifying the uncertainty on the projections of sea-level rise, due to uncertainties in the data and in the models,
+is an extremely challenging task. The large dimensionality of the parameter space, and high computational cost of
+ice sheet models make Bayesian inference and uncertainty quantification infeasible, despite the large computational
+resources available. While there are efficient ways to perform Bayesian inference under certain approximations
+[4, 5], previous attempts to quantify the uncertainty on sea level rise (e.g., [6, 7, 8]) perform drastic reductions of
+the dimensionality of the parameter space that are often dictated by feasibility reasons rather than by physical or
+mathematical arguments.
+Several efforts [9, 10, 11, 12, 13, 14, 15, 16, 17, 18] over the last decades focused on efficiently solving the steady
+state Stokes-like flow equations governing the ice flow, which still represents the most computationally expensive
+part of an ice flow model. Flow equations need to be solved at each time step. While time steps can be as little as a
+week, typical temporal periods of interest range from a few decades to centuries, to millennia. In this work we aim at
+replacing the most expensive part of an ice sheet model, the Stokes-like flow equations, with a deep learning surrogate
+that is order of magnitudes faster than the finite element based implementation. A similar idea has been pursued by
+Jouvet et al. [19], where a deep learning model was used to accelerate ice sheet modeling of Paleo simulations. A key
+requirement for our surrogate, that sets it apart from [19], is that it depends on high-dimensional parameter spaces
+(parameter fields), such as the basal friction coefficient that determines the basal sliding or the bed topography. This
+allows us to use the model for inference and for uncertainty quantification. We also note that in paleo simulations
+Preprint submitted to Elsevier
+January 30, 2023
+arXiv:2301.11402v1  [physics.comp-ph]  26 Jan 2023
+
+most of the uncertainty comes from the climate forcing, whereas in the simulations in which we are interested here,
+that spans approximately half a century, model error is a significant source of uncertainty [8], which forces us to
+have very accurate models. Another related problem, where deep learning models have been used to approximate the
+parameter-to-velocity map in ice-sheet problems, is presented in [20]. In that work, the authors first find a basis of the
+operator using principal component analysis, and then use a residual neural network to compute the basis coefficients
+as a function of the parameters. In contrast to our problem, in [20] only a handful of parameters are considered.
+We represent our deep learning surrogate with Deep Operator Networks (DeepONets) [21], which have proven
+to work well in learning operators in a wide range of applications ranging from fracture mechanics to combustion
+problems [22, 23, 24, 25, 26]. In its vanilla formulation, a DeepONet contains two deep neural networks, referred to as
+the branch network and the trunk network. The trunk network takes as input spatial coordinates whereas the branch
+network takes as input the input fields evaluated at a fixed set of points. DeepONets approximate operators as a linear
+combination of “basis functions” generated by the trunk network, with coefficients generated by the branch network.
+The mathematical foundations of DeepONets are based on the universal approximation theorem [27, 28], and, under
+mild assumptions, it has been proven that DeepONets can approximate with given accuracy any operator [21]. Our
+DeepONet surrogate takes as input fields the ice thickness and the basal friction field and computes the depth-averaged
+ice velocity field.
+We use the trained DeepONet to build a fast hybrid ice-flow model, where the evolution of the ice thickness is
+discretized with a classic finite element method, and, at each time step, the ice velocity field (as a function of the ice
+thickness and the basal friction field) is computed by the DeepONet. A finite element implementation of the ice-flow
+model is used as the “reference model” and also used to generate data to train the the DeepONet model. We demonstrate
+our approach on two ice sheet problems: 1. a synthetic ice sheet problem for exploring different hyper-parameters of
+the DeepONet and for studying the impact of mesh resolution and correlation length on the DeeoONet training and
+accuracy, and 2. a realistic simulation of the Humboldt glacier, which is one of the largest glaciers in Greenland and
+one that is expected to greatly contribute to sea-level rise in this century [29]. We show how our DeepONet surrogate
+can approximate the ice velocity computed by the finite element model very accurately (relative error of 0.4%) and at a
+fraction of the cost of the finite element model. The hybrid model produces accurate results for the ice thickness (˜2%
+relative error over a span of 100 years). We also show how the mass loss of the Humboldt glacier, computed using the
+hybrid model, is an accurate representation of the finite element model and can be used for computing statistics of sea
+level rise, yielding a 10 fold speed-up.
+In Section 2 we present the mathematical equations that we use to compute the ice thickness and velocity and
+the probability distribution of the basal friction parameter. In section 3 we introduce the hybrid model, focusing in
+particular on its DeepONet component. In Section 4 we present the result of training the DeepOpNet for a synthetic
+test case, studying how the resolution of the input data and the correlation length of the basal friction distribution affect
+the accuracy and training of the DeepONet. Finally in Section 5 we target the Humboldt glacier and show how hybrid
+model can be effectively used for computing the statistics of the glacier mass loss. We conclude in Section 6 with a
+summary.
+2. Ice Sheet Models
+In this section, we briefly introduce the ice sheet models considered in this work, as depicted in Fig. 1.
+Let x and y denote the horizontal coordinates and z the vertical coordinate, chosen such that the sea level corresponds
+to z = 0. The ice domain, at time t, can be approximated as a vertically extruded domain Ω defined as
+Ω(t) := {(x, y, z) s.t. (x, y) ∈ Σ, and l(x, y, t) < z < s(x, y, t)},
+where Σ ⊂ R2 is the horizontal extension of the ice. Γl(t) := {(x, y, z) s.t. z = l(x, y, t)} denotes the lower surface of the
+ice at time t, and Γs(t) := {(x, y, z) s.t. z = s(x, y, t)} denotes the upper surface of the ice1. The bed topography, which
+we assume constant in time, is given by Γb := {(x, y, z) s.t. z = b(x, y)}. In general, the ice sheet can have ice shelves
+where the ice is floating. We hence partition the lower surface of the ice Γl in the grounded part Γg = Γl ∩ Γb (here,
+1For simplicity here we assume that Σ does not change in time. This implies that the ice sheet cannot extend beyond Σ but it can become thicker
+or thinner (to the point of disappearing in some regions).
+2
+
+l(x, y, t) = b(x, y)) and the floating part Γ f under the ice shelf. We partition the lateral boundary of Ω in Γm, denoting
+the ice sheet margin (either terrestrial or marine margin), and, when we only consider a portion of the ice sheet, in Γd,
+denoting an internal (artificial) boundary often chosen in correspondence of the ice divides.
+Figure 1: Cartoon of an ice sheet in the x − z plane.
+The thickness of the ice, given by H(x, y, t) := s(x, y, t) − l(x, y, t), is defined on Σ × [0, t f ] and evolves according to
+∂tH + ∇ · (¯uH) = fH
+(1)
+where ¯u := 1
+H
+� s
+l
+u dz is the depth-integrated velocity and fH is an accumulation rate, accounting for accumulation
+(e.g., due to snow precipitations) and melting at the upper surface and accumulation/melting at the base of the ice sheet.
+We need to constrain H to be non-negative, as there is no guarantee that fH, typically coming from climate models, is
+consistent with the ice thickness equation.
+Ice sheets behave as a shear thinning fluid and can be modeled with the nonlinear Stokes equation [30]. In this work
+we use simplifications of Stokes equations that are less expensive to solve and that are obtained with scaling arguments
+based on the fact that glaciers and in particular ice sheets are typically shallow. We consider two such simplifications:
+the mono-layer higher-order approximation (MOLHO) and the shallow shelf approximation (SSA). The MOLHO
+model [31] is suitable for both frozen and thawed beds, whereas the simpler SSA model [32, 33] works well only
+for grounded ice with significant sliding at the bed or for ice shelves where the ice is floating over the water. In the
+following we detail the Stokes model and its approximations.
+2.1. Stokes model
+We denote with u, v and w the x, y and z components of the ice velocity, respectively, and the ice velocity vector is
+denoted by u := (u, v, w). Denoting the pressure with p, and the ice density with ρ, the Stokes equation reads
+−∇ · σ = ρg
+(2)
+∇ · u = 0
+(3)
+with stress tensor σ = 2µD − pI, and strain rate tensor Di j(u) = 1
+2
+�
+∂ui
+∂xj + ∂uj
+∂xi
+�
+. The non-linear viscosity is given by
+µ = 1
+2A(T)−q De(u)q−1
+(4)
+with q ≤ 1. In this work we take q = 1
+3, a typical choice. A is the ice flow factor that depends on the ice temperature
+T. The effective strain rate De(u) is given by De(u) =
+1√
+2|D(u)|, where | · | denotes the Frobenius norm. The Stokes
+3
+
+Td
+2
+Ta
+Ig
+ice
+Tm
+If
+bedrock
+Fb
+oceanequation is accompanied by the following boundary conditions:
+�������������������
+σn = 0
+on Γs
+stress free, atmospheric pressure neglected
+σn = ρw g min(z, 0)n
+on Γm
+boundary condition at the ice margin
+u = ud
+on Γd
+Dirichlet condition at internal boundary
+u · n = 0, (σn)∥ = βu∥
+on Γg
+impenetrability + sliding condition
+σn = ρw g z n
+on Γf
+back pressure from ocean under ice shelves
+Here β(x, y) is the sliding (or friction) coefficient, ρw is the density of the ocean water and n the unit outward-pointing
+normal to the boundary. The boundary condition at the margin includes the ocean back-pressure term, when the margin
+is partially submerged (z < 0). For terrestrial margin, z > 0, hence the term becomes a stress-free condition. The
+friction term β can also depend on u, depending on the choice of the sliding law.
+2.2. Mono-layer higher-order (MOLHO)
+The MOLHO model [31] is based on the Blatter-Pattyn approximation [34] that can be derived neglecting the terms
+wx and wy in the strain-rate tensor D and, using the continuity equation, replacing wz with −(ux + vy):
+D =
+��������������
+ux
+1
+2(uy + vx)
+1
+2uz
+1
+2(uy + vx)
+vy
+1
+2uz
+1
+2uz
+1
+2vz
+−(ux + vy)
+��������������
+.
+(5)
+This leads to the following elliptic equations in the horizontal velocity (u, v)
+− ∇ · (2µ ˆD) = −ρg∇s
+(6)
+with
+ˆD =
+� 2ux + vy
+1
+2(uy + vx)
+1
+2uz
+1
+2(uy + vx)
+ux + 2vy
+1
+2vz
+�
+.
+(7)
+Here the gradient is two-dimensional: ∇ = [∂x, ∂y]T. The viscosity µ is given by (4) with the effective strain rate
+De =
+�
+u2x + v2y + uxvy + 1
+4(uy + vx)2 + 1
+4u2z + 1
+4v2z.
+The boundary conditions reads
+���������������������
+2µ ˆD n = 0
+on Γs
+stress free, atmospheric pressure neglected
+2µ ˆD n = ψn
+on Γm
+boundary condition at at ice margin
+u = ud
+on Γd
+Dirichlet condition at internal boundary
+2µ ˆD n = βu∥
+on Γg
+sliding condition
+2µ ˆD n = 0
+on Γ f
+free slip under ice shelves
+where ψ = ρg(s − z)n + ρw g min(z, 0)n, which can be approximated with its depth-averaged value ¯ψ = 1
+2gH(ρ − r2ρw),
+r being the the submerged ratio r = max
+�
+1 − s
+H , 0
+�
+; u∥ is the component of the velocity u tangential to the bed.
+MOLHO consists of solving the weak form of the Blatter-Pattyn model, with the ansatz that the velocity can be
+expressed as :
+u(x, y, z) = ub(x, y) + uv(x, y)
+�
+1 −
+� s − z
+H
+� 1
+q +1�
+.
+The problem is then formulated as a system of two two-dimensional partial differential equations (PDEs) for ub and uv
+(for a detailed derivation see [31].) Note that the depth-averaged velocity is given by ¯u = ub + (1+q)
+(1+2q) uv.
+4
+
+2.3. Shallow Shelf Approximation (SSA)
+The shallow shelf approximation [32] is a simplification of the Blatter-Pattyn model, assuming that the velocity is
+uniform in z, so u = ¯u. It follows that uz = 0 and vz = 0, giving:
+D =
+����������
+ux
+1
+2(uy + vx)
+0
+1
+2(uy + vx)
+vy
+0
+0
+0
+−(ux + vy)
+���������� ,
+ˆD =
+� 2ux + vy
+1
+2(uy + vx)
+0
+1
+2(uy + vx)
+ux + 2vy
+0
+�
+,
+(8)
+and De =
+�
+u2x + v2y + uxvy + 1
+4(uy + vx)2. The problem simplifies to a two-dimensional PDE in Σ
+−∇ ·
+�
+2µH ˆD(¯u)
+�
++ β¯u = −ρgH∇s,
+in Σ
+with ¯µ = 1
+2 ¯A(T)− 1
+n De(¯u)
+1
+n −1, where ¯A is the depth-averaged flow factor and with boundary conditions:
+�
+2µ ˆD(¯u) n = ¯ψn
+on Γm
+boundary condition at ice margin
+¯u = ¯ud
+on Γd
+Dirichlet condition at internal boundary
+Recall that ¯ψ = 1
+2gH(ρ − r2ρw), r being the the submerged ratio r = max
+�
+1 − s
+H , 0
+�
+. With abuse of notation, here Γm
+and Γd are intended to be subsets of ∂Σ.
+2.4. Distribution of basal friction field
+The basal friction field β is one of the main factors that control the ice velocity. It cannot be measured directly and
+it is typically estimated by solving a PDE-constrained optimization problem, e.g., [35, 36], to assimilate observations
+of the surface ice velocity. As a result, the basal friction field is affected by both uncertainties in the observations and in
+the the model. While it is possible to characterize the probability distribution for β using a Bayesian inference approach,
+e.g., [37], here we adopt a simplified log-normal distribution for β. We write the basal friction field as β = exp(γ),
+where γ is normally distributed as
+γ ∼ F
+�
+log(¯β), kl
+�
+, and kl(x1, x2) = a exp
+�
+−|x1 − x2|2
+2l2
+�
+.
+(9)
+Here log(¯β) is the mean of the Gaussian process F and it is often obtained by assimilating the observed velocities [35],
+l is the correlation length and a is a scaling factor. In this work we choose values of the correlation length and of the
+scaling factor that produce reasonable results. While an in-depth validation of the chosen parameters is beyond the
+scope of this work, we explore the dependence of the accuracy of the DeepONet model as a function of the correlation
+length, as discussed in Section 4.
+3. Computational Models
+In this section we introduce the finite element ice flow model and the hybrid ice flow model. We first perform a
+semi-implicit time discretization of the ice thickness equation (1):
+�
+Hn+1
+=
+Hn − ∆t ∇ ·
+�
+¯unHn+1�
++ ∆tFn
+H
+¯un
+=
+G(β, Hn)
+(10)
+where Hn is the approximation of H at time tn = t0+n∆t, for a given time-step ∆t, and Fn
+H = FH(tn) is the corresponding
+discrete approximation of the accumulation rate fH. Here, G(·, ·) is the velocity operator that maps the basal friction
+field and the ice thickness into the depth-averaged velocity vector, based either on the SSA (Sec. 2.3) model or the
+MOLHO (Sec. 2.2) model. In this work we discretize the thickness equation (10) with finite elements, using streamline
+upwind stabilization. Similarly, we provide a classic Galerkin finite element discretization of the nonlinear operator G.
+The finite element discretization is implemented in FEniCS [38]. We use continuous piece-wise linear finite elements
+for both the thickness and the velocity fields, and solve the discretized problem with PETSc [39] SNES nonlinear
+5
+
+solvers. We refer to this finite element implementation of (10) as the finite element ice flow model that we use as our a
+reference model.
+The focus of the paper is on avoiding the high computational cost of constructing a finite element approximation of
+the nonlinear operator G, and using, instead, a DeepONet approximation of G, which, in combination with the finite
+element discretization of the first equation of (10), constitutes the hybrid ice flow model. The DeepONet implementation
+and training are performed using JAX [40]. At each time step, the FEniCS finite element code calls theJAX DeepONet
+code to compute an approximation of G(β, Hn). In the next sections we describe in detail the DeepONet architecture
+and its training.
+3.1. DeepONet approximation
+As briefly discussed in the introduction, the main idea of DeepONet is to learn, in general nonlinear, operators
+mapping between infinite-dimensional function spaces via deep neural networks [21]. Inspired by the universal
+approximation theorem for operators [27], DeepONet’s architecture consists of two neural networks: one is used to
+encode the input function sampled at fixed sensor points (branch net) whereas the other inputs the location coordinates
+to evaluate the output function (trunk net). It has been shown that this architecture of two sub-networks can substantially
+improve generalization compared to fully connected neural networks [21]. In this study, a DeepONet denoted by Gθ is
+used as a surrogate for the nonlinear operator G in Eq. (10),
+Gθ(β, Hn)(x) ≈ G(β, Hn)(x),
+(11)
+where θ represents the collection of trainable parameters in DeepONet, and the approximated velocity components are
+¯un
+x ≈ Gx
+θ(β, Hn)(x) =
+p
+�
+m=1
+bm(β, Hn)tm(x),
+¯un
+y ≈ Gy
+θ(β, Hn)(x) =
+2p
+�
+m=p+1
+bm(β, Hn)tm(x),
+(12)
+where bm and tm denote the outputs of the branch net and the trunk net, respectively. The details of the DeepONet
+model is shown in the schematic of Fig. 2. In this setting, the input functions, i.e., the friction β and thickness Hn at the
+moment tn, evaluated at finite locations (sensors), X = {x1, x2, ..., xN}, are mapped as embedded coefficients through
+the branch net, while the trunk net learns a collection of space-dependent basis functions that are linearly combined
+with the branch coefficients to approximate the velocity components. Note that the learned operator Gθ(β, Hn) is a
+continuous function with respect to coordinates x, which are the inputs to the trunk net. For brevity, we denote the
+DeepONet approximated velocity as ¯uNN.
+3.2. DeepONet training
+The trainable parameters, i.e., θ, associated with the DeepONet model are obtained by minimizing the loss function
+L(θ) =
+1
+NβNT
+Nβ
+�
+i=1
+NT
+�
+j=1
+�
+x∈Y
+wi j(x)|¯u(x, t j; βi) − Gθ(βi, H j)(x)|2,
+(13)
+where wij(x) are weights corresponding to each data point, Nβ is the number of friction fields {βi}
+Nβ
+i=1 used for different
+training simulations, NT is the number of time steps within each simulation to sample the velocity and thickness,
+¯u(x, t j; βi) is the target velocity solution, and Gθ(βi, H j)(x) is the predicted value obtained from DeepONet. Both target
+solution ¯u(x, t j; βi) := G(βi, H j)(x) and DeepONet prediction Gθ(βi, H j)(x) are evaluated at the set of locations Y. The
+input functions βi and H j of the branch network are discretized at the fixed set of sensor points X (see Fig. 2). In this
+work it is convenient to choose X to be the set of the grid nodes used in the finite element discretization and to take
+Y = X.
+In Eq. (13), the penalizing weights wij(x) are generally related to the characteristics of training data, i.e., the friction
+field, time step, and spatial locations. For simplified cases where the target operator presents little variability with
+6
+
+𝛽
+𝛽(𝒙!)
+𝛽(𝒙")
+⋮
+𝛽(𝒙#)
+𝐻$(𝒙!)
+𝐻$(𝒙")
+⋮
+𝐻$(𝒙#)
+𝐻$
+Branch net
+𝒙 = (𝑥, 𝑦)
+Trunk net
+𝑏%&!
+⋮
+𝑏"%
+×
+𝒢'
+((𝛽, 𝐻$)(𝒙)
+𝑏!
+⋮
+𝑏%
+𝑡!
+⋮
+𝑡%
+𝑡%&!
+⋮
+𝑡"%
+×
+𝒢'
+)(𝛽, 𝐻$)(𝒙)
+𝒢'(𝛽, 𝐻$)(𝒙)
+Figure 2: Schematic representation of DeepONet. The branch net takes as inputs the functions β(x) and Hn(x) = H(tn, x) evaluated at N fixed sensor
+points X = {xi}N
+i=1 and returns the feature embedding vector b ∈ R2p as output. The trunk net takes the continuous coordinates x ∈ Y as input
+and outputs another embedding vector t ∈ R2p. The embedding vectors b and t are combined by dot product to generate the solution operator,
+Gθ(β, Hn)(x). The trainable parameters θ associated with the branch net and the trunk net are optimized by minimizing the loss function defined as a
+weighted mean square error (see Eq. 13). In this study, we set Y = X for simplicity.
+respect to the input parameters, the weights are assumed to be unity, i.e., wi j(x) ≡ 1. However, it is observed in our
+numerical investigation that using nonuniform (space-dependent) weights can lead to better generalization. To this
+end, we use the self-adaptive weight estimation approach [41, 42] to adjust the weight parameters through gradient
+descent along with the network parameters. Assuming that the weights depend only on the space coordinates, i.e.,
+wi j(x) = w(x), the loss function (13) is modified as
+L(θ, λ) =
+1
+NβNT
+Nβ
+�
+i=1
+NT
+�
+j=1
+�
+x∈Y
+w(x)|¯u(x, t j; βi) − Gθ(βi, H j)(x)|2,
+(14)
+where w(x) is further defined as m(λ(x)) in which λ = {λ(x)}x∈Y are the trainable self-adaptive weight parameters
+dependent on locations x, and m(λ) is a mask function defined on [0, ∞] to accelerate convergence [41]. The mask
+function needs to be differentiable, nonnegative, and monotonically increasing. The polynomial mask m(λ) = λq for
+q = 1, 2, ... is adopted in this study.
+The key feature of self-adaptive DeepONet training is that the loss L(θ, λ) is simultaneously minimized with respect
+to the network parameters θ but maximized with respect to the self-adaptive parameters λ, i.e.,
+min
+θ max
+λ
+L(θ, λ).
+(15)
+If one uses the gradient descent method, the updated equations of the two sets of parameters at v iteration are:
+θv+1 = θv − ηθ∇θL(θv, λv),
+λv+1 = λv + ηλ∇λL(θv, λv),
+(16)
+where ηθ and ηλ are the learning rates for updating θ and λ, respectively. The employment of self-adaptive weights can
+significantly improve the prediction accuracy at the localized features in the solution by properly balancing the terms
+via the corresponding weights [41, 43].
+3.3. Data preparation & training details
+To generate sufficient training data, we perform simulations of the finite element ice flow model (10) based on
+either SSA or MOLHO and considering Nβ basal friction samples, βi(x), i = 1, ..., Nβ, taken from distribution (9). For
+7
+
+each sample βi, we compute the thickness and depth-integrated velocity using the finite element flow model and store
+their values {H j
+i }NT
+j=1 and {¯u j
+i }NT
+j=1 at times t j, j = 1, 2, ..., NT and grid points xi ∈ X.
+In training the DeepONet, the input functions, β(x) and Hn(x), as well as the DeepONet operator Gθ are evaluated
+at points X = {x1, x2, ..., xN}, as described in Fig. 2. Therefore, a DeepONet training dataset is expressed as a triplet of
+the form,
+��
+[β(k), H(k)]
+�NβNT
+k=1 ,
+�
+Y(k)�NβNT
+k=1 ,
+� ¯U(k)�NβNT
+k=1
+�
+,
+(17)
+where
+[β(k), H(k)] = [βj(x1), βj(x2), ..., β j(xN), H j
+i (x1), H j
+i (x2), ..., H j
+i (xN)],
+Y(k) ≡ X = {x1, x2, ..., xN},
+¯U(k) = [¯uj
+i (y1), ¯uj
+i (y2), ..., ¯u j
+i (yNu)].
+(18)
+Here, the superscript k is defined as k = (i − 1)NT + j with i = 1, ..., Nβ and j = 1, ..., NT, denoting the index of input
+parameters associated with time steps and friction samples.
+Regarding the basal friction fields, we adopt the following procedure to split the training and testing data: if Nb
+friction fields are generated from the Gaussian process described in Section 2.4, the simulation solutions associated
+with the first 20 fields, {βi}20
+i=1, are exclusively used for testing, while the rest Nβ = Nb − 20 fields, {βi}
+20+Nβ
+i=21 , are selected
+for training the DeepONet model. Unless stated otherwise, for the given training basal friction fields the finite element
+solutions at time steps t = 1, 2, ..., 100 (i.e., NT = 100) are used for the training.
+In the following tests, the default training scheme uses the Adam optimizer with a learning rate 1 × 10−3. ReLU is
+selected as the activation function, and the batch size is 200. The architecture of both the branch net and the trunk net is
+a fully connected neural network consisting of 4 hidden layers and 300 neurons per layer (denoted as 4 × 300). To
+mitigate possible overfitting in training, we also introduce an ℓ2 regularization in (14) with a small penalty coefficient
+5 × 10−5. However, we note that we did not observe any signs of conventional overfitting during our numerical tests,
+and the additional regularization has a negligible impact on the DeepONet accuracy.
+4. Synthetic Ice-Sheet Problem
+In this section we apply our approach to a well-known benchmark in ice sheet modeling, the MISMIP problem
+[44]. We use this problem to explore how hyper-parameters affect the training of the DeepONet and the accuracy of the
+hybrid model.
+The problem geometry is defined by a marine ice stream that is partially floating. The ice domain is 640 km long
+and 80 km wide (Ω = [0, 640 km] × [0, 80 km]). The bed topography is provided in [44]. We consider an initial
+thickness (note that this is different from the one in [44]):
+H(x, y) = 100 m
+�3
+2 + 1
+2 tanh
+�400 km − x
+100 km
+��
+.
+We prescribe the normal velocity at the upstream boundary (x = 0 km) and lateral boundaries (y = 0 km and y = 80 km)
+to be zero, and free-slip conditions in the direction tangential to these boundaries. We prescribe stress-free conditions
+at the outlet boundary (x = 640 km). No boundary conditions are prescribed for the thickness equation, as there are
+no inflow boundaries. We use a constant mean basal friction field ¯β = 5000 Pa yr/m and a scaling factor a = 0.2 in
+(9). As described in Section 3.2, for each sample β from (9), we run the finite-element ice flow model for 100 years,
+using a constant forcing fH = 0.3 m/ yr, and compute the ice thickness H. We then use the thickness data to train the
+DeepONet.
+For ease of analysis, the mean squared error (MSE) and relative squared error (RSE), given as follows, are used to
+evaluate the DeepONet performance:
+eMS E = 1
+N
+N
+�
+i=1
+||ui − u∗
+i ||2,
+eRS E =
+�N
+i=1 ||ui − u∗
+i ||2
+�N
+i=1 ||u∗
+i ||2
+8
+
+Figure 3: The loss plots of DeepONet training for the MISMIP testcase with SSA model under different correlation lengths: (a) l = 80 km; (b) l = 40
+km; (c) l = 20 km. The simulation data associated with {βi}300
+i=21 is used as the training data while {βi}20
+i=1 is used as testing data. At the final epoch
+(300, 000), the training MSEs are 2.60 × 10−6, 1.03 × 10−5, and 3.33 × 10−5, respectively.
+Table 1: MISMIP test case with the SSA and MOLHO models. The mean square errors of the DeepONet training with different training dataset sizes
+under various correlation lengths l. The testing error is evaluated on the same size of testing data of {βi}20
+i=1.
+l = 80 km
+l = 40 km
+l = 20 km
+Training dataset
+SSA
+MOLHO
+SSA
+MOLHO
+SSA
+MOLHO
+{βi}200
+i=21
+7.37 × 10−5
+7.31 × 10−5
+2.09 × 10−4
+1.77 × 10−4
+2.92 × 10−4
+3.04 × 10−4
+{βi}300
+i=21
+4.84 × 10−5
+4.72 × 10−5
+1.32 × 10−4
+1.47 × 10−4
+2.54 × 10−4
+2.11 × 10−4
+{βi}400
+i=21
+3.62 × 10−5
+4.09 × 10−5
+1.05 × 10−4
+0.97 × 10−4
+2.28 × 10−4
+1.97 × 10−4
+where ui and u∗
+i denote the prediction and reference values, respectively, and N is the number of data.
+Table 1 shows that DeepONet converges well with respect to the size Nβ of the training dataset and that using more
+training data enhances generalization capacity. The table also shows the impact of the correlation length magnitude on
+the approximation accuracy. As expected, in order to maintain the same level of accuracy, larger training datasets are
+required for smaller correlation lengths. Another important piece of information from the table is that DeepONets can
+approximate with a similar accuracy both the lower-fidelity SSA model and higher-fidelity MOLHO model.
+Taking the case with {βi}300
+i=21 as an example, the curves of training and testing losses are plotted in Fig. 3. The result
+shows that the DeepONet models converge stably for all three different correlation lengths, and the prediction accuracy
+on testing cases reaches a plateau after 50000 epochs. It is observed that the generalization gap2 remains nearly the
+same for the data with different correlation lengths when the size of the training dataset is fixed.
+The trained DeepONet model Gθ(β, H j)(x) is able to predict the velocity field ¯uNN(x) at any time t j for the given
+friction field β and thickness field H j. The DeepONet predictions at t = 99 yr for an exemplary training case
+corresponding to correlation lengths l = 20, 40, 80 km are presented in Fig. 4. The results in Fig. 4(g)-(i) show that
+more localized features appear in the velocity solution with a smaller correlation length, e.g., the case of l = 20 km.
+The RSEs between the predicted and reference velocity fields at t = 99 yr are 3.61 × 10−4, 2.57 × 10−3, and 7.96 × 10−3
+for the correlation lengths l = 80, 40, and 20 km, respectively, indicating the excellent learning capacity of DeepONet
+on the training velocity fields.
+To examine the generalization performance, we test the trained DeepONet on an unseen test case (β6) with l = 20
+km at two different time instances, as shown in Fig. 5. The relative squared errors at t = 18 and t = 94 yr are 5.67×10−2
+and 4.88 × 10−2, respectively. We observe that the DeepONet accuracy does not depend significantly on the time t at
+which the input thickness is evaluated.
+Lastly, we investigate the effect of mesh resolution on the DeepONet performance. We use the same 4 × 300
+DeepONet architecture as before, but we change the size of the input layer to accommodate input data of different
+resolutions. Table 2 presents the relative squared errors of the DeepONet model against the training dataset {βi}300
+i=21
+and testing dataset {βi}20
+i=1 under different mesh resolutions of 36 × 9, 60 × 15, and 100 × 25. Overall, the accuracy of
+2The difference between a model’s performance on training data and its performance on unseen testing data drawn from the same distribution.
+9
+
+10 -2
+training
+training
+ training
+(a)
+(b)
+(c)
+testing
+testing
+testing
+10 -3
+10 ~3
+10 -3
+SSOL
+loss
+MSE loss
+MSE
+MSI
+10 -4
+10 -4
+10°
+10 ~5
+10 -5
+10 ~5
+0
+500
+1000
+1500
+2000
+2500
+3000
+0
+500
+1000
+1500
+2000
+2500
+3000
+0
+500
+1000
+1500
+2000
+2500
+3000
+epoch (×100)
+epoch (×100)
+epoch (×100)Figure 4: The DeepONet prediction at t = 99 yr for an exemplary training case (β25) corresponding to correlation lengths l = 20, 40, 80 km. (a) - (c):
+log10(β); (d) - (f): the thickness H; (g)-(i): The modulus of the predicted velocity |¯uNN|. The relative squared errors are 3.61 × 10−4, 2.57 × 10−3, and
+7.96 × 10−3 for the correlation lengths l = 80, 40, and 20 km, respectively. The simulation data associated with {βi}300
+i=21 is used as the training data.
+DeepONet remains comparable for the various mesh resolutions. The training time for DeepONet under different mesh
+resolutions is also provided in Table 2, indicating a linear relation between the training time and the size of meshes (i.e.,
+the size of the dataset).
+Table 2: MISMIP testcase with the SSA model under different mesh resolutions. The relative squared errors of the DeepONet model against the
+training dataset {βi}300
+i=21 and testing dataset {βi}20
+i=1 under various correlation lengths l. The clock time used to train DeepONet based on a given mesh
+resolution remains the same for different correlation lengths.
+l = 80 km
+l = 40 km
+l = 20 km
+Mesh resolution
+Time
+training
+testing
+training
+testing
+training
+testing
+36 × 9
+1.13 hrs
+2.97 × 10−4
+8.02 × 10−3
+0.90 × 10−3
+2.70 × 10−2
+5.28 × 10−3
+6.19 × 10−2
+60 × 15
+2.80 hrs
+3.03 × 10−4
+5.70 × 10−3
+1.14 × 10−3
+1.99 × 10−2
+5.48 × 10−3
+4.25 × 10−2
+100 × 25
+7.24 hrs
+4.55 × 10−4
+4.02 × 10−3
+1.56 × 10−3
+2.82 × 10−2
+5.44 × 10−3
+4.60 × 10−2
+5. Hybrid Modeling of Humboldt Glacier
