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+Damage Preserving Transformation for Materials with Microstructure
+Philip P. Müllera, Falk K. Wittela, David S. Kammera,∗
+aInstitute for Building Materials (IfB), ETH Zuerich, Laura-Hezner-Weg 7, 8093, Zuerich, Switzerland
+Abstract
+The failure of heterogeneous materials with microstructures is a complex process of damage nucleation, growth and
+localisation. This process spans multiple length scales and is challenging to simulate numerically due to its high com-
+putational cost. One option is to use domain decomposed multi-scale methods with dynamical refinement. If needed,
+these methods refine coarse regions into a fine-scale representation to explicitly model the damage in the microstructure.
+However, damage evolution is commonly restricted to fine-scale regions only. Thus, they are unable to capture the full
+complexity and breath of the degradation process in the material. In this contribution, a generic procedure that allows
+to account for damage in all representations is proposed. The approach combines a specially designed damage law,
+with a scheme to generate pre-damaged fine-scale microstructures. Results indicate that the damage approximation for
+the coarse representation works well. Furthermore, the generated fine-scale damage patterns are overall consistent with
+explicitly simulated damage patterns. Minor discrepancies occur in the generation but subsequently vanish when explicit
+damage evolution continuous; for instance under increased load. The presented approach provides a methodological basis
+for adaptive multi-scale simulation schemes with consistent damage evolution.
+Keywords:
+Lattice, Continuum damage mechanics, Microstrutured disordered material, Anisotropic damage,
+Multi-scale simulation, Harmonic decomposition, Damage modelling
+∗Corresponding author
+Email addresses: [email protected] (Philip P. Müller),
+[email protected] (Falk K. Wittel), [email protected] (David S.
+Kammer)
+
+Contents
+1
+Introduction
+3
+2
+Materials and Methods
+4
+2.1
+Generic Damage Transforming Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.2
+Continuum Representation of 2D Isotropic Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.3
+Exemplary Material Motive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2.4
+Determining the Damage Law for the Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.5
+Process for the Construction of a Damaged Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.6
+Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.6.1
+The UniformSim Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.6.2
+The MultiLoadSim Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.6.3
+The ReconstrSim Simulation Setup
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3
+Results
+8
+3.1
+Details of a Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3.2
+Estimation of the Damage Law �D(#–κ)
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3.3
+Test of the Damage Evolution Law �D(#–κ)
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+3.4
+Estimation of the Transfer Function #–�r (#–κ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+3.5
+Tests of the Reconstruction Process
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+4
+Summary and Conclusion
+15
+5
+CRediT
+15
+6
+Declaration of Competing Interest
+15
+7
+Data Availability
+15
+Appendix A
+Parameters of �rx(κx) and �ry(κy)
+16
+References
+17
+2
+
+1. Introduction
+At a certain scale even heterogeneous materials will
+appear homogeneous and some can even be considered
+isotropic.
+Among others, this is true for concrete, one
+of the most widely used commodity on earth, a mixture
+made of sand, aggregates, cement, water and chemical ad-
+mixtures. The growth of damage inside concrete is highly
+affected by the particular microstructure, where, depend-
+ing on the scale, aggregates or even sand grains act either
+as focal points for stresses or obstacles for damage.
+Damage initiates at very small scales, long before the
+macroscopic structure itself will fail or crack. Instead, the
+damage leads to a reduction of the material’s stiffness.
+Nevertheless, at one point the accumulated damage be-
+comes so widespread, that even its smallest increase, will
+trigger the previously isolated nuclei to merge. This leads
+to a cascade of increasingly larger defects, culminating in
+the emergence of a macroscopic crack.
+Continuum based methods are the methods of choice
+if large structures should be simulated, due to their com-
+putational efficiency. For taking into account intrinsic de-
+generative processes, constitutive laws are used. One of
+the earliest, but still widely used laws for modelling dam-
+age in concrete was proposed by Mazars (Lemaître, 2001;
+Mazars and Lemaître, 1985).
+It employs a scalar dam-
+age variable to degrade the material’s stiffness. However,
+even if the material was initially isotropic, damage will
+induce anisotropy into the material’s behaviour. Clearly
+any scalar damage variable is inherently unable to capture
+this. During the years, a variety of anisotropic damage
+models were proposed to address this issue (Brancherie
+and Ibrahimbegovic, 2009; Braun et al., 2021; H. Chen et
+al., 2016; Delaplace and Desmorat, 2008; Desmorat et al.,
+2007; Gaede et al., 2013). All of them consider the ac-
+cumulated effects of the damage’s growth, represented by
+internal state variables at the material points and by that
+disregard the actual microstructure, whose degeneration is
+the actual cause for the emerging damage.
+To overcome this deficiency, the entire microstructure
+could be explicitly represented and simulated. Unfortu-
+nately, even with today’s fast computers, this is only pos-
+sible for small sizes. A way to overcome this barrier are
+multi-scale methods. They allow to invest computational
+power exactly where it is needed, by combining different
+representations. Although, many different methods were
+proposed over the past years, they can be classified to be
+either of hierarchical or of concurrent nature (Liu, 2018;
+Zhang et al., 2012).
+Hierarchical methods are characterised by a full sepa-
+ration of scales, which allows to treat every level indepen-
+dently from each other. Thus, the information is passed
+between the different levels as one serves as input for the
+hierarchically higher level.
+Opposed to this, concurrent methods lack the full sepa-
+ration of scales and typically decompose the computational
+domain into different regions. Imagine a typical setting
+where high accuracy is only needed inside a small part of
+the computational domain, for example around a crack tip.