+In this section we consider the Humboldt glacier, one of the largest glaciers in Greenland. In Fig. 6, we report the
+Humboldt bed topography, ice surface elevation and ice thickness obtained from observations, refer to [29] for details
+on how these fields are collected and processed. These fields will be use to determine the problem geometry and the
+initial ice thickness H0. The mean value ¯β of the basal friction in (9) is obtained with a PDE-constrained optimization
+approach [35] where the mismatch between the computed and observed surface velocities are minimized. Fig. 7 shows
+¯β together with a couple of samples of the basal friction from (9).
+Similarly to the MISMIP case, for each sample of β, obtained from (9) with correlation length l = 50 km and scaling
+a = 0.2, the ice finite element flow model is run forward in time for 100 yr, using a climate forcing generated according
+to the Representative Concentration Pathway 2.6 (see [29] for the problem definition and the data used including the
+mean basal friction ¯β). The collected thickness and velocity simulation data are used to train the DeepONet model.
+5.1. DeepONet Training
+We first evaluate the performance of DeepONet for different ice approximation models (MOLHO and SSA). Figs.
+8a-c present the plots of training and testing errors corresponding to three different DeepONet cases, i.e., training with
+1) simulation data obtained from the SSA ice model, 2) simulation data obtained from the MOLHO ice model, and 3)
+simulation data obtained from the MOLHO ice model together with the self-adaptive scheme described in (14)-(16). At
+10
+
+(b)
+3.0
+3.5
+4.0
+4.5
+50 
+50 
+50
+0 -
+0-
+0 -
+0
+100
+100
+0
+100
+600
+200
+300
+400
+500
+600
+0
+200
+300
+400
+500
+600
+200
+300
+400
+500
+[km] 
+[km]
+[km] 
+(e)
+100
+200
+300
+50 -
+50
+50
+0-
+0:
+01
+500
+100
+200
+300
+600
+0
+100
+200
+300
+400
+500
+600
+0
+100
+200
+300
+400
+600
+0
+400
+500
+[km]
+[km]
+[km]
+20 
+(h)
+40 
+50
+50
+50
+0
+0
+100
+0
+200
+300
+400
+500
+600
+100
+200
+300
+400
+500
+600
+0
+100
+200
+300
+400
+500
+0
+600
+[km]
+[km] 
+[km] (a)
+(b) 
+(c)
+(d)
+(e)
+Figure 5: The DeepONet prediction for an exemplary test case (β6) with the correlation length l = 20 km: (a) log10(β); (b) and (c) are the maps
+of reference velocity modulus |¯u| at t = 18 and t = 94 yr, respectively; (d) and (e) are the point-wise errors of the velocity modulus between the
+reference and DeepONet predictions |¯u − ¯uNN| at t = 18 and t = 94 yr, respectively, where the corresponding relative squared errors are 5.67 × 10−2
+and 4.88 × 10−2.
+Figure 6: Observations for Humboldt glacier for the initial year 2007. Left: bed topography [m] from [45], Center: ice surface elevation [m], Right:
+thickness [m]. Additional details on the collection and processing of these data can be found in [29].
+the last epoch (300, 000), the relative squared errors of these three DeepONet models on the testing data are 3.74 × 10−3,
+3.59 × 10−3, and 2.16 × 10−3, respectively. The comparison of results in Figs. 8a and b shows that training DeepONet
+with MOLHO simulation data yields higher prediction accuracy than the low-order SSA data, which is consistent with
+our observation for the MISMIP testcase. In the following, we will only consider the MOLHO model, given that it
+better describes the ice sheet dynamics compared to the SSA model, and it can be well approximated by our DeepONet
+model. We also observe in Fig. 8c that the employment of the self-adaptive weighting scheme significantly improves
+the training and testing performance in the Humboldt glacier testcase, reducing the testing error by 40%.
+We further study the impact of using self-adaptive weights in Figs. 9 and 10, where we show the prediction errors
+at different β samples for different choices of adaptive weighting schemes. In Fig. 9 we report the results for a β sample
+taken from the training dataset, whereas in Fig. 10 we consider a sample from the testing dataset. In both cases, the
+DeepONet model trained with the self-adaptive weighting scheme with m(λ) = λ4 yields the best performance, which
+is consistent with the results in Fig. 8. The self-adaptive weighting scheme especially helps mitigate the prediction
+errors in the interior of the domain and the region at the outlet (i.e., northwest) region. Given the improved prediction,
+in the following sections we will present the DeepONet models trained with the self-adaptive weighting scheme with
+m(λ) = λ4.
+11
+
+0
+500
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+[km]1000
+2000
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+[km]1000
+2000
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+[km]0
+20
+40
+50
+0
+0
+100
+200
+300
+400
+500
+600
+[km]0
+2
+4
+50
+0
+0
+100
+200
+300
+400
+500
+600
+[km]0
+20
+40
+50
+0
+0
+100
+200
+300
+400
+500
+600
+[km]0
+2
+4
+50
+0
+0
+100
+200
+300
+400
+500
+600
+[km]3.0
+3.5
+4.0
+4.5
+50
+0
+0
+100
+200
+300
+400
+500
+600
+[km]Figure 7: Mean value of the basal friction ¯β (left) and two samples of the basal friction using (9).Units: [Pa yr / m].
+Figure 8: The loss plots of DeepONet training for different ice models: (a) SSA; (b) MOLHO; (c) MOLHO with self-adaptive (SA) weighting
+scheme (m(λ) = λ4). The simulation data associated with {βi}300
+i=21 is used as the training data while {βi}20
+i=1 is used as testing data. At the final epoch
+(300, 000), the corresponding testing MSEs of these three DeepONet models are 4.23 × 10−6, 4.02 × 10−6, and 2.42 × 10−6, indicating the enhanced
+generalization by using the self-adaptive weighting scheme.
+5.2. Hybrid model: DeepONet embedded in finite element solver
+In this section we study the accuracy and cost of the hybrid ice flow model with respect to the finite element model.
+As explained in Section 3, the hybrid model approximates at each time step the operator G with the trained DeepONet
+model Gθ. Because the DeepONet approximation is much cheaper than the finite element approximation, the hybrid
+solver is significantly more efficient than a traditional finite element solver. We study the approximation properties and
+computational savings of using the hybrid model for computing the evolution of the Humboldt glacier thickness over
+time, and then focus in particular on how well the hybrid model can approximate the glacier mass change. We finally
+show how the hybrid model can be used to produce statistics of the glacier mass loss.
+5.2.1. Thickness evolution over time
+In this section we compare the ice thickness computed with the finite-element model, and with the hybrid model.
+We take 8 samples of beta (not used to train the DeepONet) from distribution (9). We then run the finite-element and
+the hybrid models for 150 years. Results of the comparison are shown in Fig. 11. The plot on the left shows the
+variability of the thickness, over time, with respect to the samples of β, using the same model. The plot on the right
+shows the relative difference between the ice thickness computed with the finite-element model and the one computed
+with the hybrid model. The relative differences due to the models are significantly smaller than the variability with
+respect to the different samples. Moreover, for t < 100 years, which is the period used for training the DeepONet, the
+relative differences between the two models are small, 3% at most. Differences increase in the extrapolation region
+(100 − 150 years), however the increase is mostly linear, which signifies robustness of the hybrid approximation.
+12
+
+20000 40000 60000
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+-100
+[km]20000
+40000
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+-100
+[km]10-3
+10-3
+10-3
+training
+ training
+ training
+testing
+- testing
+testing
+10 -4
+10-4
+10 -4
+loss
+loss
+MSE
+MSE
+MSE
+10-5
+10-5
+10 -6
+10~6
+10-6
+0
+500
+1000
+1500
+2000
+2500
+3000
+0
+500
+1000
+1500
+2000
+2500
+3000
+0
+500
+1000
+1500
+2000
+2500
+3000
+epoch (×100)
+epoch (×100)
+epoch (×100)20000
+40000
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+[km](a)	𝛽
+(b) 𝐻
+(d) |𝒖& − 𝒖&!!|
+(c) |𝒖&|
+(e) |𝒖& − 𝒖&!!|
+(f) |𝒖& − 𝒖&!!|
+Figure 9: The DeepONet prediction for an exemplary training case (β23) at t = 99 yr: (a) basal friction β in [Pa yr/m]; (b) Thickness H in [m]; (c) the
+reference velocity modulus |¯u| in [m/yr]; (d) the point-wise errors ([m/yr]) of the DeepONet; (e) the point-wise errors ([m/yr]) of the DeepONet
+trained with self-adaptive weighting scheme m(λ) = λ2; (f) the point-wise errors ([m/yr]) of the DeepONet trained with self-adaptive weighting
+scheme m(λ) = λ4. The relative squared errors corresponding to (d)-(f) are 6.29 × 10−4, 5.00 × 10−4, and 4.18 × 10−4, respectively.
+5.2.2. Glacier mass-loss over time
+As explained in the introduction, the mass change of a glacier over the years is one of the most important quantities
+of interest in ice sheet modeling because it directly affects the net amount of water added to the oceans and hence the
+potential sea level rise. In this work, we compute the mass of the glacier only considering the ice that is above flotation,
+because changes in the mass of ice that is afloat do not affect the sea level; for details, see [46]. In Fig. 12, we show
+the mass change (mass at time t minus mass at time t0 = 0) as a function of time for the same samples of the basal
+friction used for Fig. 11. While there are some small discrepancies between the finite-element and hybrid models, the
+two model are in very good agreement overall, especially in the first 100 years, which are within the period of ice
+simulation data used for training the DeepONet model, with the largest difference being ≈ 10%. We also note that the
+qualitative behaviors of the two models are very similar in the extrapolation region (100 − 150 years).
+5.2.3. Computing statistics on quantity of interest using Hybrid model
+Finally, we demonstrate how the hybrid model can be effectively used to compute statistics of the glacier mass
+change. We take unseen 2000 samples of β from distribution (9), and run both the hybrid model and the finite-element
+model for 100 years and 150 years for each sample. We then compute the glacier mass change (using only the ice
+above flotation) and show histograms (Fig. 13) of the mass-change distribution, comparing the differences between the
+reference finite element model and the hybrid model. The results demonstrate that the hybrid model can accurately
+compute the statistics of mass change, and therefore has the potential to be used to significantly accelerate the uncertainty
+quantification analysis for sea-level projections due to ice-sheet mass change. The discrepancies between the results
+computed with the reference finite element model and the hybrid model are likely small in practical applications, and,
+if needed, they can be corrected using a multifidelity approach where the hybrid model is used as low-fidelity model
+and the finite-element model as the high-fidelity model; see e.g., [47]. The figure also shows the impact of training
+the DeepONets using self-adaptive weights and uniform weights. It seems that the use of self-adaptive weights in
+training can lead to a small bias in the hybrid modeling to underestimate the mass loss. More investigation is needed
+13
+
+20000
+40000
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+-100
+[km]0
+1000
+2000
+3000
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+[km]0
+250
+500
+750
+1000
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+[km]0
+5
+10
+15
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+[km]0
+5
+10
+15
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+[km]0
+5
+10
+15
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+[km](a)	𝛽
+(b) 𝐻
+(d) |𝒖& − 𝒖&!!|
+(c) |𝒖&|
+(e) |𝒖& − 𝒖&!!|
+(f) |𝒖& − 𝒖&!!|
+Figure 10: The DeepONet prediction for an exemplary testing case (β6) at t = 92 yr: (a) β in [Pa yr/m]; (b) Thickness H in [m]; (c) the reference
+velocity modulus |¯u| in [m/yr]; (d) the point-wise errors ([m/yr]) of the DeepONet; (e) the point-wise errors ([m/yr]) of the DeepONet trained
+with self-adaptive weighting scheme m(λ) = λ2; (f) the point-wise errors ([m/yr]) of the DeepONet trained with self-adaptive weighting scheme
+m(λ) = λ4. The relative squared errors corresponding to (d)-(f) are 3.36 × 10−3, 7.66 × 10−3, and 2.88 × 10−3, respectively.
+to understand the cause of this bias and to confirm that this phenomenon is general and not specific to this particular
+glacier and the settings we used.
+5.2.4. Computational saving using Hybrid model
+Table 3 shows the computational times for running the finite element and hybrid models, when using the MOLHO
+approximation. Overall, we see almost a 5-fold speedup when using the hybrid model over the finite element model.
+The total computational costs includes time to allocate memory, initialize data and for intput/output. If we only consider
+the time to solve the coupled model system (10), we have a 11-fold speedup. The evaluation of the DeepONet takes
+only 4.99s of the 9.46s taken to solve the hybrid model. We believe that there is margin for improvement in real
+applications. While we trained the DeepONet on GPUs, our prototype FEniCS code can only run on CPUs, therefore
+the results in this section refers to simulations run on CPUs. We expect that the DeepONet would benefit more from
+running on GPUs than a the classic finite element model, because it is still challenging to efficiently run implicit
+nonlinear solvers on GPUs (see [48], in the context of a production ice sheet model), whereas modern machine learning
+code can take full advantage of GPUs. The cost of the finite element solver scales with increasing mesh resolutions
+whereas DeepONet can maintain the same level of predictive accuracy and efficiency for various mesh resolutions (as
+shown in Table 2). Moreover, we expect that an hybrid model would be significantly more efficient, compared to the
+corresponding finite element code, when higher-order approximations of the velocity solver are considered. In fact, a
+Stokes solver can be an order of magnitude slower than the MOLHO model considered here, whereas we expect the
+cost of the DeepONet to be fairly independent from the model chosen for the velocity solver, as we observed when
+comparing the SSA and the MOLHO DeepONet models.
+14
+
+20000 40000 60000
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+-100
+[km]0
+1000
+2000
+3000
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+[km]0
+5
+10
+15
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+[km]0
+5
+10
+15
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+[km]0
+5
+10
+15
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+[km]0
+250
+500
+750
+1000
+-1050
+-1100
+-1150
+-1200
+-1250
+-1300
+-1350
+-600
+-500
+-400
+-300
+-200
+[km]0
+50
+100
+150
+Time [yr]
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+||H
+ - H mean || 2 / ||H mean || 2
+10
+11
+12
+13
+14
+15
+16
+17
+0
+50
+100
+150
+Time [yr]
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+||H Hyb  - H FE|| 2 / ||H FE|| 2
+10
+11
+12
+13
+14
+15
+16
+17
+Figure 11: Left: relative difference over time between the ice thickness Hβ associated to sample β and the mean ice thickness. Right: relative
+difference over time between the ice thickness computed with the finite element model and the hybrid model.
+.
+0
+50
+100
+150
+Time [yr]
+-2000
+-1800
+-1600
+-1400
+-1200
+-1000
+-800
+-600
+-400
+-200
+0
+Mass change [gigatons]
+Glacier mass change [gigatons], FE model
+10
+11
+12
+13
+14
+15
+16
+17
+0
+50
+100
+150
+Time [yr]
+-2000
+-1800
+-1600
+-1400
+-1200
+-1000
+-800
+-600
+-400
+-200
+0
+Mass change [gigatons]
+Glacier mass change [gigatons], Hyb model
+10
+11
+12
+13
+14
+15
+16
+17
+Figure 12: Mass change [gigatons] over time for different samples of the basal friction coefficient computed using the finite element model (left) and
+the hybrid model (right).
+6. Summary
+We developed a hybrid model for ice sheet dynamics by combining a classic finite-element discretization for the
+ice thickness equation with a DeepONet approximation of the ice momentum equation, which is the most expensive
+part of a traditional ice sheet computational model. A distinctive feature of our hybrid model is that it can handle
+high-dimensional parameter spaces, which is critical for accounting for the uncertainty in parameter fields like the basal
+friction coefficient. We demonstrated that the hybrid model can accurately compute the dynamics of a real glacier an
+order of magnitude faster than a traditional ice sheet model. As explained in Section 5.2.4, the computational savings
+are likely to be larger when using production ice-sheet codes. Moreover, we showed that the hybrid model produces
+accurate statistics of the mass loss of the Humboldt glacier over a period of one hundred years and can therefore be used
+to accelerate uncertainty quantification analysis of sea-level projections due to ice sheets. Future research directions
+include scaling up our approach to target larger problems, such as using higher-resolution data or targeting the evolution
+Table 3: Comparison of computational time per sample between the finite-element and hybrid models when using MOLHO model for the velocity
+solver. The provided average times are estimated based on 50 simulations with different friction fields.
+Times per sample (s)
+Total
+Solving Eq. (10)
+Finite-element model
+123.30
+105.20
+Hybrid model
+24.15
+9.46
+Ratio
+19.59 %
+8.99%
+15
+
+Figure 13: Histogram of the distribution of the Humboldt mass change over a period of 100 years and 150 years. The histogram has been generated
+by simulating the mass change corresponding to 2000 samples from distribution (9). The DeepONet used for the results on the right (b and d) has
+been trained using self-adaptive weights, whereas uniform weights have been used for the results on the left (a and c).
+of the entire Greenland ice sheet, and performing uncertainty quantification analysis using the hybrid model.
+7. Acknowledgements
+The authors wish to thank L. Lu for helpful discussions, K. C. Sockwell for co-developing the ice-sheet code, and
+T. Hillebrand for generating the Humboldt grid.
+The work is supported by the U.S. Department of Energy, Advanced Scientific Computing Research program, under
+the Physics-Informed Learning Machines for Multiscale and Multiphysics Problems (PhILMs) project and under the
+SciDAC-BER Probabilistic Sea Level Projections from Ice-Sheets and Earth System Models (ProSPect) partnership.
+The authors also acknowledge the support from the UMII Seed Grant and the Minnesota Supercomputing Institute
+(MSI) at the University of Minnesota for providing resources that contributed to the research results reported within
+this paper.
+Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and
+Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S.
+Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.
+Pacific Northwest National Laboratory (PNNL) is a multi-program national laboratory operated for the U.S.
+Department of Energy (DOE) by Battelle Memorial Institute under Contract No. DE-AC05-76RL01830. The
+computational work was performed with resources from PNNL Institutional Computing at Pacific Northwest National
+Laboratory.
+This paper describes objective technical results and analysis. Any subjective views or opinions that might be
+expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States
+16
+
+400
+400
+IFEM
+IFEM
+Hybrid
+IHybrid
+350
+350
+300
+300
+250
+250
+200
+200
+150
+150
+100
+100
+50
+50
+0
+0
+-1500
+-1000
+-500
+0
+-1500
+-1000
+-500
+0
+400
+400
+FEM
+IFEM
+Hybrid
+IHybrid
+350
+350
+300
+300
+250
+250
+200
+200
+150
+150
+100
+100
+50
+50
+0
+0
+-2000
+-1500
+-1000
+-500
+0
+-2000
+-1500
+-1000
+-500
+0Government.
+17
+
+References
+[1] I. P. on Climate Change, Climate Change 2013 – The Physical Science Basis: Working Group I Contribution to the Fifth Assessment Report of
+the Intergovernmental Panel on Climate Change, Cambridge University Press, 2014.
+[2] A. Levermann, R. Winkelmann, T. Albrecht, H. Goelzer, N. R. Golledge, R. Greve, P. Huybrechts, J. Jordan, G. Leguy, D. Martin, et al.,
+Projecting Antarctica’s contribution to future sea level rise from basal ice shelf melt using linear response functions of 16 ice sheet models
+(LARMIP-2), Earth System Dynamics 11 (1) (2020) 35–76. doi:10.5194/esd-11-35-2020.
+[3] T. L. Edwards, S. Nowicki, B. Marzeion, R. Hock, H. Goelzer, H. Seroussi, N. C. Jourdain, D. Slater, F. Turner, C. J. Smith, C. M. McKenna,
+E. Simon, A. Abe-Ouchi, J. M. Gregory, E. Larour, W. H. Lipscomb, A. J. Payne, A. Shepherd, C. Agosta, P. Alexander, T. Albrecht,
+B. Anderson, X. Asay-Davis, A. Aschwanden, A. Barthel, A. Bliss, R. Calov, C. Chambers, N. Champollion, Y. Choi, R. Cullather, J. Cuzzone,
+C. Dumas, D. Felikson, X. Fettweis, K. Fujita, B. K. Galton-Fenzi, R. Gladstone, N. R. Golledge, R. Greve, T. Hattermann, M. J. Hoffman,
+A. Humbert, M. Huss, P. Huybrechts, W. Immerzeel, T. Kleiner, P. Kraaijenbrink, S. Le clec’h, V. Lee, G. R. Leguy, C. M. Little, D. P. Lowry,
+J.-H. Malles, D. F. Martin, F. Maussion, M. Morlighem, J. F. O’Neill, I. Nias, F. Pattyn, T. Pelle, S. Price, A. Quiquet, V. Radi´c, R. Reese, D. R.
+Rounce, M. Ruckamp, A. Sakai, C. Shafer, N.-J. Schlegel, S. Shannon, R. S. Smith, F. Straneo, S. Sun, L. Tarasov, L. D. Trusel, J. V. Breedam,
+R. van de Wal, M. van den Broeke, R. Winkelmann, H. Zekollari, C. Zhao, T. Zhang, T. Zwinger, Projected land ice contributions to 21st
+century sea level rise, Nature 593 (2021) 74–82. doi:10.1038/s41586-021-03302-y.
+[4] T. Isaac, G. Stadler, O. Ghattas, Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction
+for large-scale problems, with application to flow of the antarctic ice sheet, Journal of Computational Physics 296 (2015) 348–368. doi:
+https://doi.org/10.1016/j.jcp.2015.04.047.
+URL https://www.sciencedirect.com/science/article/pii/S0021999115003046
+[5] D. J. Brinkerhoff, Variational inference at glacier scale, Journal of Computational Physics 459 (2022) 111095. doi:https://doi.org/10.
+1016/j.jcp.2022.111095.
+URL https://www.sciencedirect.com/science/article/pii/S0021999122001577
+[6] A. Aschwanden, M. A. Fahnestock, M. Truffer, D. J. Brinkerhoff, R. Hock, C. Khroulev, R. Mottram, S. A. Khan, Contribution of the greenland
+ice sheet to sea level over the next millennium, Science Advances 5 (6) (2019). arXiv:https://www.science.org/doi/pdf/10.1126/
+sciadv.aav9396, doi:10.1126/sciadv.aav9396.
+URL https://www.science.org/doi/abs/10.1126/sciadv.aav9396
+[7] K. Bulthuis, M. Arnst, S. Sun, F. Pattyn, Uncertainty quantification of the multi-centennial response of the antarctic ice sheet to climate change,
+The Cryosphere 13 (4) (2019) 1349–1380. doi:10.5194/tc-13-1349-2019.
+URL https://tc.copernicus.org/articles/13/1349/2019/
+[8] F. Lehner, C. Deser, N. Maher, J. Marotzke, E. M. Fischer, L. Brunner, R. Knutti, E. Hawkins, Partitioning climate projection uncertainty with
+multiple large ensembles and cmip5/6, Earth System Dynamics 11 (2) (2020) 491–508. doi:10.5194/esd-11-491-2020.
+URL https://esd.copernicus.org/articles/11/491/2020/
+[9] E. Bueler, J. Brown, Shallow shelf approximation as a “sliding law” in a thermomechanically coupled ice sheet model, Journal of Geophysical
+Research 114 (F3) (2009) 1–21.
+[10] D. N. Goldberg, A variationally derived, depth-integrated approximation to a higher-order glaciological flow model, Journal Of Glaciology
+57 (201) (2011) 157–170.
+[11] M. Perego, M. Gunzburger, J. Burkardt, Parallel finite-element implementation for higher-order ice-sheet models, Journal of Glaciology
+58 (207) (2012) 76–88. doi:10.3189/2012JoG11J063.
+[12] W. Leng, L. Ju, M. Gunzburger, S. Price, T. Ringler, A parallel high-order accurate finite element nonlinear Stokes ice sheet model and
+benchmark experiments, Journal of Geophysical Research 117 (F1) (Jan. 2012).
+[13] E. Larour, H. Seroussi, M. Morlighem, E. Rignot, Continental scale, high order, high spatial resolution, ice sheet modeling using the Ice Sheet
+System Model (ISSM), Journal of Geophysical Research 117 (F1) (2012) F01022. doi:10.1029/2011JF002140.
+[14] S. L. Cornford, D. F. Martin, D. T. Graves, D. F. Ranken, A. M. Le Brocq, R. M. Gladstone, A. J. Payne, E. G. Ng, W. H. Lipscomb, Adaptive
+mesh, finite volume modeling of marine ice sheets, Journal of Computational Physics 232 (2013) 529–549.
+[15] O. Gagliardini, T. Zwinger, F. Gillet-Chaulet, G. Durand, L. Favier, B. De Fleurian, R. Greve, M. Malinen, C. Martin, P. Råback, J. Ruokolainen,
+M. Sacchettini, M. Sch¨afer, H. Seddik, J. Thies, Capabilities and performance of Elmer/Ice, a new-generation ice sheet model, Geoscientific
+Model Development 6 (4) (2013) 1299–1318.
+[16] D. J. Brinkerhoff, J. V. Johnson, Data assimilation and prognostic whole ice sheet modelling with the variationally derived, higher order, open
+source, and fully parallel ice sheet model VarGlaS, The Cryosphere 7 (4) (2013) 1161–1184. doi:10.5194/tc-7-1161-2013.
+[17] I. K. Tezaur, M. Perego, A. G. Salinger, R. S. Tuminaro, S. Price, Albany/FELIX: a parallel, scalable and robust, finite element, first-order Stokes
+approximation ice sheet solver built for advanced analysis, Geoscientific Model Development 8 (2015) 1–24. doi:10.5194/gmd-8-1-2015.
+[18] M. J. Hoffman, M. Perego, S. F. Price, W. H. Lipscomb, T. Zhang, D. Jacobsen, I. Tezaur, A. G. Salinger, R. Tuminaro, L. Bertagna,
+Mpas-albany land ice (mali): a variable-resolution ice sheet model for earth system modeling using voronoi grids, Geoscientific Model
+Development 11 (9) (2018) 3747–3780. doi:10.5194/gmd-11-3747-2018.
+URL https://www.geosci-model-dev.net/11/3747/2018/
+[19] G. Jouvet, G. Cordonnier, B. Kim, M. L¨uthi, A. Vieli, A. Aschwanden, Deep learning speeds up ice flow modelling by several orders of
+magnitude, Journal of Glaciology (2021) 1–14doi:10.1017/jog.2021.120.
+[20] D. Brinkerhoff, A. Aschwanden, M. Fahnestock, Constraining subglacial processes from surface velocity observations using surrogate-based
+bayesian inference, Journal of Glaciology 67 (263) (2021) 385–403. doi:10.1017/jog.2020.112.
+[21] P. J. Lu Lu, Z. Z. Guofei Pang, G. E. Karniadakis, Learning nonlinear operators via deeponet based on the universal approximation theorem of
+operators, Nature Machine Intelligence 3 (2021) 218––229.
+[22] C. Lin, M. Maxey, Z. Li, G. E. Karniadakis, A seamless multiscale operator neural network for inferring bubble dynamics, Journal of Fluid
+Mechanics 929 (2021).
+18
+
+[23] R. Ranade, K. Gitushi, T. Echekki, Generalized joint probability density function formulation inturbulent combustion using deeponet, arXiv
+preprint arXiv:2104.01996 (2021).
+[24] S. Goswami, M. Yin, Y. Yu, G. Karniadakis, A physics-informed variational deeponet for predicting the crack path in brittle materials, arXiv
+preprint arXiv:2108.06905 (2021).
+[25] P. C. Di Leoni, L. Lu, C. Meneveau, G. Karniadakis, T. A. Zaki, Deeponet prediction of linear instability waves in high-speed boundary layers,
+arXiv preprint arXiv:2105.08697 (2021).
+[26] M. Sharma Priyadarshini, S. Venturi, M. Panesi, Application of deeponet to model inelastic scattering probabilities in air mixtures, in: AIAA
+AVIATION 2021 FORUM, 2021, p. 3144.
+[27] T. Chen, H. Chen, Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to
+dynamical systems, IEEE Transactions on Neural Networks 6 (4) (1995) 911–917.
+[28] A. D. Back, T. Chen, Universal approximation of multiple nonlinear operators by neural networks, Neural Computation 14 (11) (2002)
+2561–2566.
+[29] T. R. Hillebrand, M. J. Hoffman, M. Perego, S. F. Price, I. M. Howat, The contribution of humboldt glacier, northern greenland, to sea-
+level rise through 2100 constrained by recent observations of speedup and retreat, The Cryosphere 16 (11) (2022) 4679–4700. doi:
+10.5194/tc-16-4679-2022.
+URL https://tc.copernicus.org/articles/16/4679/2022/
+[30] K. Cuffey, Paterson, The Physics of Glaciers, 4th Edition, Butterworth-Heinneman, Amsterdam, 2010.
+[31] T. Dias dos Santos, M. Morlighem, D. Brinkerhoff, A new vertically integrated mono-layer higher-order (molho) ice flow model, The
+Cryosphere 16 (1) (2022) 179–195. doi:10.5194/tc-16-179-2022.
+[32] L. W. Morland, I. R. Johnson, Steady motion of ice sheets, Journal of Glaciology 25 (92) (1980) 229–246.
+doi:10.3189/
+S0022143000010467.
+[33] M. Weis, R. Greve, K. Hutter, Theory of shallow ice shelves, Continuum Mechanics and Thermodynamics 11 (1) (1999) 15–50. doi:
+10.1007/s001610050102.
+[34] J. K. Dukowicz, S. F. Price, W. H. Lipscomb, Consistent approximations and boundary conditions for ice-sheet dynamics from a principle of
+least action, Journal of Glaciology 56 (197) (2010) 480–496. doi:10.3189/002214310792447851.
+[35] M. Perego, S. Price, G. Stadler, Optimal initial conditions for coupling ice sheet models to earth system models, Journal of Geophysical
+Research: Earth Surface 119 (9) (2014) 1894–1917.
+[36] M. Perego, Large-scale PDE-constrained Optimization for Ice Sheet Model Initialization, https://sinews.siam.org/Details-Page/
+large-scale-pde-constrained-optimization-for-ice-sheet-model-initialization (2022).
+[37] N. Petra, J. Martin, G. Stadler, O. Ghattas, A computational framework for infinite-dimensional bayesian inverse problems, part ii: Stochastic
+newton mcmc with application to ice sheet flow inverse problems, SIAM Journal on Scientific Computing 36 (4) (2014) A1525–A1555.
+doi:10.1137/130934805.
+[38] M. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes, G. N. Wells, The fenics project version
+1.5, Archive of Numerical Software 3 (100) (2015).
+[39] S. Balay, W. Gropp, L. C. McInnes, B. F. Smith, Petsc, the portable, extensible toolkit for scientific computation, Argonne National Laboratory
+2 (17) (1998).
+[40] J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, et al.,
+Jax: composable transformations of python+ numpy programs, Version 0.2 5 (2018) 14–24.
+[41] L. McClenny, U. Braga-Neto, Self-adaptive physics-informed neural networks using a soft attention mechanism, arXiv preprint
+arXiv:2009.04544 (2020).
+[42] S. Goswami, A. Bora, Y. Yu, G. E. Karniadakis, Physics-informed neural operators, arXiv preprint arXiv:2207.05748 (2022).
+[43] K. Kontolati, S. Goswami, M. D. Shields, G. E. Karniadakis, On the influence of over-parameterization in manifold based surrogates and deep
+neural operators, arXiv preprint arXiv:2203.05071 (2022).
+[44] S. Cornford, H. Seroussi, X. Asay-Davis, R. Arthern, C. Borstad, J. Christmann, T. Dias dos Santos, J. Feldmann, D. Goldberg, M. Hoffman,
+A. Humbert, T. Kleiner, G. Leguy, W. Lipscomb, N. Merino, G. Durand, M. Morlighem, D. Polllard, M. R¨uckamp, H. Yu, Results of the third
+Marine Ice Sheet Model Intercomparison Project (MISMIP+), The Cryosphere Discussions (2020) 1–26doi:10.5194/tc-2019-326.
+[45] M. e. a. Morlighem, Icebridge bedmachine greenland, version 3 (2017). doi:10.5067/2CIX82HUV88Y.
+URL https://nsidc.org/data/IDBMG4/versions/3
+[46] H. Goelzer, V. Coulon, F. Pattyn, B. de Boer, R. van de Wal, Brief communication: On calculating the sea-level contribution in marine ice-sheet
+models, The Cryosphere 14 (3) (2020) 833–840. doi:10.5194/tc-14-833-2020.
+URL https://tc.copernicus.org/articles/14/833/2020/
+[47] B. Peherstorfer, K. Willcox, M. Gunzburger, Optimal model management for multifidelity monte carlo estimation, SIAM Journal on Scientific
+Computing 38 (5) (2016) A3163–A3194. arXiv:https://doi.org/10.1137/15M1046472, doi:10.1137/15M1046472.
+URL https://doi.org/10.1137/15M1046472
+[48] J. Watkins, M. Carlson, K. Shan, I. Tezaur, M. Perego, L. Bertagna, C. Kao, M. J. Hoffman, S. F. Price, Performance portable ice-sheet
+modeling with mali, arXiv preprint arXiv:2204.04321 (2022).
+19
+

6dE2T4oBgHgl3EQf7QhB/content/tmp_files/2301.04207v1.pdf.txt

@@ -0,0 +1,2567 @@
+Vehicle Utilization in Hub Network Design:
+Exploiting Economies of Scale in Transportation
+Mohammad Saleh Farham
+Lazaridis School of Business & Economics, Wilfrid Laurier University [email protected]
+Borzou Rostami
+Alberta School of Business, University of Alberta, [email protected]
+Michael Haughton
+Lazaridis School of Business & Economics, Wilfrid Laurier University [email protected]
+We study a vehicle-based hub network design problem (HNDPv) with the main applications in freight
+distribution and parcel delivery systems, where the economies of scale stem from the effective utilization
+of vehicles that move consolidated freight. The HNDPv is a generalization of the classical single allocation
+hub location problem, in which the transportation costs are stepwise functions of the number (and type) of
+vehicles that move the demand. We present the quadratic mixed-integer programming formulation of the
+problem and its linear reformulation. Exploiting the special structures of the linearized model, we develop
+a branch-and-cut method based on Benders decomposition with solely feasibility subproblems. We derive
+closed-form solutions for the extreme rays of the feasibility subproblems that improve the efficiency of
+the proposed algorithm through generating stronger feasibility cuts. We also address the HNDPv under
+demand uncertainty and show the flexibility of our solution methodology in handling the stochastic variant
+of the problem. To evaluate the efficiency of our models and solution approaches, we perform extensive
+computational experiments on uncapacitated and capacitated instances of the problem derived from the
+classical Australian Post dataset. The results show a considerable advantage of using HNDPv compared
+to the classical HLP with constant discount factors in terms of vehicle utilization and total transportation
+costs. Our computational experiments also demonstrate the efficiency of our proposed solution method in
+solving large-scale problem instances.
+Key words : Hub location problem; vehicle utilization; economies of scale; demand uncertainty; Benders
+decomposition
+1.
+Introduction
+Consolidation-based freight transportation is used for systems where several freight loads
+of different demand nodes are aggregated to be transported in a less costly manner. When
+the direct shipments between the origin and destination of demands are not economically
+1
+arXiv:2301.04207v1  [math.OC]  10 Jan 2023
+
+2
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+justifiable or even feasible, such systems provide a profitable balance between economy-of-
+scale-based costs and high service quality.
+Postal and parcel delivery companies, less-than-truckload (LTL) motor carriers, railroads
+or maritime liner navigation companies, and air/land or water/land-based intermodal car-
+riers use consolidation centers, called hub facilities, to centralize commodity handling and
+sorting operations, reduce setup costs, and achieve economies of scale on transportation
+costs by consolidating flows. This builds a hierarchical network called a hub network, where
+at the access-level, individual demand nodes connect to the hubs, and at the hub-level,
+interconnected hubs send and receive consolidated flows. The objective is to minimize
+the cost of locating hubs and transporting origin-destination (OD) flows on access and
+hub-level links.
+Economies of scale in freight transportation networks are directly related to the level at
+which hub/vehicle capacities are utilized to handle/transport large volumes of loads. At
+the hub-level, loads are consolidated and can be moved in bulks. Therefore, vehicles that
+travel on the inter-hub links are commonly large and are made to transport high-volume
+of loads over long distances efficiently (e.g., cargo jets, trains with freight wagons, etc.).
+Vehicles that are operated at the access-level are different. In a postal delivery system,
+for instance, small or medium-size trucks are used to transport parcels from hubs (e.g.,
+airports) to non-hubs (e.g., regional collection/distribution centers). Although the cost of
+using inter-hub vehicles might be larger, they are capable of moving bulks of consolidated
+freight more efficiently. Therefore, properly utilizing the capacity of inter-hub vehicles loads
+to a smaller unit transportation cost on hub arcs compared to the access arcs. This enables
+transportation carriers to exploit economies of scale and achieve lower transportation costs
+by consolidating loads into larger vehicles.
+In this paper, we consider vehicle-related decisions and utilization costs in the hub
+network design problem to determine the resources required to route the flow through
+the network. More specifically, we study the vehicle-based hub network design problem
+(HNDPv) with the single-assignment strategy in which each demand node is assigned to
+precisely one hub. The objective is to select hub nodes, assign demand nodes to the selected
+hubs, and determine the number and type of vehicles that travel on hub-level and access-
+level networks to minimize the total location and vehicle cost. We present mathematical
+programming models of the HNDPv and solve large-scale instances of the problem by
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+3
+developing an exact branch-and-cut solution method based on Benders decomposition.
+We show the advantages of using HNDPv compared to the classical HLP with constant
+discount factors in terms of vehicle utilization and total transportation costs.
+The HNDPv is a strategic problem where the long-term location and allocation decisions
+determine the underlying network structure and the fleet size decisions determine the
+investment that should be made to move the flow through the network. However, since
+selecting the optimal hub location and an appropriate vehicle fleet management are directly
+related to the amount of anticipated OD shipping volumes, a poor demand estimation may
+lead to an unsustainable or infeasible solution under the actual demand. To address the
+demand uncertainty, we extend the mathematical programming models of HNDPv in a
+stochastic environment and show how to adjust our branch-and-cut solution method to
+handle the demand uncertainty.
+1.1.
+Related Literature
+Despite the important economic impact of vehicle utilization in hub network design prob-
+lems, it has not gained enough attention in the literature. The large majority of in the
+hub location problems (HLPs) in the literature model the transportation cost as a linear
+function of volume transported on the network links. To address economies of scale, the
+transportation cost on the inter-hub links are multiplied by a fixed constant α called the
+discount factor (see Alumur et al. 2021, for a recent review). While this approach leads
+to a tractable (linear) cost function to minimize, it does not provide an adequate repre-
+sentation of economies of scale in consolidation-based transportation systems. Considering
+a fixed discount factor on all inter-hub links may lead to an oversimplified modeling of
+economies of scale and produce a poor estimation of the real savings. This simplification
+may result in solutions that grant discounted transportation on the inter-hub links, even
+though the flow on these links are poorly consolidated (Kimms 2006, Real et al. 2021).
+In addition, it requires advanced knowledge about the technologies used in the system to
+estimate potential α values in advance.
+To reflect economies of scale more realistically, one can consider either a decreasing unit
+cost for increasing transport volumes or a function that reflects the actual cost of vehicle
+utilization on the inter-hub links (Alumur et al. 2021). The former leads to a nonlinear
+concave cost function of the flow on an inter-hub link (Horner and O’Kelly 2001). Such
+an approach, however, has two drawbacks. First, it does not consider the technology (i.e.,
+
+4
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+Flow
+Cost
+0
+ch1h2
+αch1h2
+Regular
+Discounted
+Flow
+Cost
+0
+Qh1h2
+2Qh1h2
+3Qh1h2
+4Qh1h2
+Ch1h2
+2Ch1h2
+3Ch1h2
+4Ch1h2
+Figure 1
+Transportation cost function over an inter-hub arc.
+Note. Left: Linear discount-based cost function. Right: Stepwise vehicle-based cost function.
+the means of transport) used on the inter-hub links. Hence, vehicle utilization cost and
+capacity are ignored. Second, considering concave functions in mathematical modeling
+brings new computational challenges. As a result, researchers often consider representing
+such functions by their linear approximation (O’Kelly and Bryan 1998, Racunica and
+Wynter 2005, de Camargo et al. 2009, Rostami et al. 2022).
+Models with vehicle-based cost functions, on the other hand, take the vehicle char-
+acteristics into account by measuring the transportation costs per vehicle (as in freight
+transportation networks) or per capacitated link (as in telecommunication networks). Note
+that in the context of telecommunication, vehicle capacities translate to link capacities
+and the corresponding problem is called HLP with modular links (HLPm), which decides
+the number of capacitated links to build between each pair of nodes in the network. Such
+cost functions often lead to a stepwise linear cost function. Figure 1 plots a conventional
+discount-based and a vehicle-based cost function that calculates the transportation cost
+on an inter-hub arc (h1,h2). In the constant-discount models, the transportation cost per
+flow unit, say ch1,h2, is reduced by a factor in the range (0,1). The cost grows linearly with
+the amount of flow on the arc. The vehicle-based function calculates the transportation
+cost according to how vehicles are utilized. If each vehicle on arc (h1,h2) has a capacity
+Qh1,h2 and a fixed utilization cost Ch1,h2, then the cost of transporting loads on that arc is
+represented by a step-wise function of the number of vehicles in use.
+The main difficulty of the HNDPv (and the HLPm), in addition to the natural complexity
+of the design problem, is dealing with the integer number of vehicles (capacitated links) in
+the mathematical programming models. Different exact and approximation methodologies
+have been presented in the literature to address this challenge.
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+5
+In the steam of exact solution approaches, Yaman and Carello (2005) presents a branch-
+and-cut method for the HLPm. They solve a set of modified Australian Post (AP) instances
+with up to 20 nodes to optimality and provide feasible solutions for larger instances with up
+to 50 nodes using a heuristic. Rostami and Buchheim (2015) formulate the uncapacitated
+p-Hub median problem with vehicle-based transportation costs and develop a branch-
+and-bound algorithm where lower bounds are computed using a Lagrangian relaxation
+algorithm. The authors report the exact solutions for modified AP instances with up to
+50 nodes. Tanash et al. (2017) develop a Lagrangian-based branch-and-bound algorithm
+for the HLPm with an incomplete inter-hub network structure. In such problems, the OD
+paths are allowed to visit more than two hubs, and the design of the inter-hub network is
+part of the decision process. They solve AP instances with up to 40 nodes to optimality.
+In the stream of heuristics, different solution methods based on local search, iterated
+greedy search combined with memory strategies, and variable neighborhood search have
+been proposed in the literature (see Carello et al. 2004, Corber´an et al. 2016, Hoff et al.
+2017, Serper and Alumur 2016, Keshvari Fard and Alfandari 2019). In particular, Keshvari
+Fard and Alfandari (2019) proposes an approximation method to transform the stepwise
+vehicle-based cost function to a linear function of the flow. The authors claim that by
+choosing the proper intercept and slope parameters of the linear function, one can obtain a
+solution close to the one found for the original stepwise cost function. The authors provide
+the inexact to a set of problem test instances from CAB, AP, and Turkish datasets with
+up to 50 nodes. While this approach is more efficient, its solution quality highly depends
+on the estimated parameters of the generalized linear function and the capacity of the
+inter-hub vehicles. This approach may lead to inferior solutions when the flow on some
+inter-hub links is small or the vehicle capacities are large.
+Table 1 lists recent research in the literature on HNDPv and the HLPm with the single-
+allocation strategy. For each study, the table specifies the considered inter-hub network
+structure, whether the demand uncertainty is addressed, whether all vehicles (including
+the access-level vehicles) are capacitated, the proposed solution approach, and the largest
+instance size which could be solved exactly. Although there are a few other studies that
+consider HLP variants with step-wise cost functions (e.g., Kimms 2006, Sender and Clausen
+2011, Baumung and G¨und¨uz 2015), we do not list them in this table as they only introduce
+
+6
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+Table 1
+Related literature to the vehicle-based HLP.
+Article
+Inter-hub
+network structure
+Demand
+uncertainty
+Capacitated
+vehicles
+Solution approach
+Problem type
+(optimal size)
+Carello et al. (2004)
+Complete
+No
+Yes
+Heuristic
+CHLP
+Yaman and Carello (2005)
+Complete
+No
+Yes
+Branch-and-cut
+CHLP(20)
+Rostami and Buchheim (2015)
+Complete
+No
+Yes
+Branch-and-bound
+p-HLP(50)
+Corber´an et al. (2016)
+Complete
+No
+Yes
+Heuristic
+CHLP
+Serper and Alumur (2016)
+incomplete
+No
+Yes
+Heuristic
+CHLP
+Hoff et al. (2017)
+Complete
+No
+No
+Heuristic
+CHLP
+Tanash et al. (2017)
+incomplete
+No
+Yes
+Branch-and-bound
+HLP(40)
+Keshvari Fard and Alfandari (2019)
+Complete
+No
+Yes
+Approximation
+p-CHLP
+This study
+Complete
+Yes
+Yes
+Branch-and-cut
+CHLP(200†, 75‡)
+CHLP: capacitated HLP, p-HLP: p-hub median problem, p-CHLP: capacitated p-HLP.
+† Deterministic instance size.
+‡ Stochastic instance size.
+problem formulations, and do not present any problem-specific solution algorithms. The
+terms that are made bold in Table 1 share similarities with our problem.
+The available solution approaches for the HDNPv are either approximation methods or
+exact algorithms that are only capable of solving small-size problem instances. Moreover,
+despite the fact that the demand uncertainty directly affects the flow through the network,
+hence vehicle utilization, no research has been carried out to investigate the HNDPv with
+stochastic demand. Previous HLPs studies only consider sources of uncertainty in the
+discount-based models (see, for example, Alumur et al. 2012, Qin and Gao 2017, Tran
+et al. 2017, Wang et al. 2020, Hu et al. 2021, Rostami et al. 2021), and leave the stochastic
+HNDPv unexplored.
+1.2.
+Our Contribution
+While vehicle-based cost functions provide an adequate description of scale economies,
+they make the already challenging hub network design problem even more difficult to solve
+due to the additional integer variables that determine the number of inter-hub vehicles.
+Incorporating demand variability, which is necessary to make reliable location/allocation
+and fleet-size decisions, also adds another layer of complexity. In this study, (i) we develop
+an exact solution method based on Benders decomposition. While a natural approach to
+tackling the problem’s difficulty is to project out the integer vehicle variables, we propose
+an alternative where all the main decisions are made in the master problem. This leads
+to a branch-and-cut algorithm with feasibility-checking subproblems. (ii) To generate fea-
+sibility cuts more efficiently, we derive the extreme rays of the feasibility subproblem in
+an analytical way that, in addition to preventing solving many linear programs within the
+search tree, also provides an opportunity to add multi cuts in each call to the subproblem.
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+7
+(iii) We address the problem with demand uncertainty under the stochastic programming
+framework and show the potential of our solution methodology in solving the determinis-
+tic equivalent formulation of the problem. (iv) Extensive computational experiments are
+conducted to evaluate the potential, robustness, and efficiency of our models and solution
+methodologies on uncapacitated and capacitated instances derived from the classical Aus-
+tralian Post dataset. The results show a considerable advantage of using HNDPv compared
+to the classical HLP with constant discount factors in terms of vehicle utilization and total
+transportation costs. The proposed solution algorithm is able to solve large-scale deter-
+ministic HNDPv instances with up to 200 nodes for the first time in the literature. We
+also show the capability of the proposed solution methodology in solving problems with
+uncertain demands and general (or incomplete) inter-hub network structures.
+The remainder of the paper is organized as follows. Section 2 introduces mathematical
+formulations for the deterministic HNDPv and presents some valid inequalities. In Sec-
+tion 3, we propose our solution approaches for the HNDPv. The HNDPv under demand
+uncertainty and its solution method is discussed in Section 4. Our computational study
+in Section 5 investigates the performance of our solution algorithm in solving a set of
+benchmark problem test instances and provides managerial insights. In this section, we
+also explain how our solution approach can be adopted to solve problems with general
+inter-hub network structure. Section 6 concludes the paper and highlights future research
+directions.
+2.
+Problem Statement and Formulation
+The HNDPv is defined over a many-to-many network G = (N,A), where N represents
+the set of demand points (including hubs) and A is the set of (directed) arcs. The set
+of candidate hub nodes is denoted as H ⊂ N. Each node in N can potentially be the
+sender/receiver of specific OD flows. That is, there exists a flow amount of wij of a single
+commodity between each pair of i and j in N. We use Oi = �
+j∈N wij and Di = �
+j∈N wji
+to denote the total amount of flow originated and destined at demand node i, respectively.
+The amount of flow that can be consolidated at a hub is usually restricted by hub capacity.
+Associated with each hub h ∈ H, we consider a limited capacity Uh (that is set to a big
+number if the hub is uncapacitated) and a fixed setup cost Fh.
+
+8
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+Node-to-node connections are established through vehicle movements. At the hub level,
+loads are consolidated and are transported by high-capacity vehicles called primary vehi-
+cles. At the access-level network, smaller vehicles, called secondary vehicles, are utilized to
+pickup/deliver demands from/to non-hub nodes. Vehicles that are available at each hub
+may differ in terms of capacity and cost factors (e.g., fixed utilization cost). Therefore,
+we identify the fleet of secondary vehicles at any hub h ∈ H by capacity qh, a fixed uti-
+lization cost gh, and unit traveling cost bh. We define chi = gh + bh d(h,i) as the one-way
+transportation cost on access arc (h,i) ∈ A, where d(i,j) is the distance between nodes i
+and j. Therefore, the cost of serving node i ∈ N by a secondary vehicle from hub h ∈ H
+is c±
+hi = chi + bh d(i,h). Similarly, associated with each primary vehicle connecting a pair
+of hubs h,k ∈ H are capacity Qhk, fixed utilization cost Ghk, and unit traveling cost Bhk.
+Therefore, we let Chk = Ghk + Bhk d(h,k) be the cost of using a primary vehicle on each
+hub-hub connection (h,k).
+Note that in many strategic problems, calculating vehicle costs are based on the uti-
+lization cost over a given distance, while the filling quota plays a minor role. Therefore,
+the vehicle utilization cost is defined as a function of a fixed cost associated with using a
+vehicle on a specific link, and the cost of traversing that link. The traveling cost is assumed
+as a function of distance and not the load. In this way, dispatching vehicles with high load
+factors leads to a lower total transportation cost.
+We assume that the fixed cost, capacity, and unit traveling costs of a primary vehicle are
+strictly greater than those of a secondary vehicle, and Chk/Qhk < chk/qh holds for (h,k) ∈ A.
+This ensures that the unit transportation cost is less on inter-hub arcs compared to the
+access arcs when vehicles are adequately utilized. Consequently, the economies of scale is
+enhanced through consolidating freight into primary vehicles.
+2.1.
+Mathematical Formulation
+The HNDPv aims to (i) select a set of hub nodes, (ii) assign non-hub nodes to the selected
+hubs by satisfying the single-assignment property, and (iii) determine the number and
+type of primary and secondary vehicles, such that the overall hub location and vehicle
+utilization costs are minimized. Therefore, the major decisions in the HNDPv involve
+hub location, demand node allocation, and vehicle fleet management. We define a binary
+decision variable xhi to indicate whether node i ∈ N is assigned to hub h ∈ H. Variable xhh
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+9
+indicates whether node h ∈ H is selected as a hub. Moreover, we define yij as an integer
+variable that determines the number of vehicles needed on arc (i,j) ∈ A.
+One of the main component of the objective cost function is the cost corresponding to
+the number of primary and secondary vehicles. Due to the single-assignment property, we
+have full information on the amount of flow on each selected access arc traversed in direct
+shipments. For example, when node i is assigned to hub h, the total flow on arc (h,i) is
+equal to Di and the total flow on arc (i,h) is equal to Oi. We can calculate the number of
+vehicles required for delivering and picking up the demand of node i as n+
+hi = ⌈Di/qh⌉ and
+n−
+ih = ⌈Oi/qh⌉, respectively, where ⌈·⌉ is the ceiling function. Therefore, the total number
+of secondary vehicles yhi that need to dispatch from hub h to serve demand node i ∈ N via
+direct shipment can be prepossessed and set to n±
+hi = max
+�
+n+
+hi,n−
+hi
+�
+. This number is based
+on the fact that vehicles are available at the hub locations and start and end their trip at
+their hosting hubs. In the HLPs with modular links, n±
+hi translates into the number of links
+with capacity qh that need to be installed in order to serve demand node i from hub h (see
+Yaman and Carello 2005, Corber´an et al. 2016). Therefore, we define the following cost
+function at the access level network
+DC(x) =
+�
+h∈H
+�
+i∈N:i̸=h
+n±
+hi c±
+hi xhi.
+(1)
+However, the decisions about the number of primary vehicles in the hub-level network
+can not be prepossessed and must explicitly be addressed by y variables. The following
+function calculates the total transportation cost at the hub level.
+HC(y) =
+�
+h∈H
+�
+k∈H:k̸=h
+Chk yhk.
+(2)
+Finally, the total hub location cost is defined as:
+LC(x) =
+�
+h∈H
+Fh xhh.
+(3)
+Using the defined decisions and their associated costs, we model the HNDPv as the
+mixed-integer quadratic formulation given below.
+Minimize
+LC(x) + DC(x) + HC(y)
+(4a)
+subject to:
+�
+i∈N
+�
+j∈N
+wij xhixkj ≤ Qhk yhk, h,k ∈ H,
+(4b)
+
+10
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+x ∈ X,
+(4c)
+y ∈ Z|H|×|H|
+≥0
+,
+(4d)
+where X is the set of constraints that ensure a feasible assignment. Objective function (4a)
+minimizes the total cost of locating hubs and utilizing vehicles to transport flow through
+the network. Constraint (4b) relates assignment and vehicle variables. It calculates the
+total flow on each hub arc and ensures that a correct number of vehicles travel on that arc.
+Constraint (4d) restricts y to nonnegative integer values. For simplicity, we denote Z|H|×|H|
+≥0
+by Y.
+Set X is well-defined in the HLP literature (see Farahani et al. 2013, for a review). A
+standard feasible set X is given as:
+X =
+�
+�
+�xhi ∈ {0,1}, h ∈ H,i ∈ N
+������
+�
+h∈H xhi = 1, i ∈ N,
+xhi ≤ xhh,
+h ∈ H,i ∈ N.
+�
+�
+�,
+(5)
+where the first constraint assigns each node to exactly one hub, and the second constraint
+restricts assignments to the selected hubs. If a p-hub median problem is targeted, equality
+constraint �
+h∈H xhh = p is added to X to ensure that exactly p hubs are selected. When
+hub h has a limited capacity Uh, X also contains the following constraint to ensure that
+the total outgoing demand from a hub does not exceed its capacity.
+�
+i∈N
+Oixhi ≤ Uh xhh, h ∈ H.
+(6)
+2.2.
+A Linear Reformulation
+The HNDPv formulation (4) is a constrained binary quadratic program, which is
+intractable for many standard solvers. One can use the “path-based” formulation of Skorin-
+Kapov et al. (1996) or the “flow-based” formulation of Ernst and Krishnamoorthy (1996)
+to obtain an equivalent MIP formulation. Although the flow-based formulation is widely
+considered as the most effective model for the classical single allocation HLPs, a crucial
+assumption for its validity is that the triangle inequality for the transportation costs holds
+(Correia et al. 2010). As the inter-hub transportation costs are vehicle-dependent in our
+application, the triangular inequality condition does not generally hold and, therefore, the
+classical flow-based technique cannot be applied. Here, we use a modified version of the
+flow-based linearization of Rostami et al. (2022) that always provides a valid lineariza-
+tion regardless of the underlying cost structure. Consider a non-negative variable zihk that
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+11
+determines the amount of node i’s demand transported from hub h to hub k, i.e., zihk =
+xhi
+�
+j wijxkj,∀i ∈ N,h,k ∈ H. Then, constraint (4b) can be replaced by
+�
+i∈N
+zihk ≤ Qhk yhk, h,k ∈ H,
+(7a)
+where variable z is determined by the following set of constraints.
+�
+k∈H
+zihk = Oi xhi,
+h ∈ H,i ∈ N,
+(7b)
+�
+h∈H
+zihk =
+�
+j∈N
+wij xkj, k ∈ H,i ∈ N,
+(7c)
+z ∈ R|N|×|H|×|H|
+≥0
+.
+(7d)
+Replacing the quadratic constraints (4b) by, new set of constraints in (7) we obtain the
+following mixed-integer linear programming reformulation
+P :
+min
+x,y,z{LC(x) + DC(x) + HC(y)|(7), x ∈ X, y ∈ Y }.
+(8)