+Ideally, one limits methods with high accuracy but large
+computational burden to these small regions, while the
+remaining part of the computational domain is described
+by much more efficient methods. The flow of information
+between the different regions must be handled by a cou-
+pling scheme such as the Arlequin method (Anciaux et al.,
+2008; Bauman et al., 2008; Guidault and Belytschko, 2007;
+Unger and Eckardt, 2011; Wellmann and Wriggers, 2012).
+Whenever the decomposition is not available in advance,
+one must resort to adaptive methods to refine regions on
+demand (e.g., P. Y. Chen et al., 2021; Evangelista, Alves,
+et al., 2020; Evangelista and Moreira, 2020; Rodrigues et
+al., 2018). However, important questions are (i) how are
+the regions that need to be refined identified, and (ii) how
+is the loading history of the coarse connected to the initial
+state of the newly created fine scale representation? Espe-
+cially (ii) does not seem to be addressed well in literature.
+Most authors assume that the coarse representation does
+not accumulate any damage before being refined (L. Chen
+et al., 2021; P. Y. Chen et al., 2021; Rodrigues et al.,
+2018; Unger, Eckardt, and Konke, 2011). Consequently,
+damage is only allowed to evolve inside of the fine scale
+representations that start off as undamaged. This is in-
+consistent because it essentially disregards the entire load
+history, including the damage, that would have degraded
+a real material.
+In this paper we propose, to the best of our knowledge,
+a generic approach for the refinement step in adaptive con-
+current multi-scale simulations, that is able to account for
+the preceding damage evolution inside the coarse repre-
+sentation. Thus, the created fine scale representation con-
+tains an initial damage that is mechanically consistent to
+the damage that has evolved inside the coarse represen-
+tation. Our solution is to equip both, the fine and coarse
+scale representations, with their own damage measure. We
+analyse these damage measures and establish a connection
+between them. Our approach is actually able to address
+both questions raised above. By interpreting the coarse
+damage measure as a “measure of suitability”, critical re-
+gions that require refinement are regions whose damage
+measure surpassed some predetermined threshold value.
+Further, the coarse damage is used to initialise the fine
+scale damage.
+While the approach is generic and rather simple, its
+practical details highly depend on the selected represen-
+tations. Thus, we demonstrate it by applying it to one
+particular test case. The reminder of this paper is organ-
+ised as follows: In Sec. 2, we explain our method in more
+detail and present the proposed techniques. In Sec. 3 we
+determine the parameters of our method and asses its ap-
+plicability, before we draw final conclusions in Sec. 4.
+3
+
+2. Materials and Methods
+The particular choice of the material’s microstructure,
+also called motive, is in general arbitrary, but should fol-
+low principles of representative volume elements (RVE)
+(Lemaître and Desmorat, 2005). The state of a discrete
+representation with its inherent characteristic structure is
+fully given by r, that describes every single discrete ele-
+ment (right side of Fig. 1). In this representation, damage
+D(r) is given by the irreversible degeneration of the con-
+stituting elements. On the left side of Fig. 1, the smeared
+continuum representation is shown, which lacks such an
+explicit microstructure and only considers cumulative ef-
+fects of the damage through internal state variables added
+to the constitutive law. Here, damage is given by D, which
+depends on the state #–κ at a particular location and is em-
+bedded in the constitutive law.
+Figure 1: The continuum damage, at a certain point, is given by D, which
+depends on the respective state #–
+κ . The discrete damage D(r) depends on
+r and hence on the state of all discrete elements of the lattice. The two
+representations are interconnected to each other by homogenisation and
+refinement processes. Scale of the lattice is exaggerated.
+Since the continuum representation loses its validity
+once cracks localise, one must refine the continuum to its
+discrete twin in such way that all important aspects of the
+fracture will be captured accurately on the fine scale. The
+key for a meaningful adaptive modelling of the damage
+evolution lies in the transformation of the continuum to
+the discrete representation, that conserves the degraded
+mechanical behaviour found inside the continuum. One
+focus of this work is an approach to construct a discrete
+representation that respects the preceding damage present
+in the continuum representation.
+Even though the procedure is generic and in principal
+not restricted to specific numerical material representa-
+tions, this paper focuses on one particular choice. How-
+ever, we will outline the generic way of working with the
+method (see Sec. 2.1), before we start with our specific
+choice.
+We exemplarily chose a two-dimensional plane
+stress, isotropic material (see Sec. 2.2) with an under-
+lying material heterogeneity represented by a triangular
+network of beam-truss elements with linear-elastic, brittle
+behaviour with quenched disorder of breaking thresholds
+(see Sec. 2.3). We then discuss the particular choice of
+the damage law as well as the reconstruction step (see
+Secs. 2.4, 2.5). To determine and test them, we use data
+obtained from numerical simulations (see Sec. 2.6).
+2.1. Generic Damage Transforming Method
+Initially, the domain is described as a continuum with-
+out any internal structure, whose state is fully described
+by the continuum state variable #–κ. In the continuum, the
+damage evolution is fully govern by the damage function
+�D(#–κ). Therefore, �D(#–κ) can be interpreted as the macro-
+scopic damage, that is expected for a hypothetical dis-
+crete representation with identical loading. Thus we can
+determine the function describing the macroscopic dam-
+age by homogenising the discrete damage D(r). This leads
+to a perspective on the damage law that is different from
+the conventional one, where the damage law is calibrated
+against a physical material. Instead, here the law is cali-
+brated against a particular numerical representation of the
+material.