+The validity of this linearization follows from the equivalence between constraints (7b)
+and (7c) and the mathematical definition of the flow variables (see Remark 3 in Rostami
+et al. 2022).
+3.
+A Benders Decomposition-based Solution Algorithm
+Model P can be solved by state-of-the-art MIP solvers. However, it is still very challenging
+due to a large number of variables and linking constraints. A natural way to handle such
+a difficulty is to apply a Benders decomposition (BD) to project out integral variables
+y and deal with them in a subproblem. However, since the subproblem involves integral
+variables, it requires special treatments, such as the integer L-shaped method (Laporte
+and Louveaux 1993), to generate optimality cuts to the Benders master problem. Given
+that integer L-shaped cuts are usually not strong, one can solve the linear programming
+(LP) relaxation of the resulting subproblems and add Benders optimality cuts to improve
+the global lower bound on the value of the subproblems (see Rostami et al. 2022, for a
+recent implementation on the integer L-shaped method to solve an HLP). Our preliminary
+experiments, however, showed that such a treatment is time-consuming and negatively
+affects the performance of the BD algorithm. Therefore, we do not provide details of the
+integer L-shaped method here.
+
+12
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+In what follows, we describe an alternative Benders decomposition in Section 3.1 in
+which the flow variables are projected out and will be handled through feasibility cuts. In
+Section 3.2, we show how to compute such feasibility cuts efficiently. Finally, in Section 3.3,
+we present some valid inequalities to enhance the algorithm.
+3.1.
+Benders Feasibility Cuts
+Consider P in 8 and project out the flow variables from the model, while keeping the
+location/allocation and integer vehicle variables. This leads to an integer Benders master
+problem (BMP) and a linear program (LP) subproblem. The Benders subproblem is defined
+for a given (x,y) ∈ X × Y in order to check whether the decided vehicle variables yield a
+feasible flow on hub-hub connections. This problem is referred to as the BSP(x,y) given
+by
+BSP(x,y) :
+min 0
+(9a)
+s.t.
+�
+i∈N
+zihk ≤ Qhkyhk,
+h,k ∈ H,
+(9b)
+�
+k∈H
+zihk = Oixhi,
+h ∈ H,i ∈ N,
+(9c)
+�
+h∈H
+zihk =
+�
+j∈N
+wijxkj, k ∈ H,i ∈ N,
+(9d)
+z ∈ R|N|×|H|×|H|
+≥0
+.
+(9e)
+By defining dual variables λ,µ,ν, we can write the dual of the BSP(x,y), called
+DSP(x,y), as the following LP formulation:
+DSP(x,y) :
+max
+�
+h∈H
+�
+k∈H
+Qhkyhk λhk +
+�
+h∈H
+�
+i∈N
+�
+Oixhi µhi +
+�
+j∈N
+wijxhj νhi
+�
+(10a)
+s.t.
+λhk + µhi + νki ≤ 0,
+h,k ∈ H,i ∈ N,
+(10b)
+λ ∈ R|H|×|H|
+≤0
+,µ,ν ∈ R|H|×|N|.
+(10c)
+The feasible set of DSP(x,y) is independent of the choice of (x,y). Therefore, if it is
+not empty, the DSP(x,y) becomes either feasible or unbounded for any arbitrary choice
+of (x,y). In the former case, no further action is required and (x,y) is feasible and hence
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+13
+optimal for the problem. In the latter case, given the set of extreme rays R of the set
+��
+λ,µ,ν
+�
+: (10b)
+�
+, there is an unbounded ray (λ
+r,µr,νr), r ∈ R for which
+Ωr =
+�
+h∈H
+�
+k∈H
+Qhkλ
+r
+hk yhk +
+�
+h∈H
+�
+i∈N
+�
+Oiµr
+hi xhi +
+�
+j∈N
+wijνr
+hi xhj
+�
+> 0.
+(11)
+We must cut solution (x,y) to restrict movement in this direction. This will result in
+the following reformulation of model P refer to master problem
+BMP :
+min LC(x) + DC(x) + HC(y)
+(12a)
+s.t. Ωr ≤ 0, r ∈ R
+(12b)
+x ∈ X, y ∈ Y.
+(12c)
+In our implementation, which is evaluated in Section 5, we solve BMP using a branch-
+and-cut framework of a state-of-the-art optimization solver. The feasibility cuts are incor-
+porated into the master problem by using callbacks, allowing to add the cutting planes
+step-by-step. A callback is executed whenever an optimal solution of the LP-relaxation is
+found at the root node of the branch-and-bound-tree or an incumbent solution at any node
+of the branch-and-bound-tree is found. For the current choice of variables (x,y), if this
+solution satisfies the following conditions, then it is also feasible to the original problem
+(4) and no further action is required.
+�
+�
+�
+�
+�
+zihk = xhi
+�
+j wijxkj,
+∀i ∈ N,h,k ∈ H,
+�
+i∈N zihk ≤ Qhk yhk,
+h,k ∈ H.
+(13)
+Otherwise, i.e., if condition (13) is violated, the feasibility cut (12b) is added to the BMP
+to cut off the current (x,y). To find the feasibility cuts, one can solve the BSP, or its dual,
+using a standard LP solver.
+Remark 1. The Benders decomposition approach developed here can also be applied
+to solve the HNDPv with the general hub network structures with small modifications.
+The general hub network structure allows the OD pairs’ demands to go through more
+than two hubs if needed Tanash et al. (2017). Therefore, the quadratic constraint (4b) and
+inequality (16c) are no longer valid. The Benders subproblem should incorporate a flow
+conservation constraint for each hub node to ensure that the flows are correctly routed
+through the network. That is, we have
+
+14
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+G-BSP(x,y) : min 0
+(14a)
+s.t. (7a),(7d),
+�
+k∈H
+zihk −
+�
+k∈H
+zikh = Oixhi −
+�
+j∈N
+wijxhj, h ∈ H,i ∈ N.
+(14b)
+Constraints (14b) are the flow conservation, which can also be obtained by subtracting
+(9d) from (9c) in the BSP for complete inter-hub networks. In the online companion, we
+provide more details on the method and its computational performance.
+3.2.
+Cut Generation Improvement
+There are two main issues with the cut generation procedure described in Section 3.1.
+First, while condition (13) might be violated for more than one hub-hub connection (h,k),
+we only add one feasibility cut. This is because BSP and DSP can not be decomposed on
+inter-hub link (h,k). Moreover, solving LP subproblems can be time-consuming, as many
+of these suproblems must be solved within the search tree (see Section 5.2). To overcome
+these challenges, in the following theorem, we show how to exploit the structure of the
+DSP to obtain an unbounded ray
+�
+λ,µ,ν
+�
+for each pair of hubs for which condition (13)
+is violated.
+Theorem 1. Let (x,y) be a feasible solution to the BMP. Let ˆh, ˆk ∈ H be an arbitrary
+pair of hubs for which condition (13) is violated. Then, given a constant Γ > 0, the vector
+�
+λ,µ,ν
+�
+with
+λhk =
+�
+�
+�
+�
+�
+−Γ
+if h = ˆh and k = ˆk,
+0
+otherwise,
+h,k ∈ H,
+(15a)
+µhi =
+�
+h′∈H
+�
+k∈H
+λh′k xh′i −
+�
+k∈H
+λhk xhi h ∈ H,i ∈ N,
+(15b)
+νki = −
+�
+h∈H
+λhk xhi
+k ∈ H,i ∈ N,
+(15c)
+is an unbounded ray for DSP(x,y).
+Proof.
+Given ˆh, ˆk ∈ H, we first show that the vector
+�
+λ,µ,ν
+�
+is feasible. Substituting
+the values in left-hand side of constraint (10b), we get
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+15
+λhk + µhi + νki =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−Γ (1 − xˆhi)
+if h = ˆh and k = ˆk,
+−Γ xˆhi
+if h ̸= ˆh and k ̸= ˆk,
+0
+otherwise,
+h,k ∈ H,i ∈ N,
+which is always lest than or equal to 0 as −Γ < 0 and xhi ≤ 1 holds for any h,k ∈ H,i ∈ N.
+We now show that the objective function is unbounded over
+�
+λ,µ,ν
+�
+. Substituting the
+values in expression (10a) yields
+�
+h∈H
+�
+k∈H
+Qhkyhk λhk +
+�
+h∈H
+�
+i∈N
+�
+Oixhi µhi +
+�
+j∈N
+wijxhj νhi
+�
+= −Γ Qˆhˆkyˆhˆk −
+�
+h:h̸=ˆh
+�
+i∈N
+Γ xˆhiOixhi +
+�
+i∈N
+�
+j∈N
+Γ wijxˆhixˆkj
+= −Γ Qˆhˆkyˆhˆk − 0 + Γ
+�
+i∈N
+xˆhi
+�
+j∈N
+wijxˆkj
+= Γ
+��
+i∈N
+ziˆhˆk − Qˆhˆkyˆhˆk
+�
+.
+Since ˆh, ˆk ∈ H violate condition (13), we have �
+i∈N ziˆhˆk −Qˆhˆkyˆhˆk > 0. Therefore, the max-
+imization problem (10) becomes unbounded as Γ → ∞.
+□
+Using Theorem 1, we can generate the feasibility cut (12b) without solving the DSP
+subproblems. More importantly, it implies that we can generate a feasibility cut whenever
+we detect a pair of hubs (ˆh, ˆk) on which there exists an insufficient number of primary
+vehicles. If there are more than one infeasible inter-hub link, one can choose an arbitrary
+pair or select an (ˆh, ˆk) equal to arg max(h,k)∈H×H
+��
+i∈N
+�
+j∈N wijxˆhixˆkj − Qhkyhk
+�
+, i.e.,
+the pair for which the highest amount of capacity violation is observed. Therefore, we
+consider a modified branch-and-cut framework, where at each node, we find all (ˆh, ˆk)
+with infeasible flows, calculate the extreme rays
+�
+λ,µ,ν
+�
+using Theorem 1, and add the
+resulting cuts to prune that node (if needed). Our computational results show that the
+multi-cut approach outperforms the single-cut version, specially for the problem under
+multiple demand scenarios (see Section 4).
+3.3.
+Valid Inequalities
+Initially, the BMP has poor information on the flows over the inter-hub links as it includes
+y variables, but not their relation to the flow variables z. Therefore, the initial bounds
+
+16
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+are usually loose, and the algorithm may go through many iterations to obtain some
+information through feasibility cuts. In a desire to give the algorithm a better warm start
+and improve its linear relaxation, we can exploit the model’s structure to generate some
+valid inequalities. In particular, we can set a lower bound on the total number of vehicles
+that arrive to and dispatch from a hub based on the total incoming and outgoing demand
+assigned to that hub. Let Qin
+h = maxk{Qkh} and Qout
+h
+= maxk{Qhk}. Then, the following
+set of inequalities provide the aforementioned bounds.
+1
+Qin
+h
+�
+i∈N
+Di xhi ≤
+�
+k∈H
+ykh, h ∈ H
+(16a)
+1
+Qout
+h
+�
+i∈N
+Oi xhi ≤
+�
+k∈H
+yhk, h ∈ H.
+(16b)
+Moreover, when there is a flow between two nodes and both nodes are selected as hubs,
+they must be connected by at least one primary vehicle. The following valid inequality
+calculates the minimum number of vehicles required to travel between two hubs based on
+their shipment volume and primary vehicle capacity.
+�whk
+Qhk
+�
+(xhh + xkk − 1) ≤ yhk, h,k ∈ H.
+(16c)
+4.
+Demand Uncertainty
+The HNDPv problem presented in Section 2 assumes that the OD demands are known in
+the planning stage. In reality, however, shipment volumes are stochastic, and long-term
+deterministic forecasts are unreliable. In this section, we address the demand uncertainty
+in HNDPv under a stochastic programming framework. The goal is to account for demand
+uncertainty in the design phase of the network in order to maintain the operational relia-
+bility of the network when the actual demand is realized.
+For each i,j ∈ N, let random variable wij(ξ) represent the flow that needs to be sent
+from node i to node j, where ξ ∈ Ξ for a given support Ξ. Define Oi(ξ) = �
+j∈N wij(ξ)
+and Di(ξ) = �
+j∈N wji(ξ) as random variables representing the total outgoing flow from
+and the total incoming flow to node i, respectively. We consider a two-stage stochastic
+program with recourse in which the location and allocation variables are dealt with in the
+first stage, while the flow variables and the required number of vehicles are determined in
+the second stage. The two-stage stochastic formulation of HNDPv is given as
+min
+x∈X LC(x) + Eξ[P(x,ξ)]
+(17)
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+17
+where Eξ denotes the mathematical expectation with respect to ξ ∈ Ξ,
+P(x,ξ) = min
+y∈Y
+�
+DC ξ(x) + HC(y)
+�����
+�
+i∈N
+�
+j∈N
+wij(ξ)xhixkj ≤ Qhk yhk, h,k ∈ H
+�
+,
+(18)
+and DC ξ(x) is the cost of direct access to non-hub node i which is defined as a function
+of random variables calculated as:
+DC ξ(x) =
+�
+h∈H
+�
+i∈N
+max
+��Di(ξ)
+qh
+�
+,
+�Oi(ξ)
+qh
+��
+c±
+hi xhi.
+(19)
+Evaluating Eξ[P(x,ξ)] in (17) is difficult and makes the optimization problem intractable.
+Therefore, following other works on stochastic hub location (see, for example,
+Alumur
+et al. 2012, Rostami et al. 2021), we assume that the random variable ξ follows a discrete
+distribution with finite support S = {s1,...,sm}, where each event s ∈ S occurs with prob-
+ability P(ξ = s) = ps. Accordingly, we use ws
+ij to denote the amount of flow from node
+i to node j for each scenario s ∈ S. Therefore, Os
+i = �
+j∈N ws
+ij and Ds
+i = �
+j∈N ws
+ji rep-
+resent the total outgoing flow from and the total incoming flow to node i, respectively.
+Since vehicle selection decisions are to be done once scenario s is realized, we redefine y
+variables as ys
+hk to indicate the number of primary vehicles traveling on hub arc (h,k) ∈
+H × H under scenario s. Moreover, for each scenario s ∈ S, we redefine flow variables as
+zs
+ihk to determine the amount of node i’s demand transported from hub h to hub k, i.e.,
+zs
+ihk = xhi
+�
+j ws
+ijxkj,∀i ∈ N,h,k ∈ H. Then, the deterministic equivalent formulation of P
+is stated as:
+DEP :
+min LC(x) +
+�
+s∈S
+ps(DC s(x) + HC s(y))
+(20a)
+s.t.
+�
+i∈N
+zs
+ihk ≤ Qhk ys
+hk,
+h,k ∈ H,s ∈ S
+(20b)
+�
+k∈H
+zs
+ihk = Os
+i xhi,
+h ∈ H,i ∈ N,s ∈ S
+(20c)
+�
+h∈H
+zs
+ihk =
+�
+j∈N
+ws
+ij xkj,
+k ∈ H,i ∈ N,s ∈ S
+(20d)
+x ∈ X, zs ∈ R|N|×|H|×|H|
+≥0
+, ys ∈ Y,s ∈ S
+(20e)
+where,
+DC s(x) =
+�
+h∈H
+�
+i∈N
+max
+��Ds
+i
+qh
+�
+,
+�Os
+i
+qh
+��
+c±
+hi xhi,
+(21)
+
+18
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+HC s(y) =
+�
+h∈H
+�
+k∈H
+Chk ys
+hk.
+(22)
+4.1.
+Solving the HNDPv Under Demand Uncertainty
+The Benders decomposition approach presented in Section 3 can be easily adjusted to solve
+DEP. For each scenario s ∈ S, and for any feasible solution (x,y) to the master problem
+at a given iteration of the algorithm, we define one scenario-based Benders subproblem
+(S-BSP) as follows.
+S-BSPs(x,y) :
+min 0
+(23a)
+s.t.
+�
+i∈N
+zs
+ihk ≤ Qhk ys
+hk,
+h,k ∈ H,
+(23b)
+�
+k∈H
+zs
+ihk = Os
+i xhi,
+h ∈ H,i ∈ N,
+(23c)
+�
+h∈H
+zs
+ihk =
+�
+j∈N
+ws
+ij xkj, k ∈ H,i ∈ N,
+(23d)
+zs ∈ R|N|×|H|×|H|
+≥0
+.
+(23e)
+Following the approach described in Section 3, one can solve the dual of S-BSPs(x,y)
+to generate feasibility cuts and apply them to the master problem whenever an infeasible
+S-BSP is observed. The Benders master problem corresponding to the S-HNDPv (i.e., the
+S-BMP) is formulated as:
+S-BMP :
+min LC(x) +
+�
+s∈S
+ps(DC s(x) + HC s(y))
+(24a)
+s.t. Ωsr ≤ 0, r ∈ Rs,s ∈ S,
+(24b)
+x ∈ X, ys ∈ Y,s ∈ S,
+(24c)
+where
+Ωsr =
+�
+h∈H
+�
+k∈H
+Qhkλ
+sr
+hk ys
+hk +
+�
+h∈H
+�
+i∈N
+�
+Os
+i µsr
+hi xhi +
+�
+j∈N
+ws
+ijνsr
+hi xhj
+�
+,
+(25)
+and λ
+sτ
+hk, µsτ
+hi, and µsτ
+hi being the unbounded rays in the set extreme rays Rs corresponding
+to constraint (23b), (23c), and (23d), respectively, in the dual space.
+Note that the results of Theorem 1 are still valid for S-BSPs(x,y) for a given scenario
+s ∈ S. Therefore, similar to the deterministic case, we may add multiple cuts for any
+scenario s with infeasible inter-hub flows.
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+19
+5.
+Computational Experiments
+In this section, we present the numerical results evaluating models and solution algorithms.
+The algorithms are coded in Python. We used Guropi optimizer v9.5 (Gurobi Optimization,
+LLC 2022) and its callback features to solve our optimization models. Experiments are
+conducted on a Linux laptop with Intel® Core™ i9-11900H CPU @ 2.50GHz and 32GB
+of RAM using up to 14 threads. All experiments are done within a time limit of 12,000
+seconds.
+In the following sections, we first present the test instances and then provide the exper-
+imental results of solving deterministic and stochastic HNDPv test instances.
+5.1.
+Problem Test Instances and Experimental Design
+To perform our experiments, we use the Australian Post (AP) dataset used by Contreras
+et al. (2009). We consider location cost and capacity values provided in the AP dataset.
+The location costs are all assumed to be tight, while both tight (T) and loose (L) values
+are tested for hub capacity levels. An uncapacitated case (U) is also considered where
+the hubs have unrestricted capacity. We assume fixed capacities Qhk = Q,∀h,k ∈ H, and
+qh = q,∀h ∈ H, and fixed unit transportation costs bij = b and Bij = B, ∀(i,j) ∈ A, for all
+primary and secondary vehicles, respectively. Inspired by Tanash et al. (2017), we set all
+vehicle fixed costs to 0 and consider the following configurations for primary and secondary
+vehicle parameters:
+• L1: (Q = 600,q = 100,B = 600,b = 260) ⇒ B/Q
+b/q ≈ 0.38
+• L2: (Q = 600,q = 150,B = 600,b = 300) ⇒ B/Q
+b/q = 0.50
+• L3: (Q = 320,q = 100,B = 500,b = 260) ⇒ B/Q
+b/q ≈ 0.60
+• L4: (Q = 320,q = 150,B = 500,b = 300) ⇒ B/Q
+b/q ≈ 0.78
+These configurations are chosen such that the unit transportation cost on hub arcs, when
+fully utilized, is smaller than the unit transportation cost on access arcs. We also indicate
+the B/Q
+b/q ratio for L1 to L4. This value is considered as the smallest discount factor that can
+be achieved on hub arcs (Tanash et al. 2017). Instances with |N| equal to 20, 25, 40, 50, 100,
+and 200 are considered for our computational tests. We refer to each instance by #1#2-#3
+notation, where #1 denotes the number of nodes in the instance (e.g., 50), #2 indicates
+the capacity configuration (i.e., T, L, or U), and #3 shows the vehicle configuration (e.g.,
+L1).
+
+20
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+500
+1000
+CPU
+4.8
+5.0
+5.2
+5.4
+5.6
+Value
+×105
+MIP
+0
+200
+CPU
+BD
+Upper bound
+Lower bound
+0
+10
+CPU
+BC
+Figure 2
+Closing optimality gap by different approaches (problem instances: 50L-L4).
+5000
+10000
+CPU
+5.5
+6.0
+6.5
+7.0
+7.5
+Value
+×105
+MIP
+0
+5000
+10000
+CPU
+BD
+Upper bound
+Lower bound
+0
+20
+40
+CPU
+BC
+Figure 3
+Closing optimality gap by different approaches (problem instances: 75L-L4).
+5.2.
+Algorithmic Efficiency
+We evaluate the performance of the proposed algorithms in terms of efficiency and compare
+them with Gurobi applied directly to solve the MIP formulation P in (8). We show by
+MIP the direct application of Gurobi to solve the MIP model, by BD the Benders-based
+branch-and-cut algorithm with single feasibility cut, and by BC, the Benders-based branch-
+and-cut algorithm with multiple feasibility cuts derived from Theorem 1. First, we compare
+the performance of these algorithms on two different instances in Figures 2 and 3. We
+then compare MIP and BC on small to medium-size instances in Table 3, and show the
+performance of the BC on large-scale instances in Table 4, and Figures 4 and 5. These
+tables and figures have been designed to give a full picture of the algorithms’ efficiency.
+The detailed performance of the algorithms can be found in the online supplement of the
+paper.
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+21
+Table 2
+Comparison of the BD and BC performances.
+Inst
+BD
+BC
+#BNodes
+#Cuts
+#Calls
+CPUc
+CPU (%Gap)
+#BNodes
+#Cuts
+#Calls
+CPUc
+CPU (%Gap)
+50L-L4
+7,561
+118
+125
+317.53
+324.61 (0.0)
+5,002
+259
+119
+2.76
+17.97 (0.0)
+75L-L4
+5,239
+168
+175
+>12,000
+>12,000 (0.88)
+10,230
+476
+220
+9.81
+39.47 (0.0)
+Figures 2 and 3 illustrate how the upper bound and the lower bound converge during
+the procedure of solving instances 50L-L4 and 75L-L4. In each figure, the x access shows
+the CPU time in seconds, while the y access shows the values for lower and upper bounds.
+To solve the 50L-L4, Gurobi takes 23 minutes, the BD takes 5 minutes, while the BC only
+takes 18 seconds. Although the initial lower bound in MIP is better, and the solver is able
+to find a good upper bound after some time, it takes a long time to close the gap. BD and
+BC start with worse lower bounds (since less information is available at the beginning),
+but they close the gap significantly faster. When the number of nodes is increased to 75
+(Figure 3), we see the same behavior for the upper and lower bounds. However, neither
+MIP nor BD was able to solve the instance within the given time limit of 12,000 seconds,
+while the BC was able to find the optimal solution in less than 40 seconds.
+Table 2 report more information on the performance of the BD and BC in solving the
+same two instances. For each instance, CPUc shows the time spent to solve the feasibility
+subproblems and generating the associated cuts, #BNodes, #Cuts, #Calls, and %Gap
+show the number of explored branch-and-bound nodes, the number of generated feasibility
+cuts, the number of time the solver has called the subproblem, and the percent relative
+optimality gap, respectively. For 50L-L4, while both call the feasibility check subproblems
+almost the same, the BC generated more than twice the cuts than the BD. However,
+the BC generates the cuts 114 times faster than the BD. For the 75L-L4, the number of
+feasibility cuts generated by BC is larger, but the total time spent to generate these cuts
+is significantly smaller. We can see that most of the time in the BD is spent in finding the
+coefficients in the feasibility cuts. Theorem 1, on the other hand, allows us to skip solving
+an optimization problem and add multiple cuts at once. That is, generating cuts in this
+way allows us to solve larger problem instances more efficiently. Therefore, in the rest of
+this section, we do not report the results of the BD.
+Next, in Table 3 we compare BC and MIP performances in solving small to medium-size
+problem instances for which both approaches could solve them to optimality within the
+time limit. For each instance, with size |N|, the hub capacity configuration (Cap), and the
+
+22
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+Table 3
+CPU values of MIP and BC in solving small and medium-size instances.
+|N|
+Cap†
+MIP
+BC
+L1
+L2
+L3
+L4
+L1
+L2
+L3
+L4
+20
+T
+7.27
+3.69
+2.97
+9.18
+0.15
+0.36
+0.29
+0.74
+L
+9.88
+6.52
+2.76
+1.99
+0.40
+0.27
+0.37
+0.44
+U
+8.28
+8.11
+3.41
+1.99
+0.20
+0.32
+0.24
+0.33
+25
+T
+16.53
+14.84
+17.79
+12.88
+0.27
+0.24
+2.71
+11.28
+L
+21.01
+24.55
+10.91
+10.18
+1.50
+1.36
+1.41
+1.12
+U
+18.41
+9.40
+4.72
+2.21
+0.69
+0.68
+0.31
+1.31
+40
+T
+266.52
+219.05
+270.53
+304.10
+0.49
+0.53
+44.16
+474.52
+L
+98.27
+44.47
+87.70
+42.07
+1.08
+3.12
+1.68
+1.70
+U
+110.98
+41.58
+71.28
+44.68
+2.70
+1.65
+1.39
+3.47
+50
+T
+9,765.28
+6,691.78
+5,218.47
+4,488.73
+2.50
+1.41
+1,327.72
+11.68
+L
+734.93
+1,459.41
+1,286.41
+1,402.19
+1.72
+0.80
+7.49
+17.97
+U
+915.23
+988.54
+766.35
+1,238.07
+2.82
+1.66
+8.26
+4.74
+Average
+997.72
+792.66
+645.28
+629.86
+1.21
+1.03
+116.34
+44.11
+† Hub capacity configuration.
+Table 4
+CPU and (%Gap) values of BC in solving large instances.
+|N|
+Cap
+Vehicle configuration
+L1
+L2
+L3
+L4
+75
+T
+2.15
+2.34
+(2.71)†
+(1.96)
+L
+2.21
+2.19
+81.24
+39.47
+U
+1.54
+2.23
+50.77
+40.26
+100
+T
+8.31
+5.57
+(0.60)
+143.39
+L
+3.14
+6.26
+29.75
+11.95
+U
+2.35
+2.86
+76.16
+28.71
+150
+T
+24.81
+23.49
+520.12
+399.36
+L
+9.40
+14.61
+722.13
+615.61
+U
+6.47
+12.79
+619.59
+421.49
+200
+T
+109.19
+106.40
+80.79
+104.07
+L
+83.46
+106.87
+(0.24)
+7,395.43
+U
+56.82
+70.47
+7,689.79
+(0.09)
+Average
+25.82 (0.00)
+29.67 (0.00)
+3,822.53 (1.18)
+2,766.65 (1.03)
+† Values in parentheses represent percentage optimality gaps.
+vehicle configurations L1 to L4, the table reports CPU times for each solution method. For
+both methods, the instances with tight capacity are the most difficult to solve. From the
+vehicle configuration perspective, the L3 and L4 are the most difficult instances for the BC,
+while this is not the case for the MIP. This will be investigated further Table 4. Overall,
+as can be seen, the BC was considerably faster than MIP, particularly when solving larger
+problems. For instances with |N| = 75 and more, the MIP could not close the gap within
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+23
+the time limit. Therefore, in Table 4, we only report the performance of the BC in solving
+larger problem instances.
+Table 4 lists the CPU time for each instance under different hub capacities and vehicle
+configurations. In cases where the time limit is reached, instead of the time, the %Gap is
+reported in parentheses. Out of 48 instances, the BC was able to find the optimal solution
+for 43 instances. These instances took, on average, about 20 seconds for |N| = 75, 30
+seconds for |N| = 100, 5 minutes for |N| = 150, and 30 minutes when |N| = 200 to be
+solved by BC. The optimality gap for unsolved instances is reported as 1.12%. We observe
+that in larger instances, the vehicle configurations have a more significant effect on the
+BC performance. The last row of Table 4 shows the average CPU time and %Gap values
+for different vehicle configurations. Vehicle configurations L3 and L4 are the most difficult
+settings to deal with. The reason is that the primary vehicle capacities are smaller in
+these settings. Hence, more effort should be made to ensure the feasibility of the flows
+on the inter-hub links, and more feasibility cuts are generated (see the online supplement
+for more details). The average CPU time and %Gap values of the L3 configuration are
+the highest among all vehicle configurations. The reason is that the L3 configuration has
+not only the smallest primary vehicle capacity, but also the smallest secondary capacity.
+Therefore, more vehicles are required on the access-level network. This increases the cost
+of assignment decisions, and, as a result, more exploration is required to find the optimal
+location and allocation decisions.
+In Figures 4 and 5, we show the effect of instance characteristics on the number of
+branch-and-bound nodes explored by the algorithm (#BNodes) and the number of gen-
+erated feasibility cuts (#Cuts). Figure 4 illustrates the average values for different hub
+capacity configurations and instance sizes. For instances with size 100 or smaller, the algo-
+rithm explores significantly more #BNodes and outputs more #Cuts when hub capacities
+are tight. We observed that in the AP instances with |N| ≥ 150, although there exists a
+large number of OD pairs, shipment volumes are smaller than those in smaller instances.
+Therefore, when hub capacities get large, more assignment options become available for the
+demand nodes, and more demands can get consolidated at hubs. This leads to more con-
+solidated flows on the inter-hub links, which in turn requires more cuts to ensure feasibility
+on those links.
+
+24
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+25
+50
+75
+100
+150
+200
+|N|
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+×106
+#BNodes (avg)
+Cap
+T
+L
+U
+25
+50
+75
+100
+150
+200
+|N|
+0
+1
+2
+3
+4
+×103
+#Cuts (avg)
+Figure 4
+Effect of hub capacity configurations on the number of explored branch-and-bound nodes and the