+When the continuum model experiences a certain dam-
+age limit, it is no longer suitable and has to be refined to
+a discrete representation. However, this discrete state has
+to be consistent with the previous continuum representa-
+tion. This includes stiffness and damage, which have to
+be preserved as much as possible by the transformation.
+Determining this reconstruction process challenging, since
+it is by its very nature not unique.
+2.2. Continuum Representation of 2D Isotropic Continua
+To represent a two-dimensional isotropic material un-
+der plane stress, the Finite Element Method (FEM) and
+as damage measure continuum damage mechanics (CDM)
+is used (Lemaître, 2001; Lemaître, 1996; Lemaître and
+Desmorat, 2005). We use the well known material law:
+σ = (I − D) Cε,
+(1)
+where σ and ε denote the continuum stress and strain ten-
+sors, respectively, and C is the continuum stiffness tensor
+of the undamaged material. Due to the choice of CDM, as
+continuum damage measure, the damage variable D can
+directly be identified with the damage function �D(#–κ). Fur-
+ther, we identify #–κ as the continuum state variable given
+as
+#–κ :=
+�κx
+κy
+�
+.
+(2)
+A zoning approach is used to divide the principal strain
+space along an angle χ, known as zone boundary, into an
+x- (shaded red parts) and y-zone (shaded yellow parts in
+Fig. 2). The two components κx and κy represent the max-
+imal reached principal tensile strain in x and y direction,
+respectively, i.e.
+κx := max
+�
+κx, ⟨ε1⟩+
+�
+,
+κy := max
+�
+κy, ⟨ε2⟩+
+�
+.
+While ε1 and ε2 are the eigenvalues of the strain tensor
+ε, its eigenvectors form a Givens rotation matrix of angle
+Γ, which is sometimes called eigenangle.
+The angle Γ,
+together with the boundary χ, determines which zone the
+eigenvalues are associated with, see Fig. 2.
+4
+
+Figure 2: Interpretation of the zone boundary parameter χ. While ε1
+and ε2 are the eigenvalues of ε, its eigenvectors are described by the
+value Γ. The eigenangle Γ and the zone boundary χ determines which
+eigenvalue acts in which direction.
+Since the continuum damage is only used during the
+initial phase with low damage, two assumptions are made:
+(i) It is assumed that the damage is orthotropic which re-
+duces D and �D(#–κ) to diagonal matrices. (ii) It is assumed
+that κx only acts on the x-damage while κy only affects
+the y-damage, which means that we assume no correlation
+between the directions.
+2.3. Exemplary Material Motive
+The example material motive chosen here is based on
+models proposed in Refs. (Herrmann et al., 1989; Mier,
+2017), namely a regular triangular lattice but formed by
+3rd order Reddy truss-beam elements with characteristic
+lattice size ℓ (Reddy, 1997; Reddy et al., 1997).
+Using
+beams allows to include bending properties and the re-
+sulting lattice is able to represent a Cosserat continuum
+(Ostoja-Starzewski, 2008; Vardoulakis, 2019).
+The mi-
+croscopical beams consist of an isotropic material with
+Young’s modulus Eb and Poisson’s ratio νb. A list of all
+used material parameters is given in Tab. I. In a multi-
+scale simulation, Eb has to be chosen such that the result-
+ing behaviour of the discrete structure matches the one of
+the continuum, i.e. its stiffness tensor C. However, since
+this paper studies the refinement step in isolation, with-
+out having an actual continuum phase, the choice of Eb is
+actually irrelevant.
+Lattice Geometry.
+The motive is defined by the number of
+nodes (Nx, Ny) in x- and y-direction, the spatial extension
+in x-direction Lx, with resulting characteristic lattice size
+ℓ := Lx/(Nx −1) and spatial y-extension Ly := Nyℓ
+√
+3/2.
+An out-of-plane height of H is assumed. To remove the
+symmetries of the lattice, topological disorder is intro-
+duced (Moukarzel and Herrmann, 1992; Wittel, 2006) by
+adding the random displacement
+#–x ∆
+i := a ℓ
+2
+#–x ∗
+i
+(3)
+to every internal node of the grid, where #–x ∗
+i is a random
+vector sampled uniformly from the unit circle (see Fig. 3a).
+Table I: Parameters of the discrete material motive.
+Property
+Value
+Unit
+Nx, Ny
+300, 346
+[−]
+Lx, Ly
+2, 1.998
+m
+H
+1
+m
+Eb
+1 × 106
+Pa
+νb
+0.3
+[−]
+kε
+3
+[−]
+λε
+0.02
+[−]
+kΦ
+3
+◦
+λΦ
+0.02
+[−]
+The distortion is controlled by parameter a ∈ [0, 1[, known
+as distortion level.
+Figure 3: (a) Distortion of the central node, ignoring the distortion of the
+surrounding nodes. The location of the distorted node (yellow circle), is
+randomly selected within the blue circle of radius aℓ/2. Afterwards, the
+length of the beams are adjusted to match the new node location (black
+lines). (b) The thickness of beam i is given as ti := A(O)
+i
+/ℓi, where ℓi is
+its length and A(O)
+i
+is the area the beam is representing. Points zL and
+zK are centres of the adjacent triangles’ incircles.
+Geometrical Properties of Beam-Truss Elements.