+number of generated feasibility cuts.
+25
+50
+75
+100
+150
+200
+|N|
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+×106
+#BNodes (avg)
+Vehicle config.
+L1
+L2
+L3
+L4
+25
+50
+75
+100
+150
+200
+|N|
+0
+1
+2
+3
+4
+5 ×103
+#Cuts (avg)
+Figure 5
+Effect of vehicle configurations on the number of explored branch-and-bound nodes and the number
+of generated feasibility cuts.
+We illustrate the effect of different vehicle configurations on #BNodes and #Cuts in
+Figure 5. We can observe that many feasibility cuts are generated for the instances with
+small primary vehicles (i.e., L3 and L4). When primary vehicles have larger capacities, the
+introduced valid inequalities, along with other cuts generated by Gurobi, help to obtain
+feasible flows on the inter-hub links with no or only a few feasibility cuts. The online
+companion of this paper provides more details about the BC performance.
+5.3.
+Managerial Insights
+The main objective of the HNDPv is to optimize vehicle utilization to reduce cost. In this
+section, we first investigate how such utilization is compared to conventional HLP with a
+constant discount factor. Then, we explore the effect of problem instance characteristics,
+i.e., size, hub capacity, and vehicle configurations on different cost factors and operational
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+25
+decisions. While solving the conventional HLP with a constant discount factor, we assume
+that the transfer (discount), collection, and distribution factors are 0.75, 3, and 2, respec-
+tively, as given in the original AP dataset. After solving each instance of the HLP and
+finding the location and allocation decisions, we assign the minimum number of vehicles
+to transport flows on the resulting network. We use the same vehicle configurations as in
+the HNDPv.
+Table 5 compares the HNDPv and the conventional HLP solutions for instances with
+|N| = 25, 40, and 50 in terms of percentage difference of the following factors: the actual
+total cost (TC), the number of utilized primary and secondary vehicles (#Veh1 and
+#Veh2, respectively), and the average capacity utilization of the primary vehicles in per-
+cent (%VUtil). We use 100 ×
+HFlow
+#Veh1×Q to compute %VUtil, where HFlow is the total flow
+on the inter-hub links. As the HLP solution opens only one hub in uncapacitated instances
+(and hence no need for inter-hub vehicles), we only report the results for the capacitated
+instances.
+As can be seen, there is only one instance (i.e., 40T-L1) where both problems provided
+the same solution for. On the rest of the instances, the HLP-based solutions lead to, on
+average, 1.04%, 0.54%, and 3.6% higher costs for instances with 25, 40, and 50 nodes,
+respectively. All HLP-based solutions used the same or a larger number of primary vehicles
+on the network. In 50-node instances with tight capacities, the HLP required up to double
+the primary vehicle fleet size. The reason is that the HLP solution opened more hubs than
+the HNDPv solution. Therefore, although slightly fewer secondary vehicles were used, more
+primary vehicles were required, leading to poor vehicle utilization. Figure 6 illustrates the
+HNDPv solution (left) and the HLP-based solution (right). The HNDPv solution opens
+one less hub and constructs a different hub topology, resulting in a less costly solution
+with better utilized vehicles at the hub level. Even when #Veh1 is the same for both, the
+HNDPv solution provides a better primary vehicle utilization, as it incorporates vehicle-
+based decisions in the network design process, which can lead to a different location or
+allocation decisions (see Figure 7). The HNDPv solution was able to provide 10.53%,
+9.33%, and 28.22% better primary vehicle utilization, for instances with 25, 40, and 50
+nodes, respectively.
+We further investigate the effect of problem instance characteristics, i.e., size, hub capac-
+ity, and vehicle configurations on different cost factors and operational decisions. Our aim
+
+26
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+Table 5
+Comparison of the HLP-based solutions and the HNDPv solutions.
+Inst
+TC (% diff.)
+#Veh1 (% diff.)
+#Veh2 (% diff.)
+%VUtil (% diff.)
+25T-L1
++0.62
+0
++2.00
+−0.27
+25T-L2
++0.28
+0
+0
+−0.27
+25T-L3
++1.88
++12.50
++2.00
+−10.91
+25T-L4
++1.65
++12.50
+0
+−10.91
+25L-L1
++1.39
++33.33
++1.96
+−25.36
+25L-L2
++1.89
++33.33
+0
+−21.93
+25L-L3
++0.25
+0
+0
+−7.28
+25L-L4
++0.32
+0
+0
+−7.28
+Average (|N| = 25)
++1.04
++11.46
++0.75
+−10.53
+40T-L1
+0
+0
+0
+0
+40T-L2
++0.25
+0
+0
+−4.35
+40T-L3
++0.70
++12.50
+0
+−11.62
+40T-L4
++1.17
++12.50
+0
+−12.20
+40L-L1
++1.00
++33.33
+0
+−22.13
+40L-L2
++1.17
++33.33
+0
+−22.79
+40L-L3
++0.01
+0
+0
+−0.76
+40L-L4
++0.02
+0
+0
+−0.76
+Average (|N| = 40)
++0.54
++11.46
++0.00
+−9.33
+50T-L1
++7.10
++100.00
+−1.61
+−44.90
+50T-L2
++7.44
++100.00
+−1.79
+−40.12
+50T-L3
++6.51
++87.50
+−1.61
+−40.10
+50T-L4
++6.45
++87.50
+−1.79
+−36.10
+50L-L1
++0.31
+0
+0
+−16.13
+50L-L2
++0.34
+0
+0
+−16.13
+50L-L3
++0.30
+0
+0
+−16.13
+50L-L4
++0.33
+0
+0
+−16.13
+Average (|N| = 50)
++3.60
++46.88
+−0.85
+−28.22
+is to identify which vehicle configuration is preferred under differ conditions. We consider
+the criteria in our discussion: The objective function value or the total cost (TC), the
+total hub location cost (LC), the total transportation cost on the hub-level network (HC),
+the total transportation cost on the access-level network (DC), the number of selected
+hubs (#Hubs), #Veh1, #Veh2, and %VUtil. Our experiments indicate that the solution
+is highly sensitive to the number of nodes in the instance. In the AP dataset, smaller
+instances are created by aggregating the nodes in larger instances. As the total OD flows
+remain constant between all instances, smaller instances have larger OD shipment volumes,
+while larger instances have more fractional wij values. To see how such a characteristic
+may affect the solution, we refer to Figure 8, where the value of each criterion is plotted
+with respect to hub capacity and vehicle configurations. Figure 8 (left) illustrates solution
+characteristics for instances with |N| = 40. Here, when L2 and L4 are selected, a smaller
+total cost is incurred under all vehicle capacity (Cap) levels. The L3 configuration leads
+to higher HC and DC values, as it offers the combination of the smallest vehicles in both
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+27
+HNDPv solution
+HLP-based solution
+Figure 6
+Illustration of the solutions to problem instance 50T-L1.
+Note. Square shapes represent open hubs, small discs show the demand nodes.
+HNDPv solution
+HLP-based solution
+Figure 7
+Illustration of the solutions to problem instance 50L-L1.
+hub and access-level networks. Therefore, more trips are required in both levels, leading
+to higher operational costs. For the 40-node instances, using larger vehicles saves costs
+through consolidation, even though the operational cost for larger vehicles are higher.
+Instances with larger set N, however, give different results. Figure 8 (right) depicts sim-
+ilar criteria for instances with |N| = 100. Larger instances have more OD pairs with less
+shipment volumes. Therefore, consolidating flows are not as straightforward. TC is larger
+for vehicle configurations (VehConfs) that offer larger secondary vehicles (i.e., L2 and L4).
+Larger secondary vehicles are more costly to operate, and since flows are more fractional
+compared to the |N| = 40 instances, we cannot fully benefit from the excess capacities on
+these vehicles. Therefore, even though less number of vehicles are used in L2 and L4, larger
+DC values are obtained. Larger primary vehicles (as in L1 and L2) also leads to smaller
+#Veh1.
+
+28
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+4.25
+4.50
+4.75
+5.00
+5.25
+×105
+TC
+0.6
+0.8
+1.0
+1.2
+×105
+LC
+2
+3
+4
+5
+6
+×104
+HC
+3.2
+3.4
+3.6
+3.8 ×105
+DC
+T
+L
+U
+Hub capacity config.
+4
+6
+8
+#Veh1
+T
+L
+U
+Hub capacity config.
+45
+50
+55
+#Veh2
+Vehicle config.
+L1
+L2
+L3
+L4
+0.85
+0.90
+0.95
+1.00
+×106
+TC
+1.5
+2.0
+2.5
+×105
+LC
+0.8
+1.0
+1.2
+1.4
+×105
+HC
+5.5
+6.0
+×105
+DC
+T
+L
+U
+Hub capacity config.
+6
+8
+10
+12
+14
+#Veh1
+T
+L
+U
+Hub capacity config.
+1.02
+1.04
+1.06
+1.08
+1.10
+×102
+#Veh2
+Vehicle config.
+L1
+L2
+L3
+L4
+Figure 8
+Effect of different hub and vehicle configurations on the final solutions (Left: 40-node instances, Right:
+100-node instances).
+For both 40 and 100-node instances, we observe that hub capacities have a direct impact
+on TC and LC. The higher the Cap level, the lower TC and LC values are, regardless of
+the VehConf choice. Larger capacities also result in lower HC values and fewer number
+of primary vehicles, as more OD pairs can be linked through a single hub. However, DC
+values might increase as less number of hubs can be opened when the capacities are large.
+Overall, tight hub capacities lead to higher total costs, resulting from higher location
+and inter-hub transportation costs. When capacities are restrictive, either more hubs are
+selected or larger but more expensive hubs are opened. This may increase the total distance
+travelled between the hubs, hence more inter-hub transportation cost is incurred. When
+hub capacities are nonrestrictive, there is more flexibility in making location and allocation
+decisions, and this allows us to come up with a less costly solution.
+In Table 6, we analyze the effect of vehicle configurations on the solution characteristics in
+more details. Here, we see that if shipment volumes are larger, as seen in smaller instances,
+high-capacity secondary vehicles (i.e., in L2 and L4) helps to save costs, even though
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+29
+Table 6
+Effect of instance size and vehicle configuration on the final solution.
+|N|
+VehConf†
+TC (avg)
+LC (avg)
+HC (avg)
+DC (avg)
+#Hubs
+(avg)
+#Veh1
+(avg)
+#Veh2
+(avg)
+%VUtil
+(avg)
+25
+L1
+4.42e+5
+1.21e+5
+3.57e+4
+2.85e+5
+2.33
+4.00
+50.67
+74.37
+L2
+3.93e+5
+9.69e+4
+2.41e+4
+2.72e+5
+2.00
+3.00
+37.67
+69.07
+L3
+4.47e+5
+9.69e+4
+3.04e+4
+3.19e+5
+2.00
+4.33
+51.67
+91.35
+L4
+3.98e+5
+9.69e+4
+3.04e+4
+2.70e+5
+2.00
+4.33
+37.67
+91.35
+40
+L1
+4.77e+5
+8.59e+4
+2.97e+4
+3.62e+5
+2.33
+4.00
+58.67
+74.06
+L2
+4.46e+5
+7.84e+4
+3.11e+4
+3.36e+5
+2.33
+4.00
+45.67
+75.45
+L3
+4.85e+5
+8.59e+4
+3.71e+4
+3.62e+5
+2.33
+6.00
+58.67
+92.07
+L4
+4.53e+5
+8.59e+4
+3.71e+4
+3.30e+5
+2.33
+6.00
+45.67
+92.26
+50
+L1
+5.31e+5
+1.27e+5
+3.14e+4
+3.72e+5
+2.33
+3.67
+62.67
+77.66
+L2
+5.57e+5
+1.43e+5
+3.93e+4
+3.74e+5
+2.33
+3.67
+56.67
+75.93
+L3
+5.39e+5
+1.27e+5
+4.14e+4
+3.71e+5
+2.33
+5.67
+62.67
+89.50
+L4
+5.66e+5
+1.43e+5
+4.92e+4
+3.74e+5
+2.33
+5.67
+56.67
+87.62
+75
+L1
+6.18e+5
+1.24e+5
+5.81e+4
+4.36e+5
+3.00
+6.00
+86.00
+60.25
+L2
+6.64e+5
+1.24e+5
+5.81e+4
+4.82e+5
+3.00
+6.00
+80.00
+60.25
+L3
+6.29e+5
+1.16e+5
+5.91e+4
+4.54e+5
+3.00
+8.67
+86.00
+83.03
+L4
+6.74e+5
+1.24e+5
+6.63e+4
+4.84e+5
+3.00
+8.67
+80.00
+76.22
+100
+L1
+8.63e+5
+1.87e+5
+1.04e+5
+5.72e+5
+3.33
+8.00
+109.67
+45.82
+L2
+9.04e+5
+1.87e+5
+1.04e+5
+6.13e+5
+3.33
+8.00
+101.67
+45.70
+L3
+8.61e+5
+1.87e+5
+1.02e+5
+5.72e+5
+3.33
+10.00
+109.67
+66.39
+L4
+9.01e+5
+1.87e+5
+9.96e+4
+6.14e+5
+3.33
+9.67
+101.67
+68.07
+150
+L1
+9.31e+5
+1.10e+5
+1.09e+5
+7.12e+5
+4.00
+12.00
+158.00
+34.43
+L2
+1.02e+6
+1.10e+5
+1.09e+5
+7.97e+5
+4.00
+12.00
+151.00
+34.42
+L3
+9.20e+5
+1.10e+5
+9.50e+4
+7.15e+5
+4.00
+12.67
+158.00
+60.43
+L4
+1.01e+6
+1.10e+5
+9.50e+4
+8.01e+5
+4.00
+12.67
+151.00
+60.79
+200
+L1
+1.22e+6
+1.09e+5
+1.56e+5
+9.51e+5
+4.33
+14.67
+206.67
+28.89
+L2
+1.34e+6
+1.11e+5
+1.61e+5
+1.06e+6
+4.33
+14.67
+199.67
+29.76
+L3
+1.19e+6
+1.09e+5
+1.32e+5
+9.53e+5
+4.33
+15.00
+206.67
+52.63
+L4
+1.31e+6
+1.30e+5
+1.75e+5
+1.01e+6
+5.00
+20.00
+199.00
+41.28
+† Vehicle configuration.
+they are more expensive to operate. For larger instances, where more trips of secondary
+vehicles are required, their operational costs dominate their capacity benefits. Therefore, on
+average, L2 and L4 become expensive choices. Larger instances also require more primary
+vehicles to use, leading to lower %VUtil values. Therefore, it is always best to go with an
+option that provides the most vehicle utilization for the operational costs we pay at the
+hub-level network. That is why in large instances with |N| ≥ 100, the L3 configuration is
+the cost-efficient choice. For the smaller instances with more aggregated flows, choosing
+larger primary vehicles helps to save costs. VehConf has no significant effect on the number
+of selected hubs in most of the instances. We observed that for small-size instances with
+|N| = 25 and large hub capacities, the solution tries to open more hubs when larger primary
+vehicles are available in order to exploit the economies of scale by consolidating demand
+on the inter-hub links and save costs on the access-level network. Another exception is
+observed for the 200-node instance under L and U capacity settings. This problem requires
+
+30
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+many secondary vehicles. Therefore, when vehicle configuration is L4, one additional hub
+is opened to save on DC. This did not happen for L2 since the additional hub for L2
+would require more primary vehicle trips which would be more expensive due to the higher
+operational cost of primary vehicles in L2 compared to L4. More detailed numerical results
+are provided in the online supplement.
+5.4.
+HNDPv Instances With Stochastic Demands
+To evaluate the BC in solving the stochastic version of the HNDPv, we use the dataset
+introduced in Rostami et al. (2021) with |N| ∈ {25,40,50,75}. The demand value for an
+OD pair (i,j) in a stochastic scenario is chosen from a Poisson distribution with event
+rate πiπjwij, where wij are the demand values of the underlying AP dataset instance, and
+πi denotes the deviation from the base case being uniformly distributed in the interval
+[0.5,1.5]. Similar to Alumur et al. (2012) and Rostami et al. (2021), we consider five
+scenarios, each with a probability of occurrence of 0.2.
+Table 7 lists the CPU time for each instance under different hub capacity and vehicle
+configurations. In cases where the time limit is reached, the %Gap is reported in paren-
+theses. As expected, stochastic instances are more complex and difficult to solve than
+deterministic ones. Out of 48 instances, the BC was able to find the exact solution to 35
+instances within the time limit. Like the deterministic case, stochastic instances with tight
+hub capacities and small primary vehicle capacities are the most difficult ones to solve. The
+average optimality gap for unsolved instances is reported as 3.31%. The online companion
+of the paper gives the computational details of solving the stochastic HNDPv instances
+and a discussion on how the stochastic solution compares to the solution to the expected
+value problem.
+In Figure 9, we illustrate the effect of increasing the number of scenarios on the objective
+function value and computational time. Since the computational time increases very quickly
+as the number of scenarios increases for larger problems, we only consider the 25-node
+instances under 5 to 100 scenarios. Figure 9 shows that the increase in the number of
+scenarios leads to a lower TC value. When |S| increases beyond 60, no significant change
+in TC is observed. We observe that the effect of size S on the CPU time depends on the
+instance characteristics. In uncapacitated instances, increasing |S| increases the required
+computational time. This might not be the case when hubs are capacitated, specially with
+tight capacity levels, after |S| passes a threshold. We may explain this situation as follows.
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+31
+Table 7
+CPU and (%Gap) values of BC in solving stochastic instances.
+|N|
+Cap
+Vehicle configuration
+L1
+L2
+L3
+L4
+25
+T
+1.02
+4.02
+3.16
+8.38
+L
+0.94
+0.51
+2.43
+2.29
+U
+0.89
+0.38
+1.18
+1.09
+40
+T
+69.26
+4.06
+(1.98)
+(3.05)
+L
+24.01
+12.65
+54.05
+29.48
+U
+26.20
+10.00
+40.90
+49.37
+50
+T
+84.02
+85.17
+(1.41)
+(2.53)
+L
+16.19
+17.61
+615.14
+282.78
+U
+89.45
+52.17
+143.78
+238.97
+75
+T
+(4.46)
+(7.00)
+(5.03)
+(6.20)
+L
+2,547.60
+(1.17)
+(1.21)
+(3.22)
+U
+1,483.88
+979.86
+(1.57)
+(1.81)
+Average
+1,361.96 (4.46)
+2,097.20 (4.09)
+5,071.72 (2.24)
+5,051.03 (3.36)
+3.0
+3.5
+4.0
+TC
+×105Hub capacity config.: T
+Vehicle config.
+L1
+L2
+L3
+L4
+2.5
+3.0
+3.5
+4.0 ×105Hub capacity config.: L
+2.5
+3.0
+3.5
+×105Hub capacity config.: U
+5
+20
+40
+60
+80
+100
+|S|
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+CPU
+×103
+5
+20
+40
+60
+80
+100
+|S|
+5
+10
+15
+5
+20
+40
+60
+80
+100
+|S|
+2
+4
+6
+8
+10
+Figure 9
+Effect of the number of scenarios on TC and CPU (|N| = 25).
+The more scenarios we have, the more variables we need to handle in our problem, and
+the more feasibility cuts are generated. When the number of demand scenarios gets very
+large for a particular node or when the hubs have very limited capacities, the options that
+provide a feasible allocation to a capacitated hub become less. Therefore, the algorithm
+may start with better bounds and fix variables more efficiently, hence its faster termination.
+Overall, the stochastic solution may provide a better estimation of the total cost when
+the number of scenarios increases. However, it is expected that the capability of solving
+the instances with a large set S becomes prohibitively limited.
+
+32
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+6.
+Conclusion
+Incorporating decisions about the number and type of vehicles to use adds more complexity
+to the hub network design problem. Therefore, finding the optimal solution to large-scale
+problem instances remains an issue in this area. This study presented an efficient solution
+algorithm which is the first to solve large-scale benchmark instances with up to 200 nodes.
+Our solution method relies on Benders decomposition with feasibility subproblems where
+the extreme rays have been derived in a closed-form solution resulting in a multiple-cut gen-
+eration approach. Our computational experiments showed the superiority of this approach
+over the conventional Benders decomposition algorithm. To cope with more realistic situ-
+ations, we addressed the HNDPv under demand uncertainty and showed the flexibility of
+our solution methodology in handling the stochastic variant of the problem.
+While the benefits of vehicle-based hub network design problems are highlighted in this
+paper, several extensions can be investigated in the future to address different decisions
+at the tactical/operational levels. As vehicles are utilized to perform pickup/deliveries
+from/to the demand nodes, one may employ vehicle routes, instead of direct shipments,
+at the access-level network to reduce the number of trips and save costs. Furthermore, to
+deal with demand uncertainty, one possible research direction is to consider some criteria
+(e.g., in a risk-averse manner) that strike a balance between the transportation cost and
+the risk of not having enough resources to meet the demand.
+Acknowledgments
+The second author acknowledges the financial support of the Natural Sciences and Engineering Research
+Council of Canada under Discovery Grant RGPIN- 2020-05395.
+References
+Alumur SA, Campbell JF, Contreras I, Kara BY, Marianov V, O’Kelly ME (2021) Perspectives on modeling
+hub location problems. European Journal of Operational Research 291(1):1–17, URL http://dx.doi.
+org/10.1016/j.ejor.2020.09.039.
+Alumur SA, Nickel S, Saldanha-da Gama F (2012) Hub location under uncertainty. Transportation Research
+Part B: Methodological 46(4):529–543, URL http://dx.doi.org/10.1016/j.trb.2011.11.006.
+Baumung MN, G¨und¨uz HI (2015) Consolidation of residual volumes in a parcel service provider’s long-haul
+transportation network. Corman F, Voß S, Negenborn RR, eds., Computational Logistics, 437–450
+(Cham: Springer International Publishing), URL http://dx.doi.org/10.1007/978-3-319-24264-4_
+30.
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+33
+Carello G, Della Croce F, Ghirardi M, Tadei R (2004) Solving the hub location problem in telecommunication
+network design: A local search approach. Networks 44(2):94–105, URL http://dx.doi.org/10.1002/
+net.20020.
+Contreras I, D´ıaz JA, Fern´andez E (2009) Lagrangean relaxation for the capacitated hub location
+problem with single assignment. OR Spectrum 31(3):483–505, URL http://dx.doi.org/10.1007/
+s00291-008-0159-y.
+Corber´an ´A, Peir´o J, Campos V, Glover F, Mart´ı R (2016) Strategic oscillation for the capacitated hub
+location problem with modular links. Journal of Heuristics 22(2):221–244, URL http://dx.doi.org/
+10.1007/s10732-016-9308-7.
+Correia I, Nickel S, Saldanha-da Gama F (2010) The capacitated single-allocation hub location problem
+revisited: A note on a classical formulation. European Journal of Operational Research 207(1):92–96,
+URL http://dx.doi.org/10.1016/j.ejor.2010.04.015.
+de Camargo RS, de Miranda G, Luna HPL (2009) Benders decomposition for hub location problems with
+economies of scale. Transportation Science 43(1):86–97, URL http://dx.doi.org/10.1287/trsc.
+1080.0233.
+Ernst AT, Krishnamoorthy M (1996) Efficient algorithms for the uncapacitated single allocation p-hub
+median problem. Location Science 4(3):139–154, URL http://dx.doi.org/10.1016/S0966-8349(96)
+00011-3, hub Location.
+Farahani RZ, Hekmatfar M, Arabani AB, Nikbakhsh E (2013) Hub location problems: A review of
+models, classification, solution techniques, and applications. Computers & Industrial Engineering
+64(4):1096–1109, URL http://dx.doi.org/10.1016/j.cie.2013.01.012.
+Gurobi Optimization, LLC (2022) Gurobi Optimizer Reference Manual. URL https://www.gurobi.com.
+Hoff A, Peir´o J, Corber´an ´A, Mart´ı R (2017) Heuristics for the capacitated modular hub location problem.
+Computers & Operations Research 86:94–109, URL http://dx.doi.org/10.1016/j.cor.2017.05.
+004.
+Horner MW, O’Kelly ME (2001) Embedding economies of scale concepts for hub network design. Journal of
+Transport Geography 9(4):255–265, URL http://dx.doi.org/10.1016/S0966-6923(01)00019-9.
+Hu QM, Hu S, Wang J, Li X (2021) Stochastic single allocation hub location problems with balanced
+utilization of hub capacities. Transportation Research Part B: Methodological 153:204–227, URL http:
+//dx.doi.org/10.1016/j.trb.2021.09.009.
+Keshvari Fard M, Alfandari L (2019) Trade-offs between the stepwise cost function and its linear approxi-
+mation for the modular hub location problem. Computers & Operations Research 104:358–374, URL
+http://dx.doi.org/10.1016/j.cor.2018.11.014.
+Kimms A (2006) Economies of scale in hub & spoke network design models: We have it all wrong. Morlock
+M, Schwindt C, Trautmann N, Zimmermann J, eds., Perspectives on Operations Research: Essays in
+
+34
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+Honor of Klaus Neumann, 293–317 (DUV), ISBN 978-3-8350-9064-4, URL http://dx.doi.org/10.
+1007/978-3-8350-9064-4_17.
+Laporte G, Louveaux FV (1993) The integer l-shaped method for stochastic integer programs with
+complete recourse. Operations Research Letters 13(3):133–142, URL http://dx.doi.org/10.1016/
+0167-6377(93)90002-X.
+O’Kelly ME, Bryan DL (1998) Hub location with flow economies of scale. Transportation Research Part B:
+Methodological 32(8):605–616, URL http://dx.doi.org/10.1016/S0191-2615(98)00021-6.
+Qin Z, Gao Y (2017) Uncapacitated p-hub location problem with fixed costs and uncertain flows. Jour-
+nal of Intelligent Manufacturing 28(3):705–716, ISSN 1572-8145, URL http://dx.doi.org/10.1007/
+s10845-014-0990-8.
+Racunica I, Wynter L (2005) Optimal location of intermodal freight hubs. Transportation Research Part B:
+Methodological 39(5):453–477, URL http://dx.doi.org/10.1016/j.trb.2004.07.001.
+Real LB, Contreras I, Cordeau JF, de Camargo RS, de Miranda G (2021) Multimodal hub network design
+with flexible routes. Transportation Research Part E: Logistics and Transportation Review 146:102188,
+URL http://dx.doi.org/10.1016/j.tre.2020.102188.
+Rostami B, Buchheim C (2015) The uncapacitated single allocation p-hub median problem with stepwise
+cost function. Optimization Online. Preprint on webpage at https://optimization-online.org/?p=
+13555.
+Rostami B, Chitsaz M, Arslan O, Laporte G, Lodi A (2022) Single allocation hub location with hetero-
+geneous economies of scale. Operations Research 70(2):766–785, URL http://dx.doi.org/10.1287/
+opre.2021.2185.
+Rostami B, K¨ammerling N, Naoum-Sawaya J, Buchheim C, Clausen U (2021) Stochastic single-allocation
+hub location. European Journal of Operational Research 289(3):1087–1106, URL http://dx.doi.org/
+10.1016/j.ejor.2020.07.051.
+Sender J, Clausen U (2011) Hub location problems with choice of different hub capacities and vehicle types.
+Pahl J, Reiners T, Voß S, eds., Network Optimization, 535–546 (Berlin, Heidelberg: Springer Berlin
+Heidelberg), URL http://dx.doi.org/10.1007/978-3-642-21527-8_59.
+Serper EZ, Alumur SA (2016) The design of capacitated intermodal hub networks with different vehicle
+types. Transportation Research Part B: Methodological 86:51–65, URL http://dx.doi.org/10.1016/
+j.trb.2016.01.011.
+Skorin-Kapov D, Skorin-Kapov J, O’Kelly M (1996) Tight linear programming relaxations of uncapacitated
+p-hub median problems. European Journal of Operational Research 94(3):582–593, URL http://dx.
+doi.org/10.1016/0377-2217(95)00100-X.
+Tanash M, Contreras I, Vidyarthi N (2017) An exact algorithm for the modular hub location problem with
+single assignments. Computers & Operations Research 85:32–44, URL http://dx.doi.org/10.1016/
+j.cor.2017.03.006.
+
+Farham, Rostami, and Haughton: Vehicle utilization in the SAHLP
+35
+Tran TH, O’Hanley JR, Scaparra MP (2017) Reliable hub network design: Formulation and solution tech-
+niques. Transportation Science 51(1):358–375, URL http://dx.doi.org/10.1287/trsc.2016.0679.
+Wang S, Chen Z, Liu T (2020) Distributionally robust hub location. Transportation Science 54(5):1189–1210,
+URL http://dx.doi.org/10.1287/trsc.2019.0948.
+Yaman H, Carello G (2005) Solving the hub location problem with modular link capacities. Computers &
+Operations Research 32(12):3227–3245, URL http://dx.doi.org/10.1016/j.cor.2004.05.009.
+