+The
+thickness of beam i, denoted as ti, depends on the lat-
+tice’s geometry. It is given as ti := A(O)
+i
+/ℓi, where A(O)
+i
+is the area represented by the beam and ℓi its length, see
+Fig. 3b. The area A(O)
+i
+is formally defined as the set of
+points that are closer to beam i than any other beam, but
+are inside the lattice. It can be determined by finding the
+intersection of the angle’s bisectors, i.e. centre of the incir-
+cle, of the two adjacent triangles denoted as zK and zL in
+Fig. 3b. In case the beam is part of the boundary A(O)
+i
+is
+artificially doubled. This ensures that in a regular lattice
+all beams have the same axial rigidity.
+Damage Criterion Applied to the Beam-Truss Lattice.
+In
+the discrete representation, damage is the irreversible fail-
+ure of elements, namely the reduction of their contributing
+stiffness to an insignificant level. To determine if a beam
+has surpassed its loading capacity, the elliptical criterion
+� εi
+εi; th
+�2
++
+max
+����Φ(r)
+i
+��� ,
+���Φ(l)
+i
+���
+�
+Φi; th
+=: Ψi ≥ 1
+(4)
+is used, where εi;th and Φi;th are the beam’s elongation
+and bending thresholds, respectively (Herrmann et al.,
+5
+
+KicKy
+E2a
+(b)
+ZK
+a
+ZL
+Φ
+(r)1989). Both thresholds are sampled independently from
+the Weibull distributions εi;th
+iid
+∼ Weib (kε, λε) and Φi;th
+iid
+∼
+Weib (kΦ, λΦ).
+The Discrete State Variable #–r .
+The
+discrete
+state
+is
+uniquely described by r. However, for the context of this
+paper the surrogate discrete state variable
+#–r :=
+�
+rx
+ry
+�
+(5)
+is introduced and termed “discrete state variable”. Since
+#–r has only two components it does not uniquely describe
+the damaged state.
+This ambiguity will be resolved by
+the reconstruction process (see Sec. 2.5).
+#–r is a purely
+mathematical quantity designed to have certain properties.
+First, its 1-norm �r := ∥#–r ∥1 := |rx| + |rx| equals to Nf/NT,
+where Nf is the number of failed beams and NT the total
+number of beams in the lattice. �r is also called the ratio of
+failed beams (rfb). Second, its components are defined by
+associating them to the x- and y-zone, respectively, similar
+to #–κ (see Sec. 2.2). But while κx is connected to strains
+in the x-zone, rx is related to the amount of beams that
+have failed due to κx.
+2.4. Determining the Damage Law for the Continuum
+The damage function �D(#–κ) will take the role of the
+damage variable D inside the constitutive equation (1).
+Thus, �D(#–κ) has to be designed such that its evolution
+mimics the expected behaviour of D (see Sec. 3.2). For
+the extraction, which involves two steps, the Uniform-
+Sim simulation data of fully discrete lattices is used (see
+Sec. 2.6.1).
+Step 1: Effective Material Stiffness Tensor C.
+First, the
+effective stiffness tensor C is calculated by homogenisation.
+After the convergence of each loading step, the following
+seven strain-states
+�
+�
+�
+�
+�
+1
+2
+3
+�
+�,
+�
+�
+4
+5
+0
+�
+�,
+�
+�
+6
+0
+0
+�
+�,
+�
+�
+0
+7
+0
+�
+�,
+�
+�
+0
+0
+8
+�
+�,
+�
+�
+9
+0
+10
+�
+�,
+�
+�
+0
+9
+8
+�
+�
+�
+�
+� ,
+(6)
+denoted as (εxx, εyy, 2εxy)T × 10−3, were applied to the
+lattice, while blocking further damage to measure the re-
+sulting stresses. This results in an overdetermined system
+of 21 equations for the 6 unknown coefficients of C, which
+is solved by a least-square approach.
+Step 2: Determining the Damage Variable D.
+Second,
+the damage variable D is extracted from the effective stiff-
+ness tensors of the lattice. For this, a technique originally
+presented by Oliver-Leblond et al. (2021) is used. For com-
+pleteness, the relevant equations are replicated to be
+d(T ) := tr1,2[T ] = Tkkij,
+(7a)
+K := 1
+4 tr
+�
+d(C)
+�
+,
+(7b)
+D := D
+�
+C, �C
+�
+:=
+1
+2K
+�
+d(C) − d
+�
+�C
+� �
+.
+(7c)
+The tensor defined by Eq. (7a) is also known as dilatation
+second order tensor, while scalar K of Eq. (7b) is the bulk
+modulus. Eq. (7c) combines the effective C and undam-
+aged stiffness tensor �C to the damage variable D. D is
+by construction a real symmetric 2 × 2 matrix, thus fully
+characterised by its two eigenvalues d(x) and d(y) as well as
+a single scalar Γ, describing the rotation of its eigenbasis
+(see Fig. 2).
+2.5. Process for the Construction of a Damaged Lattice
+The reconstruction process, i.e. the creation of a dis-
+crete lattice with a particular damage, involves two compo-
+nents: (i) The transfer function #–�r (#–κ), which transforms
+the continuum state #–κ into the discrete surrogate state
+variable #–r . (ii) A scheme which transforms the surrogate
+state #–r into the full discrete state r. Hence, the scheme
+must be able to resolve the inherently present ambiguity in
+#–r . As direct consequence of the definition of the discrete
+state #–r (see Eq. (5)), the transfer function is given as
+#–�r (#–κ) :=
+��rx(κx)
+�ry(κy)
+�
+.
+(8)
+As for the damage function �D(#–κ), we are using data ob-
+tained from the UniformSim simulations (see Sec. 2.6.1)
+to empirically determine the function #–�r (#–κ) that approxi-
+mates #–r . Due to the nature of #–r it is impossible to mea-
+sure its components and thus to fit them directly. How-
+ever, it is easy to measure and fit the quantity �r := ∥#–r ∥1.
+Because of the specific design of the simulations and as-
+sumptions, it is possible to associate the value �r to the
+components of #–r , see Sec. 2.6.1.
+For reconstructing the full discrete state, a probabilis-
+tic scheme was devised. It starts by constructing an un-
+damaged lattice from which certain beams are removed,
+such that the resulting damage matches in a statistical
+sense the one given by #–κ. Due to the assumed decoupling
+between the x- and y-zone, it is possible to handle the two
+directions independently. For each direction α, i.e. x and
+y, the following steps must be done:
+1. From the continuum state κα the corresponding dis-
+crete state
+variable,
+rα = �rα(κα)
+is
+computed.
+Through the relationship Nα := rα·NT , it is possible
+to determine how many failed beams are associated
+to this direction.
+2. Each beam is assigned a probability defined as
+pi ∝
+1
+εi; th
+���
+�#–b i, #–t α
+����
+k
+,
+(9)
+where εi; th is the elongation threshold and #–b i the di-
+rection of the beam. The vector #–t α, called “damage
+basis”, represents the main damage direction. In our
+motive, it is either #–t x := (1, 0)T or #–t y := (0, 1)T.
+Finally, parameter k, called “directional weight”, is
+6
+
+a tuning parameter that balances the relative impor-
+tance of the two terms and needs to be determined
+(see Sec. 3.4).
+3. The Nα many beams to fail are drawn from the prob-
+ability distribution defined by Eq. (9) without re-
+placement.
+4. The selected beams are marked as failed.
+2.6. Numerical Simulations
+For estimating and testing the damage function �D(#–κ)
+and the transfer function #–�r (#–κ), a series of different numer-
+ical simulations are carried out on fully discrete lattices.
+We employ for this a customised version of the Akantu
+FEM library (Richart and Molinari, 2015).
+Due to the
+randomness of the lattice, 30 realisations were made for
+each case.
+2.6.1. The UniformSim Simulation Setup
+The first type of simulation, called UniformSim, is used
+for estimating the transfer function #–�r (#–κ) and the damage
+law �D(#–κ). These simulations realise an uni-axial strain
+Figure 4: Boundary conditions used by the UniformSim series, shown for
+the case of ϕ = 0°. In general, the boundary conditions given by Eq. (10)
+are applied to the whole boundary. Scale of the lattice is exaggerated.
+state of the lattice that is also rotated by an arbitrary but
+constant angle ϕ, called the pull direction. (see Fig. 4).
+Thus
+εϕ := Rϕ
+T
+��ε1
+0
+0
+0
+�
+Rϕ,
+(10)
+where Rϕ is the Givens rotation matrix for angle ϕ that
+is applied to the lattice’s boundary. In each loading step,
+�ε1 is increased by 0.0001 until 0.005 is reached. The limit
+is chosen to ensure that no localisation will occur and that
+damage maintains its diffuse character.
+The particular setup of the UniformSim simulations
+together with the previous assumptions on D and #–�r (#–κ)
+allows the following conclusions and simplifications:
+(i) A pull direction is either associated to the x- or y-
+zone (see Sec. 2.2).
+This allows to probe the be-
+haviour of a single zone. Which zone is probed de-
+pends on ϕ and the yet unknown zone boundary
+value χ.
+(ii) A simulation, i.e. a particular value of ϕ, will only
+affect the state of either the x- or the y-zone. Thus,
+an increase of �ε1 will only affect one eigenvalue of D
+and a single component of #–κ as well as #–r . Which
+component is affected depends on ϕ and χ.
+(iii) For the continuum state variable the relation �κ
+!=
+∥#–κ∥1
+!= |κα|
+!= �ε1 holds.
+Thus, one component
+equals the applied uni-axial strain �ε1, while the other
+is zero.
+(iv) For the discrete state variable, the relation �r
+!= ∥#–r ∥1
+!=
+|rα| holds. Thus, only one component is non zero and
+equals �r. This can be used to determine the compo-
+nents of #–r from �r, once the x- and y-zones are known
+(see Sec. 3.4).
+2.6.2. The MultiLoadSim Simulation Setup
+For testing the damage function as well as the recon-
+struction procedure, a second type of simulation is used,
+called MultiLoadSim.
+It realises a bi-axial strain state,
+imposed along the x- and y-axes (see Fig. 5). Both strains
+are increased until εxx = εyy = εfin is reached, where εfin
+is the control parameter. For each simulation, the loading
+is imposed in three different ways, but each time the same
+initial lattice is used:
+XThenYSim: εxx is increased in steps of 0.0001 until it
+reaches εfin and then maintained. Then εyy is in-
+creased by the same increment until εfin is reached.
+YThenXSim: The same as XThenYSim, however, the order
+of loading the axes is switched.
+BothXYSim: Both strains εxx and εyy are increased simul-
+taneously, in steps of 0.0001 until εfin is reached.
+All three paths reach the same final state, εxx = εyy =
+κx = κy = εfin, but via different paths. As a consequence,
+the special relation �κ := ∥#–κ∥1 = |εxx| + |εyy| holds in
+these simulations. Both, the XThenYSim and the YThenX-
+Sim loading path impose in the first half of the loading an
+uni-axial strain state and then switch to a bi-axial strain
+state for the second half, while the BothXYSim loading path
+imposes a bi-axial strain state from the beginning.