7tAzT4oBgHgl3EQf-f6Q/content/tmp_files/2301.01935v1.pdf.txt

@@ -0,0 +1,826 @@
+Topic Segmentation Model Focusing on Local Context
+Jeonghwan Lee, Jiyeong Han, Sunghoon Baek, Min Song*
+Yonsei University
+[email protected], [email protected], [email protected], [email protected]
+Abstract
+Topic segmentation is important in understanding scientific
+documents since it can not only provide better readability but
+also facilitate downstream tasks such as information retrieval
+and question answering by creating appropriate sections or
+paragraphs. In the topic segmentation task, topic coherence is
+critical in predicting segmentation boundaries. Most of the
+existing models have tried to exploit as many contexts as
+possible to extract useful topic-related information. However,
+additional context does not always bring promising results,
+because the local context between sentences becomes inco-
+herent despite more sentences being supplemented. To allevi-
+ate this issue, we propose siamese sentence embedding lay-
+ers which process two input sentences independently to get
+appropriate amount of information without being hampered
+by excessive information. Also, we adopt multi-task learn-
+ing techniques including Same Topic Prediction (STP), Topic
+Classification (TC) and Next Sentence Prediction (NSP).
+When these three classification layers are combined in a
+multi-task manner, they can make up for each other’s limi-
+tations, improving performance in all three tasks. We exper-
+iment different combinations of the three layers and report
+how each layer affects other layers in the same combination
+as well as the overall segmentation performance. The model
+we proposed achieves the state-of-the-art result in the Wiki-
+Section dataset.
+Introduction
+Nowadays, we can easily access vast amounts of scientific
+documents such as PubMed and Wikipedia. A lot of re-
+searchers are studying ways to effectively use these docu-
+ments in areas like information retrieval (IR), question an-
+swering (QA) and search engine. However, applying previ-
+ous IR models (or QA models) directly on these documents
+is impossible because most of them assume an input size of
+at most a paragraph while these documents consist of multi-
+ple paragraphs. Furthermore, extracting crucial parts of each
+document does not necessarily require the whole document
+to be used. For example, to search for similar papers on a
+topic of interest we can simplify the problem by calculating
+cosine similarity between sections of each document rather
+than full text to save resources.
+*Corresponding author.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+Which topic?
+Which topic?
+Which topic?
+Which topic?
+Which topic?
+Sentence 1
+Sentence 2
+Sentence 3
+Sentence n
+Sentence n-1
+.
+.
+.
+Same topic?
+&
+Consecutive?
+Same topic?
+&
+Consecutive?
+Same topic?
+&
+Consecutive?
+Figure 1: A window with a size of 1 slides through the entire
+sentence, predicting the topic of each sentence. At the same
+time, the model determines whether the two sentences are in
+the same topic and whether the two sentences are consecu-
+tive.
+These are where topic segmentation can be used. Topic
+segmentation divides a document into segments with respect
+to the topic coherence of each segment. A well-divided doc-
+ument according to the topics provides better readability,
+making it easier for the readers to find the desired infor-
+mation in the document. Most importantly, it can facilitate
+downstream-tasks such as IR and QA.
+Although most of the existing topic segmentation models
+take topic coherence into consideration when dividing a doc-
+ument, they don’t undergo the process of classifying topic
+labels for each sentence, even when these topic labels are
+useful for inferring topic coherence. Most importantly, in-
+spired by Neural Text Segmentation Model (Koshorek et al.
+2018), these models are designed to take block of text as in-
+put, which possibly hinders understanding local context of
+the input text (Xing et al. 2020).
+We try to tackle the above issues by adopting a siamese
+network to encode two input sentences independently and
+putting them through a multi-task learning algorithm that in-
+cludes topic classification and other auxiliary tasks. First,
+in order to deal with two input sentences independently,
+we construct our model in a siamese network with sen-
+tence embeddings from a Sentence Transformer (Reimers
+arXiv:2301.01935v1  [cs.CL]  5 Jan 2023
+
+and Gurevych 2019). This method allows our model to pre-
+serve local context between the two input sentences without
+being overwhelmed by excessive information.
+We consider topic segmentation as a Same Topic Predic-
+tion (STP) between two input sentences, following Aumiller
+et al. (2021). However, because our model processes only
+one sentence at a time to preserve its unique information, the
+model cannot observe context information across sentences.
+To alleviate this issue, we add two auxiliary tasks to cap-
+ture local context information. One of them is Topic Classi-
+fication(TC) which predicts the exact topic of the input sen-
+tences through a topic classification layer to assist STP with
+a detailed topic information. The other is a Next Sentence
+Prediction (NSP) layer (Devlin et al. 2019), which supports
+the model in understanding the relationship between con-
+secutive sentences. Figure 1 simply shows how our model
+works.
+To sum up, our model deals with two input sentences in-
+dependently via the siamese sentence embedding layer that
+preserves local context of input sentences. Also, we show
+that connecting tasks that utilize same input sentences to ex-
+tract different features in the sentence in a multi-task manner
+improves topic segmentation performance. Consequently,
+our model achieves state-of-the-art in the topic segmentation
+task using the WikiSection dataset.
+Related Work
+Topic Segmentation
+Koshorek et al. (2018) solved topic segmentation task as a
+supervised neural network model. Block of text that consists
+of several sentences is fed into the model and the model pre-
+dicts whether each sentence should be a segmentation point.
+Badjatiya et al. (2018) introduced k-sized left and right
+supporting sentences, where neighboring k number of sen-
+tences support injecting context into input sentence. How-
+ever, Xing et al. (2020) pointed out that ”local context” was
+more important than ”global context” in topic segmentation
+task, implying that excessive context might decrease the per-
+formance. The information from various sentences can hin-
+der predicting label of a single sentence due to deterioration
+in the model’s understanding of local context.
+Arnold et al. (2019) proposed Sector which includes a
+topic embedding layer in their architecture. This topic em-
+bedding layer is implemented for topic classification and the
+result of topic classification is then used for segmentation.
+Aumiller et al. (2021) treated topic segmentation as a
+Same Topic Prediction(STP) between two input paragraphs.
+STP determines whether two input paragraphs refer to the
+same topic. They also experimented diverse sampling meth-
+ods, and among these methods we adopt consecutive sam-
+pling. Details about consecutive sampling will be explained
+at section .
+Sentence Embedding
+Sentence embedding is a method of capturing the seman-
+tic relationships among words in a sentence (Conneau et al.
+2017). Quality of sentence embedding is critical especially
+in the topic segmentation task, because the task inevitably
+has to capture as much information as possible from long
+sentences as well as short ones. Koshorek et al. (2018) uti-
+lized Bi-LSTM to generate sentence embedding where word
+embedding vectors are extracted from Word2Vec with each
+word in a sentence as input, fed into the Bi-LSTM layer one
+by one, and the final sequence representation was made by
+max-pooling over the output of the LSTM.
+After the introduction of BERT, Reimers and Gurevych
+(2019) proposed Sentence BERT specialized in creating
+sentence embeddings. Sentence BERT is a fine-tuned ver-
+sion of BERT trained on NLI(Natural Language Inference)
+and STS(Semantic Textual Similarity) task. To handle input
+sentences effectively, the authors adopted siamese network
+which encodes each sentence independently and concate-
+nates the encoded sentences to be fed into a classification
+layer. Consequently, Sentence BERT has better capability of
+dealing with long sentences, because the model understands
+high-level context of these sentences.
+As another branch of Sentence BERT, SimCSE was pro-
+posed (Gao, Yao, and Chen 2021). The authors applied con-
+trastive learning to forming sentence embeddings. They tried
+unsupervised method and showed that dropout could work
+as data augmentation and this prevented representation col-
+lapse. They also empirically and theoretically proved that
+contrastive learning objective was suitable for regularizing
+anisotropic space of a language model’s embedding to be
+more uniform and it aligned positive pairs better in a super-
+vised setting as a result.
+Lukasik et al. (2020) proposed Cross-segment BERT.
+They used pre-trained BERT in which left and right context
+were separated via [SEP] token and encoded the sequence of
+word-piece tokens into sentence representations. Aumiller
+et al. (2021) used Sentence BERT for a sentence encoder
+which is known to have substantial capability of understand-
+ing high-level context.
+Proposed Approach
+Architecture
+Our model follows the typical architecture of text segmen-
+tation models: a sentence embedding layer followed by a
+segment classifier, which is replaced by a Same Topic Pre-
+diction layer in our model.
+However, our model takes two input sentences. To handle
+them independently, our model composes sentence embed-
+ding layer in siamese network form, so that the model re-
+ceives an appropriate amount of information to predict the
+label. The encoded sentences are then fed into the topic
+classification layer one by one. By passing each sentence
+through the layer, the model acquires topic-related informa-
+tion of the sentence. Also, we adopt NSP layer to capture
+semantic relationship between the two sentences. Finally,
+STP layer predicts whether the sentences belong to the same
+topic.
+We have k documents D1, ...Dk that D consist of n num-
+ber sentences s1, ..., sn, and the sentences are paired con-
+secutively; [(s1, s2), (s2, s3), ..., (sn−2, sn−1), (sn−1, sn)].
+Each si(i ≤ n) is assigned a topic label ti which describes
+topic label of ith sentence.
+
+Models
+STP Loss
+TC Loss
+NSP Loss
+STP+TC
+4
+1
+-
+STP+NSP
+1
+-
+1
+STP+TC+NSP
+4
+1
+4
+Table 1: Designated loss weights for each layer in case of
+multi-task learning
+The sentence embedding layer encodes each input sen-
+tences si and si+1 and the encoded sentences are repre-
+sented as u and v, respectively. Figure 2 shows the overview
+of our model.
+Siamese Sentence Embedding Layers from Sentence
+Transformer
+We propose siamese sentence embedding
+layer. In our model, Sentence Transformer encodes each sen-
+tence from two input sentences independently at the entry
+level. Then, the encoded sentences are concatenated before
+being fed into the STP layer. This method aims to preserve
+each sentence’s unique information while acquiring local
+context between the two sentences.
+Multi Task Learning
+Our model has a total of three clas-
+sification layers and we train them in a multi-task manner.
+Topic classification layer: Topic classification layer is
+designed to capture exact topic information of a sentence.
+Topic classification is a multi-class classification that pre-
+dicts the topic of an input sentence out of 30 labels for
+en city and 27 labels for en disease dataset (Arnold et al.
+2019) . This layer takes u and v one by one and predicts
+each topic label ti and ti+1.
+NSP layer: NSP layer is fed with u; v; |u − v| (Reimers
+and Gurevych 2019) and the layer predicts NSP label. This
+layer aims to supplement STP layer’s limitation where STP
+layer can only determine whether the two input sentences are
+in the same topic and cannot determine if the sentences are
+actually consecutive. By adding NSP, the model can capture
+the semantic relationship between the two sentences, so the
+model can figure out whether the sentences are consecutive.
+NSP layer must go with consecutive sampling which will be
+explained below.
+STP layer: STP layer is provided with u; v; |u − v| again
+and finally predicts segmentation label that is used to draw
+segmentation points in places where the two sentences be-
+long to different topics.
+Multi task learning: When the three layers are combined
+in a multi-task manner, they can make up each other’s limi-
+tations. Since STP is based on binary classification, its task
+is much simpler than Topic Classification that is based on
+multi-class classification. However, since STP cannot cap-
+ture the exact topic label of input sentences, Topic classi-
+fication provides this information to the STP layer to help
+determine segmentation boundaries. The effect of NSP is ex-
+plained above.
+Loss weight: Because the losses from each layer are
+all different, there is a need to adjust the weights among
+the losses for improved model performance. We decide the
+weights for each loss by running numerous manual experi-
+ments and calculate the total loss using a weighted sum of
+en city
+en disease
+Docs
+19,539
+3,590
+Topics
+30
+27
+Table 2: The number of documents and topics for en city
+and en disease
+the three losses from each classification layer. Table 1 sum-
+marizes how each loss is weighted.
+Consecutive Sampling
+To make the model more robust, we add negative samples
+to the dataset by adopting consecutive sampling (Aumiller
+et al. 2021). In consecutive sampling, all samples come from
+the same document.
+We have a document Da and a sentence sti
+i ∈ Da where
+the superscript ti
+refers to topic label of si. We pick one
+positive sample and two negative samples. The positive sam-
+ple sti+1=ti
+i+1
+∈ Da is consecutive to si
+and the first nega-
+tive sample stk=ti
+k̸=i+1 is from the same topic as si, but not con-
+secutive to si . Finally, the second negative sample stl̸=ti
+l
+is from different topics, which is naturally considered not
+consecutive to si .
+Experiment
+Dataset
+We use WikiSection (Arnold et al. 2019) for training and
+evaluating our model. WikiSection covers two distinct do-
+mains: city and disease. Each domain has 19,539 and 3,590
+documents, respectively, with various topics in each docu-
+ment. In total, there are 30 and 27 topics for each domain.
+The dataset is divided into 70% training, 10% validation and
+20% test sets. Table 2 gives statistics of the dataset.
+Experimental Setup
+We use nltk sentence tokenizer1 to split the documents into
+sentence units and apply consecutive sampling only on the
+training dataset. Table 3 gives the data statistics after ap-
+plying sentence split and consecutive sampling. We imple-
+ment all-MiniLM-L12-v2 and from Sentence-Transformers2
+for our sentence encoder. We set the maximum epoch size to
+14 but save the model only when the validation Pk scores
+best. Batch size is 48, learning rate is 1e−6 and LinearLR
+scheduler is applied with the default parameters setting.
+Metric
+For
+a
+comprehensive
+evaluation,
+we
+used
+Pk,
+WindowDiff and micro F1 score to evaluate our models.
+We use Pk score for making comparisons between our
+models and all other baseline models, and WindowDiff
+is used to evaluate ours and Cross-segment BERT that we
+implemented. F1 score is used for the purpose of ablation
+study on our models.
+1NLTK :: Natural Language Toolkit
+2Pretrained Models — Sentence-Transformers documentation
+(sbert.net)
+
+𝑠!
+𝑠!"#
+Pooling
+Pooling
+𝑢
+𝑣
+Softmax Classifier
+(NSP)
+𝑢; 𝑣; |𝑢 − 𝑣|
+0 or 1
+Sentence Transformer
+Sentence Transformer
+𝑡!
+𝑡!"#
+Softmax Classifier
+(Topic Classification)
+Softmax Classifier
+(STP)
+0 or 1
+Figure 2: The overview of our model. Two input sentences which are considered consecutive are fed into Sentence Transformer
+independently and encodes each input sentence. Each max-pooled encoded sentence, represented by u and v respectively, is fed
+into Topic classification layer. Before being fed into NSP layer and STP layer, we make concatenated feature u; v; |u−v|. Using
+u; v; |u − v|, NSP layer predicts if the two sentences are consecutive and STP layer finally determines whether they belong to
+same topic.
+79.5
+95.1
+95.3
+95.4
+95.8
+72.7
+73.2
+73.3
+85.7
+86.5
+40
+50
+60
+70
+80
+90
+100
+TC only
+STP only
+STP+TC
+STP+NSP
+STP+TC+NSP
+F1
+Micro F1 Scores of STP, TC and NSP
+(en_city) 
+STP
+TC
+NSP
+54.8
+87.6
+88
+88.3
+88.4
+45.8
+45.9
+46.2
+64
+64.5
+40
+50
+60
+70
+80
+90
+100
+TC only
+STP only
+STP+TC
+STP+NSP
+STP+TC+NSP
+F1
+Micro F1 Scores of STP, TC and NSP
+(en_disease) 
+STP
+TC
+NSP
+Figure 3: Figure of F1 scores with combination of different task layers. In case of TC-only model, STP output is 1 if the results
+of topic classification on each sentence refer to the same topic otherwise 0.
+en city
+en disease
+Train
+1,690,103
+336,459
+Valid
+85,072
+16,285
+Test
+168,924
+31,110
+Table 3: The number of rows after applying nltk sentence
+tokenizer and consecutive sampling. Consecutive sampling
+is applied only on the trainset.
+Pk
+Pk (Beeferman, Berger, and Lafferty 1999) is a proba-
+bility that a segmentation model performs an incorrect seg-
+mentation. While a sliding window of size k passing over
+the sentences, the status (0 or 1) is determined by whether
+the two ends of the window are in the same segment or in
+different segments. Pk is calculated by counting unmatched
+cases between the ground truths and predicted values. As in
+many previous studies, we set the window size k to half the
+average segment length of the ground truths.
+
+Dataset
+en city
+en disease
+Metric
+Pk
+WinDiff
+Pk
+WinDiff
+SEC>T+emb
+15.5
+-
+26.3
+-
+Transformer2
+BERT
+8.2
+-
+18.8
+-
+BiLSTM + BERT
+9.3
+-
+28.0
+-
+Cross-segment BERT n context = 2
+15.4
+27.4
+33.9
+59.0
+Cross-segment BERT n context = 4
+18.3
+32.2
+34.8
+60.3
+Cross-segment BERT n context = 6
+45.1
+50.0
+34.0
+57.1
+TC-only
+15.0
+17.8
+41.5
+45.4
+STP-only
+5.1
+5.8
+14.8
+15.8
+STP + TC
+5.0
+5.7
+14.0
+15.0
+STP + NSP
+4.9
+5.6
+14.1
+15.1
+STP + TC + NSP
+4.6
+5.2
+13.7
+14.7
+Table 4: Test Pk and WindowDiff scores of baseline models and our models. Note that the WinDiff metric is used only
+in our models and the Cross-segment BERT models. We reimplement Cross-segment BERT ourselves following their official
+codes.
+WindowDiff
+WindowDiff
+(Pevzner
+and
+Hearst
+2002) is an improved metric from Pk in that it alleviates
+the impact of false negative penalty and segment size
+distribution.
+Similar to Pk, WindowDiff score also uses sliding win-
+dow and compares the ground truths with the predicted val-
+ues. However, this metric also takes the number of bound-
+aries into consideration. It is closer to the ground truth when
+the models get a lower score in both Pk and WindowDiff.
+Baseline Models
+We compare our model with competitive neural text seg-
+mentation baselines 1) SEC>T+emb (Arnold et al. 2019),
+2) Transformer2
+BERT (Lo et al. 2021), a framework based
+on two transformers, where one is a pre-trained transformer
+for encoding sentences and the other is a transformer for seg-
+mentation, 3) Bi-LSTM + BERT (Xing et al. 2020), that is
+based on a hierarchical attention Bi-LSTM network, and 4)
+Cross-segment BERT (Lukasik et al. 2020), which handles
+left and right context simultaneously using a BERT encoder.
+We adopt the results of SEC>T+emb from Arnold
+et al. (2019), Transformer2
+BERT and Bi-LSTM + BERT
+from Xing et al. (2020). We implement Cross-segment
+BERT ourselves following their official code while apply-
+ing diverse size of context.
+Results and Analysis
+We report evaluation results on Figure 3 and Table 4. Fig-
+ure 3 summarizes how combination of each classification
+layer affects their F1 scores. Table 4 shows performance
+comparison between our model and other baseline models
+in Pk and WindowDiff, respectively. Our proposed mod-
+els, except for TC-only model, outperform all the baseline
+models by a large margin.
+Effect of MTL
+Figure 3 shows F1 scores derived from
+combinations of tasks mentioned above. We can see that
+MTL is effective in improving the performance, which ap-
+plies to not only the performance of STP that is responsible
+segmentation but also the performances of TC and NSP. This
+is believed to be because, as we pointed out in the section ,
+the layers make up for each other’s limitations by extract-
+ing different features from same input sentences that assist
+understanding semantic information.
+STP-only vs TC-only
+In order to verify the effectiveness
+of STP layer, we also experiment TC-only model, which is
+close to Sector in that segmentation is performed only using
+topic labels.
+Pk and WindowDiff of TC-only model are much
+higher than those of STP-only model. Poor classification
+performance of Topic classification directly causes this phe-
+nomenon. Figure 3 indicates that F1 scores of TC-only
+model are significantly lower than those of STP-only. Be-
+cause topic classification layer is based on multi-class classi-
+fication, which is more difficult than the binary classification
+of STP-only.
+STP vs NSP
+Although STP and NSP have the same ar-
+chitecture, STP’s F1 scores are always higher than NSP’s in
+both datasets. We assume that this difference is derived from
+the difference in the information that STP and NSP focus on.
+STP-only determines whether the two sentences belong to
+the same topic, so it only pays attention to topic differences
+between two sentences. In other words, due to the nature of
+the task, STP does not consider the relationship between two
+input sentences. However, in the NSP task, the layer faces
+difficulties as two sentences may not be consecutive even
+if they belong to the same topic because of our consecutive
+sampling. Thus, NSP must find the semantic relationship be-
+tween the two sentences as well as topic coherence, which
+makes the task tricky.
+How the number of contexts affects the performance
+To show the importance of local context, we implement
+Cross-segment BERT (Lukasik et al. 2020) by applying di-
+verse size of context on the model. Table 4 shows that raising
+context size rather deteriorates the performance. We conjec-
+ture that this is because the more sentences there are, the
+more likely for different topics to be mingled, which likely
+interferes the model from understanding local context with
+
+Left context
+𝑠!
+𝑠!"#
+𝑠!$#
+𝑠!$%
+𝑠!$&
+𝑠!"%
+𝑠!"&
+𝑠!"'
+n_context = 1
+𝑠!
+𝑠!"#
+𝑠!$#
+𝑠!$%
+𝑠!$&
+𝑠!"%
+𝑠!"&
+𝑠!"'
+n_context = 4
+𝑠!
+𝑠!"#
+𝑠!$#
+𝑠!$%
+𝑠!$&
+𝑠!"%
+𝑠!"&
+𝑠!"'
+n_context = 2
+𝑠!
+𝑠!"#
+𝑠!$#
+𝑠!$%
+𝑠!$&
+𝑠!"%
+𝑠!"&
+𝑠!"'
+n_context = 3
+Right context
+Left context
+Right context
+Left context
+Right context
+Left context
+Right context
+Figure 4: Effect of context size on prediction. The color of the box represents the topic of the sentence and the red line represents
+supporting context. The vertical dotted line represents a segmentation point between si and si+1 while the dotted box describes
+that the two sentences are not segmented. si
+and si+1
+should be divided, since they belong to different topics, but are not
+segmented in cases of n context = 3 and n context = 4 .
+overflow of noise. Because processing multiple sentences si-
+multaneously using a left and right context structure rather
+adds noise to the contexts, we choose to encode the two in-
+put sentences independently.
+Figure 4 explains this local context capturing error. The
+models in Figure 4 are all expected to create a segmentation
+point between si and si+1 , but models with larger con-
+text sizes fail to split the two sentences, because segmenta-
+tion only takes into account the overall context of each side.
+As the context size increases, the model suffers from gener-
+alization and interprets left and right contexts as similar even
+when the two specific sentences refer to different topics and
+hence should be segmented.
+Also, Cross-segment BERT encodes left and right context
+simultaneously. Because Cross-encoder inevitably makes
+context of one input sentence influence the other (Humeau
+et al. 2019), the unique information of each context can
+change unexpectedly. Therefore, we process each sentence
+independently via siamese sentence embedding in order to
+preserve the original local context.
+Dealing with Scientific Documents
+We can also find that
+the scores for en disease are underperforming compared to
+that for en city. We assume that this result due to the fact
+that en disease is more science domain specific (i.e. biol-
+ogy) while en city covers relatively general topics. Arnold
+et al. (2019) commented that documents in en disease are
+described in a precise language, but on the other hand those
+in en city are described in a common language. Consider-
+ing that our backbone, miniLM was pre-trained on general
+documents like Wikipedia, the result seems natural.
+To improve the performance on en disease, we implement
+SPECTER (Cohan et al. 2020) which was trained on scien-
+tific papers using Sci-BERT as the backbone model.
+As shown in table 5, Pk and WinDiff improved by 1.6
+and 1.8 respectively in en disease compared to the miniLM
+based model. We attribute the improvement to SPECTER’s
+understanding of scientific documents. We expect the scores
+Pk
+WinDiff
+en city
+4.6
+5.2
+en disease
+12.1
+12.9
+Table 5: Test Pk and WindowDiff scores of SPECTER
+based Topic Segmentation Model with STP+TC+NSP
+for en disease to be improved if we use more biology do-
+main specific model like BioBERT as the backbone.
+Interestingly, although the number of parameters of
+SPECTER was twice as large as that of miniLM (i.e. 768
+vs 384), there was no improvement in the performance in
+en city. From this result, we can again confirm that domain
+knowledge is critical to the performance.
+Conclusion and Future Work
+In this work, we propose our topic segmentation model
+which consists of siamese sentence embedding layer from
+Sentence Transformer and three classification layers. With
+several different experiments, we show that our proposed
+model outperforms all the existing models. We also find that
+combining Same Topic Prediction, Topic Classification and
+Next Sentence Prediction in a multi-task manner increases
+segmentation performance.
+Moreover, we empirically show the importance of local
+context in topic segmentation task. Contrary to the popu-
+lar belief, increasing the number of context can rather de-
+grade the performance due to generalization of local context.
+Our experiment indicates that narrowing context through our
+siamese sentence embedding layer can be effective in pre-
+serving local context.
+Future work can highlight on the theoretical approach to
+local context. Although we empirically showed the influence
+of context size to the model performance in this paper, we
+did not concentrate on how we can determine which input
+sentences can provide substantial information in performing
+segmentation tasks. If we can infer each sentence’s signifi-
+
+cance in prediction, we expect the model to capture the im-
+portant sentences autonomously, consequently making the
+model agnostic to the context size.
+Acknowledgments
+This work was supported by Institute of Information &
+communications Technology Planning & Evaluation (IITP)
+grant funded by the Korea government(MSIT) (No. 2020-
+0-01361, Artificial Intelligence Graduate School Program
+(Yonsei University)).
+References
+Arnold, S.; Schneider, R.; Cudr´e-Mauroux, P.; Gers, F. A.;
+and L¨oser, A. 2019. SECTOR: A Neural Model for Coherent
+Topic Segmentation and Classification.
+Aumiller, D.; Almasian, S. S.; Lackner, M.; and Gertz. 2021.
+Structural text segmentation of legal documents. In Proceed-
+ings of the Eighteenth International Conference on Artificial
+Intelligence and Law, 2–11. Association for Computing Ma-
+chinery.
+Badjatiya, P.; Kurisinkel, L. J.; Gupta, M.; and Varma, V.
+2018. Attention-based Neural Text Segmentation.
+Beeferman, D.; Berger, A.; and Lafferty, J. 1999. Statistical
+Models for Text Segmentation. Machine Learning, 34: 177–
+210.
+Cohan, A.; Feldman, S.; Beltagy, I.; Downey, D.; and Weld,
+D. 2020. SPECTER: Document-level Representation Learn-
+ing using Citation-informed Transformers. In Proceedings
+of the 58th Annual Meeting of the Association for Com-
+putational Linguistics, 2270–2282. Online: Association for
+Computational Linguistics.
+Conneau, A.; Kiela, D.; Schwenk, H.; Barrault, L.; and Bor-
+des, A. 2017. Supervised Learning of Universal Sentence
+Representations from Natural Language Inference Data. In
+Proceedings of the 2017 Conference on Empirical Methods
+in Natural Language Processing, 670–680. Association for
+Computational Linguistics.
+Devlin, J.; Chang, M.-W.; Lee, K.; and Toutanova, K. 2019.
+BERT: Pre-training of Deep Bidirectional Transformers for
+Language Understanding. In Proceedings of the 2019 Con-
+ference of the North American Chapter of the Association
+for Computational Linguistics: Human Language Technolo-
+gies, volume 1, 4171–4186. Association for Computational
+Linguistics.
+Gao, T.; Yao, X.; and Chen, D. 2021. SimCSE: Simple Con-
+trastive Learning of Sentence Embeddings. In Proceedings
+of the 2021 Conference on Empirical Methods in Natural
+Language Processing, 6894–6910. Online and Punta Cana,
+Dominican Republic: Association for Computational Lin-
+guistics.
+Humeau, S.; Shuster, K.; Lachaux, M.-A.; and Weston, J.
+2019. Poly-encoders: Transformer Architectures and Pre-
+training Strategies for Fast and Accurate Multi-sentence
+Scoring.
+Koshorek, O.; Cohen, A.; Mor, N.; Rotman, M.; and Berant,
+J. 2018. Text Segmentation as a Supervised Learning Task.
+In Proceedings of the 2018 Conference of the North Ameri-
+can Chapter of the Association for Computational Linguis-
+tics: Human Language Technologies, volume 2, 469–473.
+Association for Computational Linguistics.
+Lo, K.; Jin, Y.; Tan, W.; Liu, M.; Du, L.; and Buntine, W.
+2021. Transformer over Pre-trained Transformer for Neu-
+ral Text Segmentation with Enhanced Topic Coherence. In
+Findings of the Association for Computational Linguistics:
+EMNLP 2021, 3334–3340. Association for Computational
+Linguistics.
+Lukasik, M.; Dadachev, B.; Papineni, K.; and Sim˜oes, G.
+2020. Text Segmentation by Cross Segment Attention. Pro-
+ceedings of the 2020 Conference on Empirical Methods in
+Natural Language Processing (EMNLP), 4707–4716.
+Pevzner, L.; and Hearst, M. A. 2002. A critique and im-
+provement of an evaluation metric for text segmentation.
+Computational Linguistics, 28(1): 19–36.
+Reimers, N.; and Gurevych, I. 2019.
+Sentence-BERT:
+Sentence Embeddings using Siamese BERT-Networks. In
+Proceedings of the 2019 Conference on Empirical Meth-
+ods in Natural Language Processing and the 9th Interna-
+tional Joint Conference on Natural Language Processing
+(EMNLP-IJCNLP), 3982–3992. Association for Computa-
+tional Linguistics.
+Xing, L.; Hackinen, B.; Carenini, G.; and Trebbi, F. 2020.
+Improving Context Modeling in Neural Topic Segmenta-
+tion. In Proceedings of the 1st Conference of the Asia-Pacific
+Chapter of the Association for Computational Linguistics
+and the 10th International Joint Conference on Natural Lan-
+guage Processing, 626–636. Association for Computational
+Linguistics.
+

8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:09d491c8123a3924adcc5030ab425821666a8ef1c2371e3f220dc03ab729bac7
+size 2141015

8NAyT4oBgHgl3EQfp_h6/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ea857be78444fa9fd163820b6743cee71f8302602a027b4318d33e26bdc32235
+size 5439533

8NAyT4oBgHgl3EQfp_h6/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7d048605ff2e18089f2a0a214313fa0b65e1de2fb31bce834aab3e5474f9e85c
+size 199211

8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7aadeb9d55ec348beda38760c0406eaca9baf3581fdd97a215e2b919c9508ebc
+size 1525780

8NE4T4oBgHgl3EQfdAy1/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:8a41ce89ef02ccd4970bab589013d37de7009009ba9aedc19b9e3d50aa18930c
+size 3801133

8NE4T4oBgHgl3EQfdAy1/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:546e18d6e0c84af4932a299f0ad7d0a275fd1c65bc7fdad66ed2a7c4282253e7
+size 127132

8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:fc72b4a9610cbfd05a8aaa8d5c7a7ff479ed9b37e36db119b64798f7a472e8e1
+size 410955

8tAyT4oBgHgl3EQfc_f6/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e87237db7d5beddb9e4c5789a1d35c78413481ae7b6ccd2ca2b303f05f5a4dfb
+size 1245229

8tAyT4oBgHgl3EQfc_f6/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7c20e40c4b1f86aa81286156d4f3a01f9b15fdb86f9f7116d0b726a4ee8b9ef2
+size 52061

9NAzT4oBgHgl3EQfFPpp/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:98dcd71477c861adb16594f83ae54f5e48b8c9310789d3c7477dbec3a1691273
+size 196134