+Figure 5: Boundary conditions used in the verification simulations. Scale
+of the lattice is exaggerated.
+7
+
+2.6.3. The ReconstrSim Simulation Setup
+For testing the reconstruction process (see Sec. 2.5), a
+third type of simulation is used, called ReconstrSim. The
+basic setup is equivalent to UniformSim, but for ϕ = 0°.
+Further, a lattice with a certain initial damage is used.
+This damage was constructed to match the continuum
+state #–κ = (�ε, 0)T, i.e.
+the damage created by an uni-
+axial strain of �ε applied along the x-direction. Another
+important difference is, that the loading does not start at
+zero but at �ε. In case of a perfectly working reconstruc-
+tion process, one would expect no additional damage for
+strains ≤ �ε.
+Unlike the MultiLoadSim tests, which focuses on the
+value of the reconstructed damage, these tests focus on
+how the reconstructed lattices behave after their recon-
+struction. In essence, this test simulates the exchange of
+the continuum representation with the discrete represen-
+tation, i.e. the refinement process, which is the core ap-
+plication of the proposed method.
+3. Results
+Our proposed method relies on the damage law �D(#–κ)
+used inside the continuum and the reconstruction process.
+First, we discuss how we will use the techniques introduced
+in Sec. 2 to process the data that we have collected from
+the numerical simulations. Then, we determine the dam-
+age law �D(#–κ) and the zone boundary value χ (see Sec. 3.2)
+followed by an assessment of its accuracy (see Sec. 3.3).
+Thereafter, we repeat the process to determine the trans-
+fer function #–�r (#–κ) and the directional weight parameter k
+(see Sec. 3.4). Finally, we demonstrate the applicability of
+our method (see Sec. 3.5).
+3.1. Details of a Numerical Simulation
+Let’s consider a setting similar to UniformSim but with
+ϕ = 0° (see Fig. 6a). Only one realisation is simulated and
+the loading goes beyond �ε1 = 0.005.
+We measure the normalised stresses (solid lines) and
+compare them with the expected ones in case of suppressed
+damage, i.e. undamaged case (dashed lines) (Fig. 6b). As
+expected, initially the stresses behave predominantly lin-
+ear. However, once a strain of about 0.003 is reached, we
+observe that �σxx starts to deviate from the undamaged
+case.
+While this deviation increases with further load-
+ing, we cannot observe it for �σyy, that is much less af-
+fected by the loading. At one point, we observe that both
+stresses suddenly drop. This is caused by the emergence
+of a macroscopic crack, which is the expected behaviour
+for a brittle disordered material, such as concrete.
+Using the techniques presented in Sec. 2.4, it is possible to
+extract the macroscopic damage variable D for the lattice,
+at any loading step. In Fig. 6c, we show the eigenvalues of
+the extracted damage variables where we can see that d(x)
+exceeds d(y). This also explains why �σxx deviates much
+more from the undamaged behaviour when compared to
+�σyy. The underlying reason of this difference are the hor-
+izontal beams. They experience much larger strains than
+inclined beams, since they align with the loading and thus
+fail at much larger number. From Fig. 6c, we can also see
+that d(y) drops for a strain at around 0.002. This is a non-
+physical behaviour as damage should always increase. It is
+caused by some numerical issues during the determination
+of the stiffness tensor (see Sec. 2.4) and the sensitivity of
+damage extraction process.
+Fig. 6d shows the ratio of failed beams (rfb), �r := ∥#–r ∥1 :=
+Nf/NT, where Nf is the total number of number of failed
+beams and NT the total beams in the lattice. We will use
+it to indirectly estimate the transfer function #–�r (#–κ).
+Figs. 6e-h show snapshots of the lattice’s underlying
+microstructure. While taken at different loading steps (see
+Fig. 6d), they always show the same set of nodes, located
+roughly at the lattice’s centre. While the exact damage
+pattern depends on the realisation of the lattice and lo-
+cations where the snapshot was taken, statistically they
+all look the same.
+It is this statistical damage pattern
+that we want to capture by the transfer function #–�r (#–κ)
+and recreate by the reconstruction process. Whereas the
+damage law �D(#–κ) captures the accumulated effects on the
+lattice’s macroscopical stiffness.
+3.2. Estimation of the Damage Law �D(#–κ)
+We now study the behaviour of the damage variable D
+that we have extracted from the data of the UniformSim
+simulations. From these observations, we will determine
+the damage function �D(#–κ) as well as the zone boundary
+value χ (see Fig. 2).
+Functional Form of �dx(κx) and �dy(κy).
+Since we have
+assumed an orthotropic damage variable (see Sec. 2.2), we
+have to assume the same for the damage function. Thus
+arriving the tentative form of the damage function is given
+by
+�D(#–κ) :=
+�dxx(#–κ)
+0
+0
+dyy(#–κ)
+�
+.
+To account for deviations from this assumption, we will
+connect the two diagonal elements of the damage func-
+tion with the eigenvalues of the measured damage vari-
+able. Thus, we have only two functions that we need to
+determine.
+Fig. 7 shows the evolution of the eigenvalues d(x) and
+d(x) from the extracted damage variable for the pull di-
+rections ϕ ∈ { 0°, 60° } at various distortion levels.
+For
+ϕ = 0°, the eigenvalue d(x) is much larger than d(y), while
+for ϕ = 60° the opposite is observed. Later, we will use
+this to determine the zone boundary value χ. Most im-
+portantly, the figures show that both eigenvalues follow a
+power law, irrespective of the pull direction and distortion.
+Thus we approximate the diagonal elements/eigenvalues of
+8
+
+Figure 6: Representative simulation example.
+(a) Schematics of the model configuration, scale of the lattice is exaggerated.
+(b-d) Evolution of
+continuum and microstructure properties of the lattice. Dotted lines denote strains beyond the limit of 0.005 used in UniformSim. (b) Normalised
+measured stresses. Dashed lines represent the behaviour in case of suppressed damage. (c) Eigenvalues of the extracted damage variable D, see Eq. (1).
+(d) Ratio of failed beams �r in the specimen. (e-h) Snapshots of a small section of the microstructure. Colours indicate the remaining load carrying
+capacity of the beams �
+Ψi := 1 − Ψi, where Ψi is defined by Eq. (4). Associated states are indicated in (d) by markers. Bending of beams is not shown.
+�D(#–κ) as:
+dxx ≈ �dx(κx; a, ϕ) := α(x)
+a,ϕ · κx
+β(x)
+a,ϕ,
+(11a)
+dyy ≈ �dy(κy; a, ϕ) := α(y)
+a,ϕ · κy
+β(y)
+a,ϕ.
+(11b)
+The parameters of these approximations depend on the
+distortion level a and the pull direction ϕ. Later, we will
+eliminate their dependency on ϕ and obtain the final pa-
+rameters that only depend on a, which is constant. Fur-
+ther, this choice guarantees that the damage is strictly
+increasing.
+Because of our previous assumption about the indepen-
+dence of the directions, the approximations of the eigenval-
+ues only depend on a single component of the continuum
+state #–κ (Sec. 2.2). While this could be justified due to their
+large differences, that we can see in Fig. 7, we clearly see
+that even for ϕ = 0°, there is a certain coupling between
+d(x) and d(y). To handle this, we use a simple coupling
+scheme, which leads to the final damage function
+�D(#–κ) :=
+(12)
+�
+�
+�
+�
+max
+�
+�dx(κx), �
+dy(κy)
+η
+�
+0
+0
+max
+�
+�dy(κy), �
+dx(κx)
+η
+�
+�
+�
+�
+�,
+where �dx(κx) and �dy(κy) are the approximations of the
+eigenvalues defined by Eq. (11) but without the depen-
+dence on ϕ. The coupling ensures that the eigenvalues of
+the damage function �D(#–κ) will at most differ by a factor
+of η, which is exactly what we see in the case of uni-axial
+loading (see Fig. 7). Here, we will assume that the em-
+pirical parameter η equals 10 in all cases. We will later
+give a justification of the form and value of the proposed
+coupling. It is important to notice that this coupling is
+designed for the uni-axial case. However, a more elabo-
+rated coupling might be needed, depending on the details
+of other material motives.
+Parameters of �dx(κx) and �dy(κy).
+Since the data, espe-
+cially for the non-dominant eigenvalue shows strong vari-
+ation for small strains, only data points corresponding to
+9
+
+a)
+ouy
+0.005
+<6
+0.000
+(
+d(c)
+d(y)
+-4
+-6
+d
+0.02
+(h)
+ 0.01
+(g)
+(f)
+(e)
+0.00
+0.000
+0.002
+0.004
+0.006
+0.008
+remaining load carrying capacity  := 1 - 
+Err [-]
+>0Figure 7: Eigenvalues of the extracted damage variable D for pull direc-
+tions ϕ = 0° (a) and 60° (b). Solid lines correspond to d(x), while dashed
+lines correspond to d(y). Colours indicate different distortion levels a of
+the underlying lattice.
+strains above 0.002 were used for the parameter estima-
+tion.
+In Figs. 8a,b, we see that for small values of ϕ,
+the α(x)
+a,ϕ-parameters are very close to each other, while for
+larger values of ϕ one observes a much larger scattering.
+Interestingly, α(y)
+a,ϕ-parameters behave inversely. Further-
+more, on Fig. 8b we can clearly observe the α(y)
+a,ϕ depen-
+dence on ϕ. We see that α(y)
+a,ϕ is small if ϕ is small too, but
+above a certain value of ϕ, the parameters become much
+larger and their scattering increases. The same, but in an
+opposite way, holds for the α(x)
+a,ϕ-parameters but in a less
+pronounced fashion.
+The estimates for the β-parameters (see Figs. 8c,d)
+show a similar behaviour with respect to ϕ.
+However,
+while we observed a significant change in the behaviour
+of the α-parameters’ values, from a particular value of ϕ
+on we just observe an increase of the variability of β.
+In summary, from Fig. 8 we can conclude that the β- and
+especially the α(y)-parameters have different regimes de-
+pending on ϕ. Further, inside such a regime, their partic-
+Figure 8: Values of the α- (a,b) and β-parameters (c,d) with respect to
+the pull direction ϕ. Colours indicate different distortion levels. Solid
+lines correspond to lg α(x)
+a,ϕ and β(x)
+a,ϕ, while dashed lines to lg α(y)
+a,ϕ and
+β(y)
+a,ϕ. Error bars indicate the 95% confidence interval.
+ular value does not depend much on ϕ.
+We also saw that the values for the β(x)-parameters for
+small values of ϕ and β(y)-parameters for large values of
+ϕ are both close to three. This means that the growth
+behaviour of �dx(κx) and �dy(κy) are very similar. This jus-
+tifies the form of the coupling used in the damage function
+in Eq. (12).
+Zone Boundary χ.