9tAyT4oBgHgl3EQfdPfZ/content/tmp_files/2301.00300v1.pdf.txt

@@ -0,0 +1,1769 @@
+arXiv:2301.00300v1  [math.AP]  31 Dec 2022
+STOCHASTIC QUANTIZATION OF LASER PROPAGATION MODELS
+SIVAGURU S. SRITHARAN1,2* AND SABA MUDALIAR1
+Abstract. This paper identifies certain interesting mathematical problems of stochastic
+quantization type in the modeling of Laser propagation through turbulent media. In some
+of the typical physical contexts the problem reduces to stochastic Schr¨odinger equation with
+space-time white noise of Gaussian, Poisson and L´evy type. We identify their mathematical
+resolution via stochastic quantization. Nonlinear phenomena such as Kerr effect can be
+modeled by stochastic nonlinear Schrodinger equation in the focusing case with space-time
+white noise. A treatment of stochastic transport equation, the Korteweg-de Vries Equation
+as well as a number of other nonlinear wave equations with space-time white noise is also
+given.
+Main technique is the S-transform (we will actually use closely related Hermite
+transform) which converts the stochastic partial differential equation with space time white
+noise to a deterministic partial differential equation defined on the Hida-Kondratiev white
+noise distribution space. We then utlize the inverse S-transform/Hermite transform known
+as the characterization theorem combined with the infinite dimensional implicit function
+theorem for analytic maps to establish local existence and uniqueness theorems for path-
+wise solutions of these class of problems. The particular focus of this paper on singular white
+noise distributions is motivated by practical situations where the refractive index fluctuations
+in propagation medium in space and time are intense due to turbulence, ionospheric plasma
+turbulence, marine-layer fluctuations, etc. Since a large class of partial differential equations
+that arise in nonlinear wave propagation have polynomial type nonlinearities, white noise
+distribution theory is an effective tool in studying these problems subject to different types
+of white noises.
+Key words: Laser propagation, stochastic nonlinear Schr¨odinger equation, stochastic quan-
+tization, space-time white noise, Wick product, paraxial equation, Korteweg-de Vries Equa-
+tion, white noise calculus, S-transform, Benjamin-Ono equation, Schr¨odringer-Hartree equa-
+tion, Zakharov system, Davey-Stewartson equation.
+Mathematics Subject Classification (2010): 60H15, 81S20, 37N20
+Contents
+1.
+Introduction
+2
+2.
+Derivation of the paraxial equation from the Maxwell equations
+3
+3.
+Mathematical background on white noise calculus
+4
+3.1.
+White noise theory: Gaussian, Poisson and L´evy
+4
+3.2.
+Hida-Kondratiev spaces
+7
+3.3.
+Wick products and properties
+8
+1 U. S. Air Force Research Laboratory, Wright Patterson Air Force Base, Ohio 45433, U. S. A.
+2 NRC-Senior Research Fellow, National Academies of Science, Engineering and Medicine, U. S.
+Air Force Research Laboratory, Wright Patterson Air Force Base, Ohio 45433, U. S. A.
+e-mail: [email protected] ∗Corresponding author.
+e-mail: [email protected]
+1
+
+2
+S. S. SRITHARAN AND SABA MUDALIAR
+3.4.
+S-transform, Hermite transform and characterization theorems
+9
+4.
+Kato theory of quasilinear abstract evolution equations
+10
+4.1.
+Analytic Mappings between Banach Spaces and Inverse Mapping Theorem
+11
+5.
+First-order PDE (transport-type)
+12
+5.1.
+First order random transport equation with temporal white noise
+12
+5.2.
+Stochastic transport model with space-time white noise
+13
+6.
+Stochastic nonlinear wave equations
+15
+6.1.
+Stochastic Korteweg De Vries equation
+15
+6.2.
+Stochastic Benjamin-Ono equation
+16
+7.
+Stochastic reaction diffusion equation and quantization
+16
+7.1.
+Stochastic heat equation with multiplicative noise and the KPZ Equation
+17
+7.2.
+Stochastic nonlinear heat equation with white noise Initial Data
+18
+8.
+Stochastic linear and nonlinear Schr¨odinger equations with space-time white noise 19
+8.1.
+Strichartz estimates
+19
+8.2.
+Stochastic linear Schr¨odinger equation with additive space-time white noise
+19
+8.3.
+Stochastic linear Schr¨odinger equation with multiplicative space-time white
+noise
+20
+8.4.
+Nonlinear Schr¨odinger equation with multiplicative space-time white noise
+21
+9.
+Concluding remarks
+21
+References
+23
+1. Introduction
+Stochastic partial differential equations of the type
+i ∂
+∂tψ(x, t, ω) + [∆ + Γ(x, t, ω)]ψ(x, t, ω) + F(ψ(x, t, ω)) = 0,
+(1.1)
+where Γ(·, ·, ·) is a random field describing the fluctuations in the medium are often en-
+countered in electromagnetic and acoustic wave propagation problems in random nonlinear
+media. They are usually called either “paraxial equation” or “parabolic equation model”
+in the engineering literature [60, 64, 32]. As we define later, here (Ω, F, m) is a complete
+probability space and ω is an S′(Rd)-valued random variable. In this paper we will study
+Gaussian, Poisson and Levy type white noise models for Γ(·, ·, ·).
+In the Gaussian case
+for example, Γ(x, t, ω) can also formally represented using generalized derivative of Rn+1-
+dimensional Brownian sheet B(x, t, ω):
+Γ(x, t, ω) =
+∂n+1
+∂x1, · · ·∂xn∂tB(x, t, ω), x = (x1, · · · , xn),
+(1.2)
+where the Brownian sheet B(x, t, ω) has covariance:
+⟨B(x, t, ·), B(y, τ, ·)⟩ = Πn
+k=1(xk ∧ yk)(t ∧ τ), x = (x1, · · · , xn), y = (y1, · · · , yn).
+(1.3)
+The space-time Gaussian white noise has covariance formally:
+⟨Γ(x, t, ·), Γ(y, τ, ·)⟩ = δ(x − y)δ(t − τ).
+(1.4)
+All three type noise structures will be developed in detail in later sections. In this paper
+we will provide a systematic treatment of Laser propagation in random media highlighting
+some of the well-known physical phenomena such as deep turbulence. In the simplest yet
+
+STOCHASTIC QUANTIZATION OF LASER PROPAGATION MODELS
+3
+mathematically nontrivial case, the problem reduces to stochastic Schrodinger equation with
+space-time white noise and we discuss its relationship to stochastic quantization. Nonlinear
+phenomena such as Kerr effect can be modeled by stochastic nonlinear Schrodinger equation
+[60] in the focusing case again with space-time white noise. Similar equations also arise in
+nonlinear fiber optics [1]. We will also discuss a number of other stochastic partial differential
+equations studied in the context of random wave propagation phenomena such as stochastic
+transport equation [52, 26] and stochastic Korteweg-de Vries equation [10]. The quantization
+method we use is in the spirit of Wick expansions used in quantum field theory (see for
+example [58]) and the white noise calculus ([35, 49]) method for stochastic partial differential
+equations is described in [36] which also gives a comprehensive discussion of the literature
+in this subject.
+2. Derivation of the paraxial equation from the Maxwell equations
+In this section we will give a heuristic derivation of the paraxial equation starting from the
+Maxwell equation. The paraxial equation we derive is a very well-known model widely used
+in the literature on acoustic wave and electromagnetic wave propagation in random media
+[65, 53, 64, 60, 32]. Let E, H, B, D denote respectively the electric field, magnetizing field,
+magnetic field and the displacement field.The Maxwell equations are:
+∇ × E = ∂B
+∂t
+(2.1)
+∇ × H = ∂D
+∂t
+(2.2)
+D = ǫE and B = µH.
+(2.3)
+Here ǫ is the permitivity and µ is the permiability of the medium. Taking curl of the first
+equation and taking time derivative of the second equation and substituting in the first we
+get upon using the vector identity ∇ × ∇ × E = −∆E + ∇(∇ · E)
+∆E − µ ∂2
+∂t2(ǫE) = ∇(∇ · E)
+(2.4)
+In the absence of free charge ∇ · D = 0 and hence E · ∇ǫ + ǫ∇ · E = 0 and substituting we
+get
+∆E − µ ∂2
+∂t2(ǫE) = −∇(E · ∇(logǫ))
+(2.5)
+We have noting ǫ(x, t) = ǫ0n2(x, t) where n is the refractive index and c2 = 1/(µǫ0) with c
+the light speed, we arrive at
+∆E(x, t) − 1
+c2
+∂2
+∂t2(n2(x, t)E(x, t)) = −2∇(E(x, t) · ∇log(n(x, t))).
+(2.6)
+We assume that the time scale of fluctuations in the medium is much slower than the light
+speed and invoke further simplifications based on this assumption. Thus neglecting the right
+hand side and also n2 term out of time derivative we arrive at
+∆E(x, t) − n2(x, t)
+c2
+∂2
+∂t2 E(x, t) = 0.
+(2.7)
+
+4
+S. S. SRITHARAN AND SABA MUDALIAR
+Substituting a plane wave solution E(x1, x2, x3, t) = ψ(x1, x2, x3, t) exp(ikx3 − iωt) and ne-
+glecting the back-scatter term ∂2ψ(x1,x2,x3,t)
+∂x2
+3
+(using simple scaling argument, see for example
+[53]) we arrive at the paraxial equation:
+2ik ∂ψ
+∂x3
++ [∆⊥ + k2{n2(x1, x2, x3, t)
+n2
+0
+− 1}]ψ = 0.
+(2.8)
+where ∆⊥ is the two dimensional Laplacian in variables x2, x3.
+Renaming the time-like
+variable x3 as t and suppressing the actual time variable t we end up with the two dimensional
+linear or nonlinear stochastic Schr¨odinger equation:
+i ∂
+∂tψ(x, t, ω) + [∆ + V (x, t, ω, ψ)]ψ(x, t, ω) = 0.
+(2.9)
+In this paper V will be modeled as a space-time Gaussian white noise[35, 49], Poisson white
+noise [38, 39, 4] or Levy type white noise[36] and the paraxial equation will then be framed
+as a stochastic quantization problem.
+3. Mathematical background on white noise calculus
+3.1. White noise theory: Gaussian, Poisson and L´evy. In this section we will recall
+some of the basic elements of white noise stochastic calculus [34, 35, 49, 4, 36] needed in
+this paper. We will start by defining the Schwartz space S = S(Rd) of rapidly decaying
+real-valued C∞ functions on Rd. This space equipped with the family of seminorms:
+∥f∥k,α := sup
+x∈Rd{(1 + |x|k)|∂αf(x)|}
+(3.1)
+is a Frechet space [56]. Here k is a non-negative integer, α = {α1, · · · , αd} is a multi-index
+of non-negative integers αi, i = 1, · · · , d and
+∂αf(x) =
+∂|α|
+∂xα1
+1 · · · ∂xαd
+d
+f(x), where |α| := α1 + · · · + αd.
+(3.2)
+The dual space of S(Rd) denoted S′ := S′(Rd) equipped with the weak topology is the space
+of tempered distributions. Let (Ω, F, m) is a complete probability space and ω is an S′(Rd)-
+valued random variable. The law µ of this S′(Rd)-valued random variable ω is characterized
+next. We recall the Bochner-Milnos theorem [56] for the existence of probability measures
+on the Borel sets B(S′):
+Theorem 3.1. A necessary and sufficient condition for the existence of a probability measure
+µ on B(S′) and a functional F on S such that
+�
+S′ ei<ω,φ>µ(dω) = F(φ), ∀φ ∈ S
+(3.3)
+is that F satisfies the following three conditions:
+(1) F(0) = 1,
+(2) F is positive definite: �n
+j,l=1 zj¯zlF(φj − φl) ≥ 0, ∀zk ∈ C, ∀φk ∈ S, k = 1, · · · , n,
+(3) F is continuous in the Frechet topology of S.
+
+STOCHASTIC QUANTIZATION OF LASER PROPAGATION MODELS
+5
+We will also utilize three particular cases:
+(i) Gaussian measure µG:
+�
+S′ ei<ω,φ>µG(dω) = exp{−1
+2∥φ∥2
+L2(Rd)}, ∀φ ∈ S,
+(3.4)
+(ii) Poisson measure µP:
+�
+S′ ei<ω,φ>µP(dω) = exp{
+�
+Rd(eiφ(x) − 1)dx}, ∀φ ∈ S,
+(3.5)
+(iii) Pure jump Levy measure µL:
+�
+S′ ei<ω,φ>µL(dω) = exp{
+�
+Rd Ψ(φ(x))dx}, ∀φ ∈ S and
+(3.6)
+Ψ(φ) =
+�
+Rd(ei<φ,z> − 1 − i < φ, z >)ν(dz).
+(3.7)
+Definition 3.1.
+(1) The continuous version of the mapping Rd ∋ x = (x1, · · · , xd) →
+Bx(ω) ∈ L2(µG) defined by
+Bx(ω) = ⟨ω, Ξ(x1) × · · · × Ξ(xd)⟩
+(3.8)
+is called the d-parameter Brownian motion (Brownian sheet). Here Ξ(·) ∈ L2(R) is
+defined as:
+Ξ(s) =
+�
+χ(0,s]
+if s ≥ 0
+−χ(s,0]
+if s < 0
+(3.9)
+where χ is the usual indicator function.
+(2) The right continuous interger-valued mapping Rd ∋ x = (x1, · · · , xd) → Px(ω) ∈
+L2(µP) defined by
+Px(ω) = ⟨ω, Ξ(x1) × · · · × Ξ(xd)⟩
+(3.10)
+is called the d-parameter Poisson process. The mapping Rd ∋ x = (x1, · · · , xd) →
+Px(ω) − Πd
+i=1xi ∈ L2(µP) is called compensated Poisson process.
+We will now describe Wiener-Ito expansions:
+Definition 3.2. Each f ∈ L2(µG) has an expansion in multiple (Brownian) Wiener integrals:
+f(ω) =
+∞
+�
+n=0
+�
+Rnd fn(x)dB⊗n
+x (ω),
+(3.11)
+where fn ∈
+ˆ
+L2(Rnd) are deterministic symmetrized functions in nd variables and
+∥f∥2
+L2(µG) =
+∞
+�
+n=0
+n!∥fn∥2
+L2(Rnd).
+(3.12)
+Similarly g ∈ L2(µP) has an expansion in multiple (Poisson) Wiener integrals:
+g(ω) =
+∞
+�
+n=0
+�
+Rnd fn(x)d(Px(ω) − Πd
+i=1xi)⊗n
+x (ω),
+(3.13)
+
+6
+S. S. SRITHARAN AND SABA MUDALIAR
+where gn ∈
+ˆ
+L2(Rnd) are deterministic symmetrized functions in nd variables and
+∥g∥2
+L2(µP ) =
+∞
+�
+n=0
+n!∥gn∥2
+L2(Rnd).
+(3.14)
+We will now recall equivalent expansions in terms of Hermite and Charlier polynomials.
+For n = 1, 2, · · · let ζn(x) ∈ S(R) be the Hermite function of order n:
+ζn(x) := π−1/4((n − 1)!)−1/2e−x2/2hn−1(
+√
+2x), x ∈ R,
+(3.15)
+where hn(x) is the n-th Hermite polynomial defined by
+hn(x) := (−1)nex2/2 dn
+dxn(e−x2/2), x ∈ R, n = 0, 1, 2, · · · .
+(3.16)
+It is well known that [56] the sequence {ζn}∞
+n=1 forms an orthonomal basis for L2(R). Hence
+the family {ηα} of tensor products
+ηα = eα1α2···αd := ζα1 ⊗ · · · ⊗ ζαd, α ∈ Nd
+(3.17)
+forms an orthonomal basis for L2(Rd). let us now assume that the family of all multi-indices
+α = (α1, · · · , αd) is given a fixed ordering
+(α(1), α(2), · · ·α(n), · · ·),
+(3.18)
+where α(k) = (α(k)
+1 , · · · , α(k)
+d ) and denote ηk = ηα(k). For a multi-index α = (α1, · · · , αn) and
+n ∈ N define the Hermite polynomial functionals as
+Hα(ω) := Πn
+j=1hαj(⟨ω, ηj⟩),
+(3.19)
+and the Charlier polynomial functionals as
+Cα(ω) := C|α|(ω;
+α1 times
+�
+��
+�
+η1, · · · , η1, · · · ,
+αn times
+�
+��
+�
+ηn, · · · , ηn),
+(3.20)
+with the convention
+Cn(ω; e1, · · · , en) :=
+∂n
+∂θ1 · · ·∂θn
+e{⟨ω,ln (1+�n
+j=1 θjej)⟩−�n
+j=1 θj
+�
+Rd ej(y)dy}|θ1=···=θn=0
+(3.21)
+We also have the following equalities:
+Hα =
+�
+Rnd eˆ⊗|α|dB⊗|α|
+x
+(ω)
+(3.22)
+and
+Cα =
+�
+Rnd eˆ⊗|α|d(Px(ω) − Πd
+i=1xi)⊗|α|
+x
+(ω)
+(3.23)
+Moreover, for any random functional f(ω) taking values on a separable Hilbert space V ,
+and square integrable in µG, f ∈ L2(µG; V ) we have the Wiener Hermite polynomial chaos
+expansion [13]:
+f(ω) =
+�
+α
+aαHα(ω), aα ∈ V,
+(3.24)
+with
+∥f∥2
+L2(µG;V ) =
+�
+α
+α!∥aα∥2
+V , where α! = α1! · · · αn!.
+(3.25)
+
+STOCHASTIC QUANTIZATION OF LASER PROPAGATION MODELS
+7
+Similarly, for any g ∈ L2(µP; V ) we have the Wiener Charlier polynomial chaos expansion
+g(ω) =
+�
+α
+bαCα(ω), bα ∈ V,
+(3.26)
+with
+∥g∥2
+L2(µP ;V ) =
+�
+α
+α!∥bα∥2
+V , where α! = α1! · · · αn!
+(3.27)
+Note also that the correspondence between Gaussian and Poisson spaces U : L2(µG; V ) →
+L2(µP; V ):
+U{
+�
+α
+aαHα(ω)} =
+�
+α
+aαCα(ω),
+(3.28)
+is unitary [39, 3].
+3.2. Hida-Kondratiev spaces. We will start with the characterization of the classical
+embedding of the Schwartz space in the space of tempered distributions as S(Rd) ⊂ L2(Rd) ⊂
+S′(Rd) using Hermite Fourier coefficients [57] and then define the Hida-Kondratiev spaces in
+an analogous way using Gaussian, Poisson and Levy measures.
+Theorem 3.2. (i) Let φ ∈ L2(Rd) so that we have the Fourier-Hermite expansion
+φ =
+∞
+�
+j=1
+ajηj,
+where aj = (φ, ηj), j = 1, 2, · · · ,
+(3.29)
+with
+ηj := ζδ(j) = ζδ(j)
+1 ⊗ · · · ζδ(j)
+d , j = 1, 2, · · · .
+(3.30)
+Here aj are the Fourier coefficients of φ with respect to the tensor product Hermite func-
+tions ηj. Then φ ∈ S(Rd) if and only if
+∞
+�
+j=1
+a2
+j(δ(j))γ < ∞,
+(3.31)
+for all d-dimensional multi-indices γ = (γ1, · · · , γd).
+(ii) A distribution T ∈ S′(Rd) is characterized by the expansion
+T =
+∞
+�
+j=1
+bjηj, with
+(3.32)
+∞
+�
+j=1
+b2
+j(δ(j))−θ < ∞,
+(3.33)
+for some d-dimensional multi-index θ = (θ1, · · · , θd).
+Definition 3.3. Let 0 ≤ ρ ≤ 1. We say f(ω) = �
+α aαHα ∈ L2(µG; V ) belongs to Gaussian
+Hida-Kondratiev stochastic test function space (S)ρ
+G(V ) if
+∥f∥2
+ρ,k =
+�
+α
+∥aα∥2
+V (α!)1+ρ(2N)αk < ∞ ∀k ∈ N,
+(3.34)
+where
+(2N)α = Πk
+j=1(2j)αj, α = (α1, · · · , αk).
+(3.35)
+
+8
+S. S. SRITHARAN AND SABA MUDALIAR
+We say f(ω) = �
+α bαHα(ω) ∈ L2(µG; V ) belongs to Gaussian Hida-Kondratiev stochastic
+distribution space (S)−ρ
+G (V ) if
+�
+α
+∥bα∥2
+V (α!)1−ρ(2N)−αq < ∞ for some q ∈ N.
+(3.36)
+This establishes the embedding
+(S)1
+X(V ) ⊂ (S)ρ
+X(V ) ⊂ (S)+0
+X (V ) ⊂ L2(µX; V ) ⊂ (S)−0
+X (V ) ⊂ (S)−ρ
+X (V ) ⊂ (S)−1
+X (V ),
+with X = G, P or L.
+Remark: The unitary correspondence U : L2(µG; V ) → L2(µP; V ):
+U{
+�
+α
+aαHα(ω)} =
+�
+α
+aαCα(ω),
+(3.37)
+can be extended as a unitary map U : (S)ρ
+G(V ) → (S)ρ
+P(V ), −1 ≤ ρ ≤ 1 ([39, 4]).
+Definition 3.4. We have the following definitions of white noise processes.
+(1) A d-parameter Gaussian white noise is defined by
+Wx(ω) =
+∞
+�
+k=1
+ηk(x)Hε(k)(ω), x ∈ Rd,
+(3.38)
+where ε(k) is the multi-index with 1 in k-th entry and zero otherwise.
+(2) A d-parameter compensated Poissonian white noise is defined by
+˙Px(ω) − 1 =
+∞
+�
+k=1
+ηk(x)Cε(k)(ω), x ∈ Rd.
+(3.39)
+Lemma 3.1. We have
+(1) Wx(ω) =
+∂d
+∂x1···∂xdBx(ω) ∈ (S)−ρ
+G , ρ ∈ [0, 1],
+(2) ˙Px(ω) − 1 =
+∂d
+∂x1···∂xd(Px(ω) − Πd
+i=1xi) ∈ (S)−ρ
+P , ρ ∈ [0, 1].
+3.3. Wick products and properties. The Wick product is defined as below:
+Definition 3.5. The Wick product F ⋄ G of two elements of (S)−1(Rn) is defined by:
+F =
+�
+α
+aαHα, G =
+�
+α
+bαHα ∈ (S)−1, with aα, bα ∈ Rn,
+(3.40)
+F ⋄ G :=
+�
+α,β
+aα · bβHα+β ∈ (S)−1(Rn).
+(3.41)
+The fact that Wick product gives a distribution F ⋄ G ∈ (S)−1(Rn) follows from the
+estimate below (see [36]). We have F = �
+α aαHα, G = �
+β bβHβ ∈ (S)−1(Rn) means there
+exists q1 ∈ N such that
+�
+α
+|aα|2(2N)−q1α < ∞
+and
+�
+β
+|bβ|2(2N)−q1β < ∞.
+Now rewriting
+F ⋄ G :=
+�
+α,β
+aα · bβHα+β =
+�
+γ
+(
+�
+α+β=γ
+aα · bβ)Hγ,
+(3.42)
+
+STOCHASTIC QUANTIZATION OF LASER PROPAGATION MODELS
+9
+and then setting Cγ = �
+α+β=γ aα·bβ and q = q1+k we have by Cauchy-Schwartz inequality:
+�
+γ
+(2N)−qγ|cγ|2 ≤
+��
+γ
+(2N)−kγ
+���
+α
+|aα|2(2N)−q1α
+���
+β
+|bβ|2(2N)−q1β
+�
+< ∞.
+The first term on the right is finite for k > 1 due to a Lemma by Zhang[71] and the other
+two terms are finite by definition of the distributions F, G ∈ (S)−1(Rn).
+3.4. S-transform, Hermite transform and characterization theorems.
+Definition 3.6. Let F = �
+α bαHα ∈ (S)−1
+G (V ). Then the Hermite transform of F denoted
+HGF is defined :
+HGF =
+�
+α
+bαzα ∈ V
+(when convergent),
+(3.43)
+where z = (z1, z2 · · · ) ∈ CN, zα = zα1 · · · zαk for α = (α1, · · · , αk).
+Similar statements for the Poisson and L´evy cases. Remark 2.7, [71] clarifies the conver-
+gence of this series.
+Lemma 3.2. If F, G ∈ (S)−1
+X (V ), with X = G, P, L (Gaussian, Poisson and Levy respec-
+tively) then
+HX(F ⋄ G)(z) = HXF(z) · HXG(z),
+(3.44)
+for all z such that HXF(z) and HXG(z) exist (convergent).
+Lemma 3.3. Suppose g(z1, z2, · · ·) is a bounded analytic function on Bq(δ) for some δ > 0,
+0 < q < ∞ where
+Bq(δ) :=
+�
+z = (z1, z2, · · · ) ∈ CN
+0 ;
+�
+α̸=0
+|zα|2(2N)αq < δ2.
+�
+(3.45)
+then there exists F ∈ (S)−1
+G (V ) and D ∈ (S)−1
+P (V ) such that HGF = g = HPD.
+See also the full statement of characterization theorem for (S)−1 for the Gaussian as well
+as Poisson and L´evy cases in [36] (Theorem 2.6.11, Theorem 5.4.19). In this paper we will
+utilize a vector-valued version (Hilbert or Banach space valued) of the Hida-Kondratiev
+spaces, Hermite transform and inverse Hermite transform (Characterization theorem) and
+these extensions are straightforward generalizations of what is stated above.
+Remark 3.1. Many authors have introduced S-transform of (S)−1(V ) -valued distributions
+[35, 49] which is closely related to the Hermite transform:
+Sφ(ζ) :=
+�
+S′ φ(ω) exp⋄⟨ω, ζ⟩dµ(ω),
+(3.46)
+where exp⋄⟨ω, ζ⟩ = �∞
+0
+1
+n!⟨ω, ζ⟩⋄n. It can be shown to be related to the Hermite transform as
+Hφ(z1, z2, · · · , zk) = Sφ(z1η1 + · · · + zkηk)
+for all z1, z2, · · · , zk ∈ Ck,
+(3.47)
+in a suitable neighborhood. We however find it more convenient to work with the Hermite
+transform as pointed out in [36].
+We also note for later use that the Hermite transform of the above white noises result in:
+
+10
+S. S. SRITHARAN AND SABA MUDALIAR
+(1) Hermite transform of d-parameter Gaussian white noise is given by:
+H[Wx](z) =
+∞
+�
+k=1
+ηk(x)zεk.
+(3.48)
+(2) Hermite transform of d-parameter compensated Poisson white noise is given by
+H[ ˙Px − 1](z) =
+∞
+�
+k=1
+ηk(x)zεk.
+(3.49)
+Similar mathematical theory for pure jump L´evy measures are given in Chapter 5 of Holden
+[36] which parallel the above developments in polynomial expansions, distributional spaces
+as well as Hermite transforms which will be utilized in our paper as well.
+4. Kato theory of quasilinear abstract evolution equations
+All the problems considered in this paper, after Wick-quantization followed by Hermit (or
+S )-transform, brought to a general class of deterministic quasilinear evolutions (complex
+in general) parameterized by an infinite sequence of complex numbers z1, z2, · · · ,. We will
+thus recall some general results developed by T. Kato [44, 45] on quasilinear evolutions on a
+Banach space X:
+du
+dt + A(t, u)u = 0, 0 ≤ t ≤ T,
+(4.1)
+u(0) = u0.
+(4.2)
+Here A(t, u) is an unbounded linear operator that nonlinearly depends on t and u. Kato also
+points out in [45] that an inhomogeneous equation with a right hand side:
+du
+dt + A(t, u)u = f(t, u)
+(4.3)
+can be recast as a homogeneous problem above by redefining the variables. Kato’s theory
+covers parabolic problems where the linearized operator generates an analytic semigroup as
+well as hyperbolic problems [42, 43] where the linearized operator generates a C0-semigroup
+which is not analytic. The main idea is to construct a mild solution w ∈ C([0, T]; X) for
+each u ∈ C([0, T]; X) for the linearized problem:
+dw
+dt + A(t, u)w = 0, 0 ≤ t ≤ T,
+(4.4)
+w(0) = u0.
+(4.5)
+This defines a correspondence u → w = F(u) in C([0,T];X). Then use fixed point theory
+to construct the solution of the original quasilinear problem. Kato’s quasilinear theory was
+extended to the stochastic case using infinite dimensional Ito calculus in[25, 51]. In this
+paper we will extend Kato’s theory to white noise calculus realm to treat problems with
+singular noises. We briefly summarize Kato’s theory below [44, 45, 55].
+Definition 4.1. Let B be a subset of a Banach space X and for every 0 ≤ t ≤ T and b ∈ B
+let A(t, b) be the infinitesimal generator of a C0-semigroup St,b(s), s ≥ 0 on X. The family
+of operators {A(t, b)}, (t, b) ∈ [0, T] × B, is stable if there are constants M ≥ 1, ω such that
+the resolvent set
+ρ(A(t, b)) ⊃]ω, ∞], (t, b) ∈ [0, T] × B
+(4.6)
+
+STOCHASTIC QUANTIZATION OF LASER PROPAGATION MODELS
+11
+and
+∥Πk
+j=1(λ − A(tj, bj))−1∥ ≤ M(λ − ω)−k
+for λ > ω
+(4.7)
+for every finite sequence 0 ≤ ti ≤ t2 · · · ≤ T, bj ∈ B, 1 ≤ j ≤ k.
+Stable family of generators {A(t, b)}, (t, b) ∈ [0, T] × B has stability estimate [44]
+∥Πk
+j=1Stj,bj(sj)∥ ≤ Meω �k
+j=1 sj
+for sj ≥ 0
+(4.8)
+for every finite sequence 0 ≤ ti ≤ t2 · · · ≤ T, bj ∈ B, 1 ≤ j ≤ k.
+Theorem 4.1. Let u0 ∈ Y and let B = {v ∈ Y ; ∥v − u0∥Y ≤ r}, r > 0. Let the stable family
+of C0-semigroup generators {A(t, b)}, (t, b) ∈ [0, T] × B satisfy the assumptions H1 − H4 of
+[55] (Section 6.4) then there is a T ′, 0 < T ′ ≤ T such that the initial value problem
+du
+dt + A(t, u)u = 0, 0 ≤ t ≤ T,
+(4.9)
+u(0) = u0,
+(4.10)
+has a unique mild solution u ∈ C([0, T ′]; X) with u(t) ∈ B for t ∈ [0, T ′].
+Details of the hypothesis of this theorem are discussed in [44, 45] and also [55]. Kato
+formulated this abstract theory to cover a large class of evolution system of physics and
+mechanics including the ones studied in the subsequent sections.
+4.1. Analytic Mappings between Banach Spaces and Inverse Mapping Theorem.
+Definition 4.2. A map Φ : Z1 → Z2 between Banach spaces Z1, Z2 is called analytic in
+Kδ = {u ∈ Z1; ∥u∥Z1 < δ} if it can be deveoped in to a series of the form:
+Φ(u) =
+∞
+�
+k=0
+Φk(u, u, · · · , u),
+(4.11)
+where Φk(·, · · · , ·) : Z⊗k
+1
+→ Z2 are symmetric multi-linear operators that are bounded:
+∥Φk(·)∥ = sup{∥Φk(u, · · · , u)∥Z2; ∥u∥Z1 ≤ 1} < ∞,
+and the series converges in Z2-norm for u ∈ K�� in the following sense:
+∞
+�
+k=0
+∥Φk∥ρk < ∞,
+for 0 < ρ ≤ δ.
+We will now state the analytic inverse function theorem [7, 69, 67]:
+Theorem 4.2. Suppose that Φ : Z1 → Z2 is an analytic map in a neighborhood of the origin
+0 ∈ Kδ ⊂ Z1 and that its Frech´et derivative at the origin DΦ(0) is an isomorphism from
+Z1 → Z2. Then locally Φ has a unique inverse operator which is analytic in the neighborhood
+of Φ(0) ∈ Z2.
+See Vallent[67] for the analytic version of the implicit function theorem as well.
+
+12
+S. S. SRITHARAN AND SABA MUDALIAR
+5.
+First-order PDE (transport-type)
+5.1. First order random transport equation with temporal white noise. Transport
+problems arise in many applications including radiative transfer [16, 17, 18]. In this section
+we will consider first order random transport equation with temporal white noise studied by
+S. Ogawa [52], T. Funaki[26], and H. Kunita [48] and briefly summarize their results. Let
+(Ω, Σ, m) be a complete probability space. Consider
+∂
+∂tu(x, t, ω) + n(x, t, ω) ∂
+∂xu(x, t, ω) = c(x, t, ω)u(x, t, ω) + d(x, t, ω).
+(5.1)
+Here the wave speed c(x, t, ω) and forcing d(x, t, ω) can be deterministic or random. We
+will first discuss the one dimensional Ogawa-Funaki temporal white noise model with C, D
+deterministic:
+∂
+∂tu(x, t, ω) +
+� d
+dtβ(t, ω) + b(t, x)
+� ∂
+∂xu(x, t, ω)
+= c(x, t)u(x, t, ω) + d(x, t).
+(5.2)
+Here β(t) is the 1-D Brownian motion.
+Funaki also considered the n-dimensional scalar random transport model and we first write
+with Smooth noise:
+∂
+∂tu(x, t, ω) +
+n
+�
+i=1
+ni(x, t, ω) ∂
+∂xi
+u(x, t, ω)
+= c(x, t)u(x, t, ω) + d(x, t).
+(5.3)
+The Funaki’s temporal white noise model takes the form:
+∂
+∂tu(x, t, ω) +
+n
+�
+i=1
+� n
+�
+j
+aij(t, x) d
+dtβj(t, ω) + bi(x, t)
+�
+∂
+∂xi
+u(x, t, ω)
+= c(x, t)u(x, t, ω) + d(x, t).
+(5.4)
+Here βj(t), j = 1, · · · , n are independent 1-D Brownian motions.
+Define Stratenovich differential equation
+dXt = a(t, Xt) ◦ dBt + b(t, Xt)dt, t ∈ (r, T),
+(5.5)
+Xr = x.
+(5.6)
+This is equivalent to Ito differential equation of the form:
+dXt = a(t, Xt)dBt + ˜b(t, Xt)dt, where
+(5.7)
+˜b(x, t) = b(t, x) + 1
+2(a′a)(t, x),
+(5.8)
+(a′a)(t, x)i =
+�
+k,j
+∂aij
+∂xk
+akj.
+(5.9)
+Lemma 5.1. For each r, t with 0 ≤ r ≤ t ≤ T, X(r, t, ·) is a homeomorphism of Rn on to
+Rn.
+
+STOCHASTIC QUANTIZATION OF LASER PROPAGATION MODELS
+13
+Define time reversed process
+Y (r, t, y) = X−1(r, t, ·)(y), 0 ≤ r ≤ t ≤ T, y ∈ Rn.
+(5.10)
+Consider:
+∂
+∂tu(x, t, ω) +
+n
+�
+i=1
+� n
+�
+j
+aij(t, x) d
+dtβj(t, ω) + bi(x, t)
+�
+∂
+∂xi
+u(x, t, ω)
+= c(x, t)u(x, t, ω) + d(x, t),
+(5.11)
+u(x, 0) = φ(x), x ∈ G and u(x, t) = ψ(x, t), (x, t) ∈ ∂G × [0, T].
+(5.12)
+The probabilistic solution to the stochastic transport equation by Funaki is:
+u(x, t, ω) = {φ(Y (0, t, x)) + ψ(Y (σ(t, x), t, x))} exp
+�� t
+0
+c(s, Y (s, t, x))ds
+�
++
+� t
+0
+d(s, Y (s, t, x)) exp
+�� t
+s
+c(r, Y (r, t, x))dr
+�
+ds.
+(5.13)
+Here σ(t, x) is the exit time for the domain G.
+5.2. Stochastic transport model with space-time white noise. We will begin with the
+following simple model (similar model with temporal white noise was considered by Funaki
+[26]) with space-time white noise Γ(x, t) as characteristic speed:
+∂
+∂tu(x, t, ω) + Γ(x, t, ω) ∂
+∂xu(x, t, ω) = 0, (x, t) ∈ R × [0, T],
+(5.14)
+u(x, 0, ω) = φ(x), x ∈ R.
+(5.15)
+We quantize this problem as:
+∂
+∂tu(x, t, ω) + Γ(x, t, ω) ⋄ ∂
+∂xu(x, t, ω) = 0, (x, t) ∈ R × [0, T],
+(5.16)
+u(x, 0, ω) = φ(x), x ∈ R.
+(5.17)
+We take Hermite transform to get (denoting [Hu] := ˜u )
+∂
+∂t ˜u(x, t, z) + [HΓ](x, t, z) ∂
+∂x ˜u(x, t, z) = 0,
+(5.18)
+˜u(x, 0, z) = φ(x).
+(5.19)
+We write down the equation of characteristic:
+d˜u
+ds = ∂˜u(x, t, z)
+∂t
+dt
+ds + ∂˜u(x, t, z)
+∂x
+dx
+ds
+(5.20)
+and arrive at
+d˜u
+ds = 0,
+dt
+ds = 1, and dx
+ds = [HΓ](x, t, z).
+(5.21)
+Solving we get x = ζt(x0) = x0 +
+� t
+0[HΓ](x, r, z)dr as the equation of characteristics and
+˜u(x, 0, z) = φ(x0) = φ(ζ−1
+t (x)):
+˜u(x, t, z) = φ(ζ−1
+t (x)) = φ(x −
+� t
+0
+[HΓ](x, r, z)dr).
+(5.22)
+
+14
+S. S. SRITHARAN AND SABA MUDALIAR
+Hence
+u(x, t, ω) = H−1φ(ζ−1
+t (x)) = H−1
+�
+φ(x −
+� t
+0
+[HΓ](x, r, z)dr)
+�
+.
+(5.23)
+We will now turn to the multidimensional stochastic transport model with space-time white
+noise:
+∂
+∂tu(x, t, ω) +
+n
+�
+i=1
+� n
+�
+j
+aij(t, x)Γj(x, t, ω) + bi(x, t)
+�
+∂
+∂xi
+u(x, t, ω)
+= c(x, t)u(x, t, ω) + d(x, t).
+(5.24)
+We will also study the quantized random transport model:
+∂
+∂tu(x, t, ω) +
+n
+�
+i=1
+� n
+�
+j
+aij(t, x)Γj(x, t, ω)
+�
+⋄ ∂
+∂xi
+u(x, t, ω) +
+n
+�
+i
+bi(x, t) ∂
+∂xi
+u(x, t, ω)
+= c(x, t)u(x, t, ω) + d(x, t).
+(5.25)
+Taking the Hermite transform: For z ∈ CN, and denoting Hu as ˜u, we get
+∂
+∂t ˜u(x, t, z) +
+n
+�
+i=1
+� n
+�
+j
+aij(t, x)[HΓ]j(x, t, z) + bi(x, t)
+�
+∂
+∂xi
+˜u(x, t, z)
+= c(x, t)˜u(x, t, z) + d(x, t),
+(5.26)
+This is a deterministic linear hyperbolic equation (for a fixed z ) and we can solve it similar
+to the simple model presented above and the apply inverse Hermite transform to obtain
+the solution for the original stochastic transport equation with space-time white noise as
+characteristic speed coefficient and spatial white noise initial data. With the help of the
+deterministic results in [23] we can also obtain the solvability theorem for stochastic transport
+equation with white noise characteristic speeds and spatial white noise initial data as
+Proposition 5.1. Suppose that a, b ∈ L1(0, T; L1
+loc(Rn)), c, d ∈ L1(0, T; L∞(Rn)) and the
+initial data is a spatial white noise with u0 ∈ (S)−1(L∞(Rn)), and the stochastic characteristic
+speed coefficients Γj ∈ (S)−1(L∞(Rn × [0, T])), j = 1, · · · , n. Then there exists a unique
+solution u ∈ (S)−1(L∞(0, T; L∞(Rn)).
+We now consider a multidimensional quasilinear stochastic transport model with space-
+time white noise:
+∂
+∂tu(x, t, ω) +
+n
+�
+i=1
+� n
+�
+j
+aij(u(t, x, ω))Γj(x, t, ω) + bi(u(x, t, ω))
+�
+∂
+∂xi
+u(x, t, ω)
+= c(u(x, t, ω)),
+(5.27)
+where aij, bi, c are all polynomials in u. We quantize this problem as:
+∂
+∂tu(x, t, ω) +
+n
+�
+i=1
+� n
+�
+j
+(aij(u(t, x, ω))⋄) ⋄ Γj(x, t, ω) + bi(u(x, t, ω))⋄
+�
+⋄ ∂
+∂xi
+u(x, t, ω)
+= c(u(x, t, ω))⋄,
+(5.28)
+
+STOCHASTIC QUANTIZATION OF LASER PROPAGATION MODELS
+15
+Applying Hermite transform we get
+∂
+∂t ˜u(x, t, z) +
+n
+�
+i=1
+� n
+�
+j
+aij(˜u(t, x, z))[HΓ]j(x, t, z) + bi(˜u(x, t, z))
+�
+∂
+∂xi
+˜u(x, t, z)
+= c(˜u(x, t, z)).
+(5.29)
+Extending the deterministic result to the Hermite transformed problem may require some
+smoothing of the white noise term Γ using an operator of the form (I −∆x,t)−γ for a suitable
+γ ≥ 0. The local solvability of this deterministic partial differential equation for a given
+fixed z can be obtained from Kato’s theory [44, 45] as ˜u ∈ C([0, T ′]; Hs(Rn)), s > n/2 + 1
+and from this we can use the characterization theorem for Hermite transform to deduce:
+Proposition 5.2. For u0 ∈ (S)−1(Hs(Rn)), s > n/2 + 1 and Γj(·, ·, ·) ∈ (S)−1(L∞
+loc(Rn ×
+R+)), j = 1, · · · , n, there is a unique local-in-time generalized solution for the Gaussian
+white noise forced quasilinear transport equation u ∈ (S)−1(C([0, T ′]; Hs(Rn))).
+For the
+Poisson and L´evy cases the unique correspondence map U gives a unique solution u ∈
+(S)−1(L∞([0, T ′]; Hs(Rn))).
+Global in time generalized solution (that accommodates shock waves) can be built by ap-
+plying Kruzkov theory [47] to the Hermite transformed problem with small bounded variation
+norm initial data:.
+Proposition 5.3. For u0 ∈ (S)−1�
+L∞
+loc(Rn) ∩ BV (Rn)
+�
+, with sufficiently small norm,
+and Γ(·, ·, ·) ∈ (S)−1(L∞
+loc(Rn × R+)) there is a unique global-in-time generalized solu-
+tion for the Gaussian/Poisson/L´evy white noise forced quasilinear transport equation u ∈
+(S)−1(L∞([0, T]; L∞
+loc(Rn))).
+6. Stochastic nonlinear wave equations
+6.1. Stochastic Korteweg De Vries equation. Let us consider the Korteweg De Vries
+equation with white noise initial data:
+∂
+∂tϕ(x, t, ω) + ϕ(x, t, ω) ∂
+∂xϕ(x, t, ω) + ∂3
+∂x3 ϕ(x, t, ω) = 0,
+(6.1)
+ϕ(x, 0, ω) = Γ(x, ω).
+(6.2)
+We quantize this problem as
+∂
+∂tϕ(x, t, ω) + ϕ(x, t, ω) ⋄ ∂
+∂xϕ(x, t, ω) + ∂3
+∂x3 ϕ(x, t, ω) = 0.
+(6.3)
+Applying Hermite transform results in:
+∂
+∂t ˜ϕ(x, t, z) + ˜ϕ(x, t, z) ∂
+∂x ˜ϕ(x, t, z) + ∂3
+∂x3 ˜ϕ(x, t, z) = 0.
+(6.4)
+˜ϕ(x, 0, z) = [HΓ](x, z).
+(6.5)
+Note that the Hermite transformed KDV equation will have infinite number of conservation
+laws as in P. D. Lax theory for deterministic KDV equation [50] some of which would be
+finite depending on the smoothness of noise . However, for the case of singular white noise,
+values of these conservation laws will be all infinity.
+
+16
+S. S. SRITHARAN AND SABA MUDALIAR
+Tsutsumi [66] has studied deterministic Korteweg-De Vries equation with bounded Radon
+measures as initial data (see also Bourgain [11] for the space periodic case) and proven exis-
+tence of a generalized solution and we can use this theorem the above Hermite transformed
+problem for a fixed z followed by inverse Hermite transform to conclude that:
+Proposition 6.1. For the Gaussian/Poisson/L´evy white noise initial data ϕ(·, 0, ·) = Γ(·, ·) ∈
+(S)−1(L∞
+loc(R)), there exists a unique solution u to the stochastic KDV equation such that:
+u ∈ (S)−1�
+L∞(0, ∞; H−1(R)) ∩ L2((0, T) × (−R, R))
+�
+,
+for any T, R > 0.
+6.2. Stochastic Benjamin-Ono equation. The stochastic Benjamin-Ono equation is mod-
+eled as:
+∂
+∂tϕ(x, t, ω) + ϕ(x, t, ω) ∂
+∂xϕ(x, t, ω) + H[ ∂2
+∂x2 ϕ(x, t, ω)] = 0,
+(6.6)
+where H[·] is the Hilbert transform defined as
+H[f](x) := PV 1
+π
+� ∞
+−∞
+f(y)
+x − ydy =
+F −1(i · ( signζ)F(f)(ζ)),
+(6.7)
+where F and F −1 denote Fourier transform and its inverse respectively. This problem is
+supplied with random initial data:
+ϕ(x, 0) = Γ(x, ω).
+(6.8)
+We quantize this equation as
+∂
+∂tϕ(x, t, ω) + ϕ(x, t, ω) ⋄ ∂
+∂xϕ(x, t, ω) + H[ ∂2
+∂x2 ϕ(x, t, ω)] = 0,
+(6.9)
+Applying Hermite transform we get:
+∂
+∂t ˜ϕ(x, t, z) + ˜ϕ(x, t, z) ∂
+∂x ˜ϕ(x, t, z) + H[ ∂2
+∂x2 ˜ϕ(x, t, z)] = 0,
+(6.10)
+˜ϕ(x, 0, z) = H[Γ](x, z)].
+(6.11)
+We can utilize the sharp results of T. Tao [63] along with a suitable smoothing of the noise
+by an operator of the form (I −∆)−γ to conclude that there is a unique solution to the above
+problem for fized z as ˜ϕ ∈ C([0, T]; Hs) for s ≥ 1. Hence using inverse Hermite transform
+we obtain:
+Proposition 6.2. For the Gaussian white noise initial data ϕ(·, 0, ·) = Γ(·, ·) ∈ (S)−1(Hs(R)),
+s ≥ 1, there exists a unique solution to stochastic Benjamin-Ono equation as ϕ ∈ (S)−1(C([0, T]; Hs)),
+s ≥ 1. For Poisson and L´evy cases the unique correspondence map U gives unique solution
+ϕ ∈ (S)−1(L∞([0, T]; Hs)), s ≥ 1.
+7. Stochastic reaction diffusion equation and quantization
+Parisi and Wu [54] initiated the subject of stochastic quantization [20] which takes the view
+that (Euclidean) quantum fields can be constructed by studying stochastic partial differential
+equations. Stochastic reaction diffusion equation with space-time Gaussian noise has been
+studied by many authors [24] with results such as construction of invariant measures [5, 27]
+as well as pathwise strong solutions[2, 21]. In this section we will treat this class of problems
+with white noise theory.
+
+STOCHASTIC QUANTIZATION OF LASER PROPAGATION MODELS
+17
+7.1. Stochastic heat equation with multiplicative noise and the KPZ Equation.
+Kardar, Parisi and Zhang [41] studied the stochastic model (which has come to be known as
+the KPZ equation):
+∂h
+∂t = ν∆h + λ
+2(∇h)2 + Γ(x, t).
+(7.1)
+As pointed out in this paper, in the case of smooth noise the above stochastic PDE can be
+formally transformed in to two other well-known stochastic PDEs (ignoring some constants).
+In fact v = −∇h gives the stochastic Burgers equation:
+∂v
+∂t + λv · ∇v = ν∆v + ∇Γ(x, t),
+(7.2)
+and the transform ϕ = exp(( λ
+2ν )h) converts the KPZ equation to
+∂
+∂tϕ(x, t, ω) = ν∆ϕ(x, t, ω) + λ
+2ν ϕ(x, t, ω)Γ(x, t, ω),
+(7.3)
+with spatial white noise initial data:
+ϕ(x, 0, ω) = Γ0(x, ω).
+We quantize this problem as (after setting ν = σ2/2 and λ = σ2 for simplicity):
+∂
+∂tϕ(x, t, ω) = 1
+2σ2∆ϕ(x, t, ω) + ϕ(x, t, ω) ⋄ Γ(x, t, ω).
+(7.4)
+Applying the Hermite transform gives
+∂
+∂t ˜ϕ(x, t, z) − [1
+2σ2∆ + [HΓ](x, t, z)] ˜ϕ(x, t, z) = 0,
+(7.5)
+˜ϕ(x, 0, z) = [HΓ0](x, z).
+We write the solution formally using the propagator Z(t, r) generated by [55] the unbounded
+time dependent operator 1
+2σ2∆ + [HΓ](x, t, z):
+˜ϕ(x, t, z) = Z(t, 0)[HΓ0](x, z).
+(7.6)
+The solution is also probabilistically expressed by the Feynman-Kac formula using a Brow-
+nian motion Bt independent of Γ(x, t, ω) and Γ0(x, ω):
+˜ϕ(x, t, z) = Ex
+�
+[HΓ0](σBt, z) exp
+�� t
+0
+[HΓ](σBs, t − s, z)ds
+��
+.
+(7.7)
+Alternatively, we can consider the solution as a fixed point problem for the heat semigroup:
+˜ϕ(x, t, z) = e
+1
+2 σ2t∆[HΓ0](x, z) +
+� t
+0
+e
+1
+2σ2(t−τ)∆H[Γ](x, τ, z) ˜ϕ(x, τ, z)dτ.
+(7.8)
+Heat equation with singular and non-autonomous potentials have been studied in the litera-
+ture [30, 31] and for a fixed z as necessary using smoothing operator of the form (I −∆x,t)−γ
+for the noise we can conclude that the above Hermite transformed problem has a unique
+solution in ˜ϕ ∈ C([0, T]; L∞) and hence by inverse Hermite transform we obtain:
+
+18
+S. S. SRITHARAN AND SABA MUDALIAR
+Proposition 7.1. For the Gaussian white noise Γ(·, ·, ·) ∈ (S)−1(L∞
+loc(Rd × R)), and initial
+data ϕ(·, 0) ∈ L∞
+loc(Rd), there exists a unique solution to the stochastic heat equation ϕ ∈
+(S)−1(C([0, T]; L∞(Rd))). For the Poisson and L´evy white noises the unique correspondence
+map U gives a unique solution to the stochastic heat equation ϕ ∈ (S)−1(L∞([0, T]; L∞(Rd))).
+The solution to the quantized problem is also probabilistically expressed by the Feynman-
+Kac formula:
+ϕ(x, t, ω) = Ex
+�
+Γ0(σBt, ω) ⋄ exp⋄
+�� t
+0
+Γ(σBs, t − s, ω)ds
+��
+,
+(7.9)
+where the Wick exponential of X ∈ (S)−1 is defined by:
+exp⋄ X =
+∞
+�
+0
+1
+n!X⋄n.
+We also note here that instead of the multiplicative noise term if we have a stochastic heat
+equation with space-time white noise as an additive noise term or initial data as a spatial
+white noise then after Hermite transform we end up with heat equation with singular initial
+data or forcing term and the problem can be resolved using the fundamental solution of the
+heat equation followed by inverse Hermite transform.
+7.2. Stochastic nonlinear heat equation with white noise Initial Data. We will now
+consider:
+∂
+∂tϕ(x, t, ω) + ϕ(x, t, ω)p = ∆ϕ(x, t, ω), p = 2, 3, x ∈ Rn,
+(7.10)
+ϕ(x, 0, ω) = Γ(x, ω), x ∈ Rn.
+We will quantize this equation as:
+∂
+∂tϕ(x, t, ω) + ϕ(x, t, ω)⋄p = ∆ϕ(x, t, ω),
+(7.11)
+Applying the Hermite transform gives
+∂
+∂t ˜ϕ(x, t, z) − [∆ − ˜ϕ(x, t, z)p−1] ˜ϕ(x, t, z) = 0,
+(7.12)
+˜ϕ(x, 0, z) = [HΓ](x, z).
+(7.13)
+Nonlinear heat equations with measure initial data has been studied in the literature with
+positive as well as negative results. When the initial data is a Dirac measure it has been
+shown in [8] that this problem with the nonlinearity selected above (p = 2 and 3) has no
+solution in any space dimension n ≥ 1. This means that we need to smooth the measure by
+an operator of the form (I −∆)−γ so that we can use results for slightly less singular but still
+measure data such as that presented in [9] with initial data in Lq, 1 ≤ q < ∞. This provides
+a unique short time solution to the Hermite transformed problem as ˜ϕ ∈ C([0, T ′]; Lq) and
+we then inverse Hermite transform to obtain:
+Proposition 7.2. There exists a unique solution ϕ ∈ (S)−1(C([0, T ′]; Lq)) to the stochastic
+nonlinear heat equation with spatial white noise initial data Γ(·, ·) ∈ (S)−1(Lq).
+
+STOCHASTIC QUANTIZATION OF LASER PROPAGATION MODELS
+19
+8. Stochastic linear and nonlinear Schr¨odinger equations with space-time
+white noise
+In this section we will address the stochastic linear and nonlinear Schr¨odinger equations
+with space-time white noise that arise in the Laser propagation problems as discussed earlier.
+Once again we will utilize the Hermite transform to convert the problem to deterministic
+linear and nonlinear Schrodinger equations parameterized by an infinite sequence of complex
+variables z. There is a wealth of literature on linear Schr¨odinger semigroup with a range of
+potentials [59, 6] and nonlinear Schr¨odinger equations [12, 62] which will enable the solvability
+of the deterministic problems obtained by Hermite transforms as we discuss below.
+8.1. Strichartz estimates. We recall the following estimates for the Schr¨odinger free prop-
+agator [61, 46]:
+Lemma 8.1. For (q, r) and (˜q, ˜r) such that for d ≥ 1 both exponent pair satisfying:
+2 ≤ q, r ≤ ∞, 1
+q = d
+2(1
+2 − 1
+r) (q, r, d) ̸= (2, ∞, 2),
+we have:
+∥eit∆u0∥Lq
+t Lrx(R×Rd) ≲ ∥u0∥L2x(Rd),
+(8.1)
+and
+∥
+� t
+0
+ei(t−s)∆F(s, ·)ds∥Lq
+tLrx(R×Rd) ≲ ∥F∥L˜q′
+t L˜r′
+x (R×Rd).
+(8.2)
+8.2. Stochastic linear Schr¨odinger equation with additive space-time white noise.
+i ∂
+∂tψ(x, t, ω) + ∆ψ(x, t, ω) = Γ(x, t, ω), (x, t) ∈ Rd × R+,
+(8.3)
+with spatial white noise initial data:
+ψ(x, 0, ω) = Γ0(x, ω), x ∈ Rd.
+(8.4)
+Applying Hermite transform we get the Schr¨odinger equation in free space:
+i ∂
+∂t
+˜ψ(x, t, z) + ∆ ˜ψ(x, t, z) = H[Γ](x, t, z), x ∈ Rd, z ∈ CN,
+(8.5)
+˜ψ(x, 0, z) = H[Γ0](x, z), x ∈ Rd, z ∈ CN.
+(8.6)
+The solution is written formally using the free Schr¨odinger propagator eit∆ as
+˜ψ(x, t, z) = eit∆H[Γ0](x, z) − i
+� t
+0
+ei(t−τ)∆H[Γ](x, τ, z)dτ.
+(8.7)
+The d-dimensional Schr¨odinger kernel is:
+Kt(x) =
+1
+(4πit)d/2ei |x|2
+4t .
+(8.8)
+Hence we also have:
+˜ψ(x, t) =
+1
+(4πit)d/2
+��
+Rd ei |x−y|2
+4t
+H[Γ0](y, z)dy − i
+� t
+0
+�
+Rd ei |x−y|2
+4(t−τ) H[Γ](x, τ, z)dydτ
+�
+.
+(8.9)
+This equation makes sense for a fixed z ∈ CN if H[Γ0](x, z) ∈ L2(Rd) and H[Γ](·, ·, z) ∈
+Lq
+tLr
+x(R × Rd) due to the Strichartz estimates recalled above. Hence inverse Hermite trans-
+form gives:
+
+20
+S. S. SRITHARAN AND SABA MUDALIAR
+Proposition 8.1. For initial data noise distribution Γ0 ∈ (S)−1(L2(Rd)) and noise forcing
+Γ ∈ (S)−1(Lq
+tLr
+x(R × Rd)) there exists a unique solution ψ ∈ (S)−1(Lq
+tLr
+x(R × Rd)) to the
+stochastic Schr¨odinger equation.
+8.3. Stochastic linear Schr¨odinger equation with multiplicative space-time white
+noise. Consider the linear stochastic Schr¨odinger equation with multiplicative space-time
+noise: For ω ∈ Ω, x ∈ Rd, d ≥ 1, t > 0
+i ∂
+∂tψ(x, t, ω) + [∆ + Γ(x, t, ω)]ψ(x, t, ω) = 0,
+(8.10)
+and spatial white noise initial data:
+ψ(x, 0, ω) = Γ0(x, ω), x ∈ Rd.
+(8.11)
+Here the time-like coordinate t is the propagation direction, ϕ is the (complex) electric field
+and the potential V in general depends on the refractive index of the medium.
+We consider the quantized linear Schr¨odinger equation with multiplicative white noise:
+i ∂
+∂tψ(x, t, ω) + ∆ψ(x, t, ω) + ψ(x, t, ω) ⋄ Γ(x, t, ω) = 0.
+(8.12)
+Applying Hermite transform we get the Schr¨odinger equation with a potential V (x, t, z) =
+H[Γ](x, t, z):
+i ∂
+∂t
+˜ψ(x, t, z) + (∆ + H[Γ](x, t, z)) ˜ψ(x, t, z) = 0.
+(8.13)
+˜ψ(x, 0, z) = H[Γ0](x, z), x ∈ Rd, z ∈ CN.
+(8.14)
+We write the solution formally using the propagator Z(t, r) generated by [59, 55] the un-
+bounded Hamiltonian time dependent (∆ + V (x, t, z)):
+˜ψ(x, t, z) = Z(t, 0) ˜ψ(x, 0, z).
+(8.15)
+Alternatively, we can consider the solution as a fixed point problem for the free propagator:
+˜ψ(x, t, z) = eit∆ ˜ψ(x, 0, z) − i
+� t
+0
+ei(t−τ)∆H[Γ](x, τ, z) ˜ψ(x, τ, z)dτ.
+(8.16)
+Schr¨odinger equations with time dependent unbounded singular potential have been studied
+in the literature [70] where it is shown that the two parameter propagator Z(t, r) is unitary
+in L2(R2) (actually K. Yajima’s results holds in any dimension.) With an introduction to a
+suitable smoothing for the noise in the form of (I−∆x,t)−γ we can fit the Hermite transformed
+problem above in Yajima’s framework and deduce a unique solution ˜ψ ∈ C([0, T]; L2(R2))
+and hence inverse Hermite transform gives:
+Proposition 8.2. For Gaussian initial data noise distribution Γ0 ∈ (S)−1(L2(Rd)) and
+Gaussian white noise forcing Γ ∈ (S)−1(L∞
+loc(R × Rd)), there exists a unique solution
+to the stochastic Schr¨odinger equation with multiplicative space-time white noise as ψ ∈
+(S)−1(C([0, T]; L2(R2))). For Gaussian and L´evy cases unique correspondence map U gives
+a unique solution ψ ∈ (S)−1(L∞([0, T]; L2(R2)))
+
+STOCHASTIC QUANTIZATION OF LASER PROPAGATION MODELS
+21
+8.4. Nonlinear Schr¨odinger equation with multiplicative space-time white noise.
+Nonlinear defocusing cubic Stochastic Schr¨odinger equation with multiplicative space-time
+white noise:
+i ∂
+∂tψ(x, t, ω) + ∆ψ(x, t, ω) +
+�
+Γ(x, t, ω) − |ψ|2�
+ψ(x, t, ω) = 0, x ∈ R3, t > 0
+(8.17)
+and spatial white noise initial data:
+ψ(x, 0, ω) = Γ0(x, ω), x ∈ R3.
+(8.18)
+We consider the quantized nonlinear Schrodinger equation with multiplicative white noise:
+i ∂
+∂tψ(x, t, ω) + ∆ψ(x, t, ω) + (Γ(x, t, ω) − ψ(x, t, ω) ⋄ ψ∗(x, t, ω)) ⋄ ψ(x, t, ω) = 0.
+(8.19)
+Here (in the Gaussian white noise case) we take
+ψ(x, t, ω) =
+�
+α
+Φα(x, t)Hα(ω) and its complex conjugate ψ∗(x, t, ω) =
+�
+α
+Φ∗
+α(x, t)Hα(ω),
+Applying Hermite transform we get a deterministic nonlinear Schr¨odinger equation:
+i ∂
+∂t
+˜ψ(x, t, z) + [(∆ + H[Γ](x, t, z) − ˜ψ(x, t, z) ˜
+(ψ∗)(x, t, z)] ˜ψ(x, t, z) = 0,
+(8.20)
+where
+˜ψ(x, t, ω) =
+�
+α
+Φα(x, t)zα and
+˜
+(ψ∗)(x, t, z) =
+�
+α
+Φ∗
+α(x, t)zα.
+˜ψ(x, 0, z) = H[Γ0](x, z), x ∈ R3, z ∈ CN.
+(8.21)
+Nonlinear Schr¨odinger equation with time dependent potential of the above type (with fixed
+z ) is studied in [15]. With an introduction to a suitable smoothing for the noise in the form
+of (I−∆x,t)−γ the Hermite transformed problem above in the framework of [15] (in particular
+Assumption 1.3 regarding the potential in that paper) and deduce a unique solution ˜ψ ∈
+C([0, T]; L2(R3)) ∩ L8/3([0, T]; L4(R3)) and hence inverse Hermite transform gives:
+Proposition 8.3. For Gaussian initial data noise distribution Γ0 ∈ (S)−1(L2(Rd)) and
+Gaussian noise forcing Γ ∈ (S)−1(L∞
+loc(R × Rd)), there exists a unique solution to the sto-
+chastic nonlinear Schr¨odinger equation with multiplicative space-time white noise as ψ ∈
+(S)−1(C([0, T]; L2(R3))∩L8/3([0, T]; L4(R3))). For Poisson and L´evy cases unique correspon-
+dence map U gives a unique solution ψ ∈ (S)−1(L∞([0, T]; L2(R3)) ∩ L8/3([0, T]; L4(R3)))
+9. Concluding remarks
+In this paper we have casted a number of Laser propagation problems in random media in
+the white noise distribution theory framework to stimulate further research in their mathe-
+matical structure. Some of the most prominent problems are selected for our discussion and
+a number of other nonlinear wave equations that arise in Laser interaction with plasma such
+as the Schrodinger-Hartree equation [33], Zakharov system [72] and Davey-Stewartson equa-
+tion [22, 14] can also be treated by the method initiated here when dealing with stochastic
+medium effects. We briefly indicate below how one may proceed with Wick quantization in
+these well-known nonlinear wave problems and for simplicity we formulate them with spa-
+tially random initial data.
+
+22
+S. S. SRITHARAN AND SABA MUDALIAR
+(1) Random (quantized) Schr¨odinger-Hartree equation:
+i ∂
+∂tϕ + ∆ϕ = ±[|x|n ⋆ (ϕ∗ ⋄ ϕ)] ⋄ ϕ, x ∈ Rd, 0 < n < d,
+(9.1)
+ϕ(x, ω, 0) = Γ(x, ω).
+(9.2)
+Upon Hermite transform this system (for a fixed z) will result in the usual deterministic
+Schr¨odinger-Hartree system measure initial data. We can apply a smoothing to the noise in
+the form (I −∆)−γ and then use a solvability theorem such as in [37] and utilize the analytic
+implicit function theorem [7, 67] and inverse Hermite transform to deduce the solvability:
+Proposition 9.1. For 0 < γ < min (2, n), and Gaussian/Poisson/L´evy white noise initial
+data Γ(·, ·) ∈ (S)−1(L2(Rn)), there is a unique solution to the (quantized) Schr¨odinger-
+Hartree equation ϕ ∈ (S)−1�
+C(R; L2(Rn)) ∩ L8/γ
+loc(R; L
+4n
+2n−γ (Rn))
+�
+.
+(2) Random (quantized) Zakharov system in Rd+1, d = 2, 3:
+i ∂
+∂tϕ + ∆ϕ = ϕ ⋄ n,
+(9.3)
+[ ∂2
+∂t2 − ∆]n = −∆(ϕ∗ ⋄ ϕ),
+(9.4)
+ϕ(x, ω, 0) = Γ1(x, ω), n(x, ω, 0) = Γ2(x, ω), and ∂
+∂tn(x, ω, 0) = Γ3(x, ω).
+(9.5)
+Upon Hermite transform this system (for a fixed z) will result in the usual deterministic
+Zakharov system measure initial data. We may have to apply a smoothing to the noise in
+the form (I −∆)−γ before we can start with a suitable solvability theorem such as in [29, 19]
+and utilize the analytic implicit function theorem [7, 67] and inverse Hermite transform [36]
+to deduce the solvability of the stochastic Zakharov model.
+Proposition 9.2. Suppose the initial data Gaussian white noise distribution satisfy (Γ1, Γ2, Γ3) ∈
+(S)−1�
+H1/2(Rd)) × L2(Rd) × H−1(Rd)
+�
+then there exists a unique local-in-time solution to
+stochastic Zakharov equation with white noise initial data such that
+(ϕ, n, ∂tϕ) ∈ (S)−1�
+C([0, T]; H1/2(Rd) × L2(Rd) × H−1(Rd))
+�
+.
+(9.6)
+(3) Random (quantized) Davey-Stewartson system in R2+1: The system was derived by
+Davey and Stewartson [22] and see [28] for a rigorous study. We consider the Wick-quantized
+problem:
+i ∂
+∂tu + δ ∂2
+∂x2 u + ∂2
+∂y2u = χ(u∗ ⋄ u) ⋄ ϕ + bu ⋄ ∂
+∂xϕ,
+(9.7)
+∂2
+∂x2 ϕ + m ∂2
+∂y2ϕ = ∂
+∂x(u∗ ⋄ u),
+(9.8)
+u(x, y, ω, 0) = Γ(x, y, ω).
+(9.9)
+The four parameters δ, χ, b, m are real, |δ| = |χ| = 1. The system is classified as elliptic-
+elliptic, elliptic-hyperbolic, hyperbolic-elliptic and hyperbolic-hyperbolic according to the
+respective signs of (δ, m) : (+.+), (+, −), (−, +), (−, −). Upon Hermite transform this sys-
+tem (for a fixed z) will result in the usual deterministic Davey-Stewartson system measure
+initial data and as discussed in the paper we can start with a suitable solvability theorem
+
+STOCHASTIC QUANTIZATION OF LASER PROPAGATION MODELS
+23
+such as in [28, 68] and utilize the analytic implicit function theorem [7, 67] and inverse
+Hermite transform [36] to deduce the solvability of the stochastic model:
+Proposition 9.3. For the Gaussian/Poisson/L´evy initial data white noise distribution Γ ∈
+(S)−1(L2(R2)), there exists a unique local in time solution in the elliptic-elliptic and hyperbolic-
+elliptic cases (m > 0):
+u ∈ (S)−1�
+C([0, T ∗[; L2(R2)) ∩ L4((0, T ∗) × R2)
+�
+,
+(9.10)
+∇ϕ ∈ (S)−1�
+L2((0, T ∗) × R2)
+�
+.
+(9.11)
+Acknowledgment: The first author’s research has been supported by the U. S. Air Force
+Research Laboratory through the National Research Council Senior Research Fellowship of
+the National Academies of Science, Engineering and Medicine.
+References
+[1] G. P. Agrawal, Nonlinear fiber optics: its history and recent progress, J. Optical Society of America B,
+Vol. 28, No. 12 (2011), A1-A10.
+[2] Albeverio, Z. Habba and S. Russo, A two-space dimensional semilinear heat equation perturbed by
+(Gaussian) white noise, Probab Theory Relat. Fields, 121, (2001), 319–366.
+[3] F. E. Benth, A White Noise Approach to a Class of Non-linear Stochastic Heat Equations, Journal of
+Functional Analysis, 146, 382-415, (1997).
+[4] F. E. Benth and J. Gjerde, A Remark on the Equivalence between Poisson and Gaussian Stochastic
+Partial Differential Equations, Potential Analysis, Vol. 8,(1998), 179–193.
+[5] R. Benzi, G. Jona-Lasinio, and A. Sutera, Stochastically perturbed Landau-Ginzburg system, J. Stat.
+Phys., vol. 55, Nos. 3-4, (1989), 506-522.
+[6] F. A. Berezin and M. A. Shubin, The Schr¨odinger Equation, Springer-Verlag, (1991).
+[7] N. Bourbaki, El´ements de Math´ematique: Variet´e diff´erentielles et analytiques, Fasc. XXXIII, Livre VI.
+Hermann, Paris (1967).
+[8] H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial data, University
+of Wisconsin, Madison, Mathematics Research Center Report No. 2277, (1981).
+[9] H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, Journal D’Analyse
+Mathematique, Vol. 68, (1996), 277-308.
+[10] A. de Bouard, A. Debussche, On the stochastic Korteweg-de Vries equation, J. Funct. Anal. 22 154,
+215–251 (1998).
+[11] J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data,Selecta mathematica,
+New ser., Vol. 3 (1997) 115 – 159.
+[12] J. Bourgain, Global Solutions of Nonlinear Schr¨odinger Equations, American Mathematical Society,
+Providence, Rhode Island, 1999.
+[13] R. H. Cameron and W. T. Martin, The orthogonal development of nonlinear functionals in series of
+Fourier-Hermite functions, Annals of Mathematics, Vol. 48, No. 2, April, 1947, 385-392.
+[14] P. Carbonaro, The Davey-Stewartson equation in a complex plasma, Waves and Stability in Continuous
+Media, pp. 68-73 (2010).
+[15] R. Carles, Nonlinear Schr¨odinger equation with time dependent potential, Commun. Math. Sci., vol. 9
+(2011), 937–964.
+[16] S. Chandrasekhar, The transfer of radiation in stellar atmospheres, Bull. Amer. Math. Soc. 53(7), 641-
+711 (1947).
+[17] S. Chandrasekhar, On the diffuse reflection of a pencil of radiation by a plane parallel atmosphere,
+Proceedings of the National Academy of Sciences, Vol. 44, (1958), 933-940.
+[18] S. Chandrasekhar, Radiative Transfer, Dover Publishers, (1960).
+[19] Z. Chen and S. Wu, Local well-posedness for the Zakharov system in dimension d = 2, 3, Communica-
+tions on Pure and Applied Analysis, Vol. 20, Issue 12: 4307-4319, (2021).
+
+24
+S. S. SRITHARAN AND SABA MUDALIAR
+[20] P. H. Damgaard and H. Huffel, Stochastic quantization, Physics Reports, Vol. 152, Nos. 5 & 6, (1987),
+227-398.
+[21] G. Da Prato and A. Debussche, Strong solution to the stochastic quantization equation, The annals of
+probability, Vol. 31, No. 4, (2003), 1900-1916.
+[22] A. Davey, K. Stewartson, “On three dimensional packets of surface waves”, Proc. R. Soc. London, Series
+A, Vol. 338 (1613), 101–110,(1974).
+[23] R. J. Diperna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces,
+Inventiones Mathematicae, 98, 511-547 (1989).
+[24] W. G. Faris and G. Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise, J. Phys.
+A, Gen. Math., Vol. 15 (1982) 3025-3055.
+[25] B. P. W. Fernando and S. S. Sritharan, Stochastic quasilinear partial differential equations of evolution,
+Infinite Dimensional Analysis, Quantum Probability and Related Topics, 18(3), 01–13 (2015).
+[26] T. Funaki, Construction of a solution of random transport equation with boundary condition, J. Math.
+Soc. Japan, Vol. 31, No. 4, 1979.
+[27] D. Gatarek and B. Goldys, Existence, uniqueness and ergodicity for the stochastic quantization equation,
+Studia Mathematica, Vol. 119, No. 2, (1996), 179-193.
+[28] J.M. Ghidaglia and J. C. Saut, On the initial value problem for the Davey-Stewartson equation, Non-
+linearity, 3 (1990) 475-506.
+[29] J. Ginibre and Y. Tsutsumi, On the Cauchy problem of the Zakharov equation, Journal of Functional
+Analysis, 151, 384-436 (1997).
+[30] A. Gulisashvili, Non-autonomous Kato classes of measures and Feynman-Kac propagators, Trans. Amer-
+ican Math. Soc., Volume 357, Number 11, (2004) 4607–4632.
+[31] A. Gulisashvili and J. A. Van Casteren, Non-autonomous Kato Classes of Measures and Feynman-Kac
+Propagators, World Scientific Publishing, Singapore (2006).
+[32] Jonathan Gustafsson, Benjamin F. Akers, Jonah A. Reeger, Sivaguru S. Sritharan, Atmospheric prop-
+agation of high energy lasers, Eng. Math. Lett., 2019 (2019), Article ID 7.
+[33] D. R. Hartree, The Calculation of Atomic Structures, John Wiley and Sons, New York, (1957).
+[34] T. Hida, Brownian Motion, Springer-Verlag, New York, (1980).
+[35] T. Hida, H-H. Kuo, J. Patthoff, and L. Streit, White Noise - an Infinite Dimensional Calculus, Springer
+Science, 1993.
+[36] H. Holden, B. Oksendal, J. Uboe and T. Zhang, Stochastic Partial Differential Equations: A Modelling
+White Noise Functional Approach, Springer-Verlag, 2010.
+[37] R. Hyakuna, On the global Cauchy problem for the Hartree equation with rapidly decaying initial data,
+Annales de l’Institut Henri Poincar´e C, Analyse non lin´eaire, Vol. 36, Issue 4, July (2019), 1081-1104.
+[38] Y. Ito, Generalized Poisson Functionals, Probab. Th. Rel. Fields, Vol. 77, (1988), 1-28.
+[39] Y. Ito and I. Kubo, Calculus on Gaussian and Poisson white noises, Nagoya Math. Journal, Vol. Ill
+(1988), 41-84.
+[40] S. Kaligotla and S. V. Lototsky, Wick Product in The Stochastic Burgers Equation: A Curse or a Cure?
+Asymptotic Analysis, Vol. 75, No 3-4, pp. 145-168, (2011).
+[41] M. Kardar, G. Parisi and Y. Zhang, Dynamic scaling of growing interfaces, Physical Review Letters,
+Vol. 56, No. 9, (1986), 889-892.
+[42] T. Kato, Linear evolution equation of “hyperbolic” type, J. Fac. Sc., University of Tokyo, Vol. 25,
+(1970), 241-258.
+[43] T. Kato, Linear evolution equation of “hyperbolic” type-II, J. Math. Soc. Japan, Vol. 25, (1973), 648-
+666.
+[44] T. Kato, Quasi-linear equations of evolution with applications to partial differential equations, Lecture
+Notes in Math. 448, Springer Verlag (1975) 25-70.
+[45] T. Kato, Linear and quasi-linear equations of evolution of hyperbolic type. (C.I.M.E.) II CicIo (1976).
+[46] M. Keel and T. Tao, End point Strichartz estimates, American Journal of Mathematics, Vol. 120,
+955-980, (1998).
+[47] S. N. Kruzkov, First order quasilinear equations of several independent variables, Math. U. S. S. R.
+Sbornik, Vol. 10 (1970), No. 2 ,217-243.
+[48] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cam-
+bridge, U.K. (1990).
+
+STOCHASTIC QUANTIZATION OF LASER PROPAGATION MODELS
+25
+[49] H-H. Kuo, White Noise Distribution Theory, CRC Press, 1996.
+[50] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Communications on Pure
+and Applied Mathematics, Vol. 21, No.5, 467–490, (1968).
+[51] M. T. Mohan and S. S. Sritharan, Stochastic quasilinear evolution equation in UMD-Banach spaces,
+Math. Nachr. 1–20 (2017).
+[52] S. Ogawa, A partial differential equation with the white noise as a coefficient, Z. Wahr. verw. Geb., 28
+(1973), 53-71.
+[53] G. C. Papanicolaou, D. McLaughlin, and R. Burridge, A Stochastic Gaussian Beam, J. Math, Phys.,
+Vol. 14, No. 1, (1973), 84-89.
+[54] G. Parisi and Y. Wu, Perturbation theory without gauge fixing, Scientia Sinica, Vol. XXIV, No.4,
+(1981), 483-496.
+[55] A. Pazy, Semigroups of Operators and applications to Partial Differential Equations, Springer-Verlag,
+New York, (1983).
+[56] M. Reed and B. Simon, Methods of Modern Mathematical Physics, V.1, Academic Press, 1980.
+[57] B. Simon, Distributions and their Hermite expansions, Journal of Mathematical Physics, Vol. 12, No.1,
+(1971), 140-148.
+[58] B. Simon, The P(Φ)2 Euclidean (Quantum) Field Theory, Princeton Series in Physics, 1974.
+[59] B. Simon, Schrodinger semigroups, Bulletin of the American Mathematical Society, Vol. 10, No. 3,
+(1982), 447-526.
+[60] P. Sprangle, J. R. Pe˜nano, and B. Hafizi, Propagation of intense short laser pulses in the atmosphere,
+Phys. Rev. E 66, Issue: 4, 2002, Paper 046418 pp.1-21.
+[61] R. S. Strichartz, Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave
+equations, Duke Mathematical Journal, Vol. 44, No.3, (1977), 705-714.
+[62] C. Sulem and P-L. Sulem, The Nonlinear Schr¨odinger Equation: Self-Focusing and Wave Collapse,
+Springer-Verlag, New York, (1999).
+[63] T. Tao, Global well-posedness of the Benjamin-Ono equation in H1(R), Journal of Hyperbolic Differen-
+tial Equations, Vol. 1, No. 1 (2004) 27–49.
+[64] Fred D. Tappert, The parabolic approximation method, in Joseph B. Keller, John S. Papadakis, Editors,
+Wave Propagation and Underwater Acoustics, Springer-Verlag, Berlin, 1977.
+[65] V. I. Tatarski, Wave Propagation in a Turbulent Medium, Dover Publishers, New York, (1961).
+[66] Y. Tsutsumi, The Cauchy problem for the Korteweg-De Vries equation with measures as initial data,
+SIAM J. Math. Analy., Vol. 20, No. 3, pp. 582-588, (1989).
+[67] T. Vallent, Boundary Value Problems of Finite Elasticity, Springer-Verlag, New York, 1988.
+[68] E. J. Villamizar-Roa and J. E. Perez-Lopez, On the Davey–Stewartson system with singular initial data,
+Comptes Rendus Mathematique, Vol. 350, Issues 21–22, (2012), 959-964.
+[69] M. J. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Aca-
+demic Publishers, (1988), Norwell, MA, U. S. A.
+[70] K. Yajima, Schr¨odinger equations with time-dependent unbounded singular potentials, Reviews in Math-
+ematical Physics, Vol. 23, No. 8 (2011) 823–838.
+[71] T. Zhang, Characterizations of the white noise test functionals and Hida distributions, Stochastics and
+Stochastic Reports , 41:1-2, 71-87 (1992).
+[72] V. E. Zakharov, “Collapse of Langmuir waves”, Soviet Journal of Experimental and Theoretical Physics,,
+Vol. 35: 908–914,(1972).
+

A9AyT4oBgHgl3EQfd_iI/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:92a61774e23aaa23d0d195367b7151223c2652e79cc03686036097b383a51193
+size 2687021

ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:bf058c9c0f80f8f6ed7f0c14fa4b4f0f549efe18c59f523c45db1e3924ad3474
+size 1239877