+In Figs. 7 and 8, we have observed
+that depending on the pull direction either d(x) or d(y) is
+dominant. We now exploit this fact to define χ. To this
+end, we define the dominance function ζ as:
+ζ(a, ϕ) := lg
+�
+d(x)
+a,ϕ; ˜κ=0.005
+d(y)
+a,ϕ; ˜κ=0.005
+�
+,
+(13)
+with d(α)
+a,ϕ; ˜κ=0.005 as the damage eigenvalue associated to
+direction α, once the uni-axial strain has reached 0.005.
+10
+
+a
+ = 0.0°
+10-2
+d(r) a = 0.0
+d(r) aα = 0.1
+d(r) a = 0.2
+10-3
+d(r) a = 0.3
+d(r) α = 0.5
+10-4
+~ d(y)
+(α)p
+10-5
+10-6
+10-7
+L
+(b)
+Φ = 60.0°
+10-2
+d(y) a = 0.1
+d(y) a = 0.2
+10-3
+d(y) a = 0.3
+d(y) α = 0.5
+10-4
+~ d(r)
+(6)p
+10-5
+10-6
+10-7
+10-4
+10-3
+ [-](a)
+5.0
+4.0
+(b)
+5.5
+[-] °
+5.0
+4.5
+4.0
+(c)
++
+a= 0.0
+α= 0.3
+4.0
+α = 0.1
+a = 0.5
+a = 0.2
+a = 0.7
+3.5
+3.0
+(d)
+4.0
+二
+3.5
+3.0
+0
+20
+40
+60
+80
+[] The most important aspects of this function are its sign
+and root, to a lesser extend its value. ζ > 0 means that
+d(x) is dominant, while ζ < 0 indicates that d(y) is domi-
+nant. Thus, χ, which might depend on the distortion a, is
+defined as ζ(a, χ)
+!= 0.
+Figure 9: Dominance function ζ(a, ϕ), Eq. 13, for different distortion
+parameters a. The x-dominated region, i.e. d(x) ≫ d(y), is defined by
+ζ > 0, while the y-dominated (grey shaded) region, i.e. d(x) ≪ d(y), is
+defined by ζ < 0. The UniformSim data was used.
+Examining Fig. 9, we see that, irrespective of the distor-
+tion, χ must lie between 30° and 45°. After some exper-
+imentation, we decided to use 40° as zone boundary, ir-
+respective of the distortion level. A closer analysis might
+yield different estimations.
+ζ can be seen as a measure of the coupling between
+d(x) and d(y). Thus, we can used it to determine the value
+of the empirical coupling parameter η, see Eq. (12). Our
+value η = 10 was selected because it is roughly the mean
+value for ϕ = 0°.
+Final Parameters of �dx(κx) and �dy(κy).
+Eliminating the
+dependency of the α- and β-parameters on the pull di-
+rection ϕ will results in parameters that are valid inside
+the entire x- or y-zone. For this, we combine the different
+estimates as:
+lg α(x)
+a
+:= 1
+|X|
+�
+ϕ∈X
+lg α(x)
+a,ϕ,
+lg α(y)
+a := 1
+|Y|
+�
+ϕ∈Y
+lg α(y)
+a,ϕ, (14a)
+β(x)
+a
+:= 1
+|X|
+�
+ϕ∈X
+β(x)
+a,ϕ,
+β(y)
+a
+:= 1
+|Y|
+�
+ϕ∈Y
+β(y)
+a,ϕ,
+(14b)
+where X contains all the pull directions associated to the
+x- and Y the ones associated to the y-zone. Parameters
+associated to the transversal directions are simply ignored,
+e.g. lg α(y)
+a,ϕ=0°. Further, the functional form of �dx(κx) and
+�dy(κy) is still given by Eq. (12), just without the depen-
+dency on ϕ. Note that Eq. (14) weights the different pull
+directions equally.
+3.3. Test of the Damage Evolution Law �D(#–κ)
+We now evaluate how well the damage function is able
+to predict the damage of a fully discrete simulation. For
+this purpose, the MultiLoadSim simulations are used.
+Figure 10: Damage eigenvalues for the three different loading paths, de-
+scribed in Sec. 2.6.2, with final strain εfin = 0.002, plotted against
+τ := ˜κ/2 εfin. Using a fully discrete simulation (solid) as reference and
+the CDM damage law �
+D(#–
+κ ) (dash-dotted).
+The colours indicates the
+three different loading paths. The distortion of the lattices was a = 0.3.
+Fig. 10 shows the results of such an experiment for
+εfin = 0.002. We can see the eigenvalues, once computed
+for the reference (solid), i.e. a fully discrete simulation,
+and alternatively computed by the damage function �D(#–κ)
+(dash-dotted), i.e.
+CDM. They are plotted against the
+normalised total strain τ := ˜κ/2 εfin. Thus, both the X-
+ThenYSim (orange) and the YThenXSim (green) load paths
+switch from an uni-axial to a bi-axial strain state at τ =
+0.5. We observe that irrespective of the loading path the
+same final damage values are reached. The value depends
+on the used method, since the final value of the CDM is
+different from the reference value. Note that this is not
+problematic since the CDM is only used during the initial
+phase.
+11
+
+2.0
+a
+=0.0
+c-dominated
+a = 0.1
+a = 0.2
+1.0
+a = 0.3
+a = 0.5
+[-] (
+a = 0.7
+a
+= 0.8
+0.0
+‘p)S
+y-dominated
+-1.0
+X 
+-2.0
+0
+20
+40
+60
+80
+6e
+10-3
+10-4
+10-5
+(a)p
+10-6
+10-7
+(b)
+10-3
+10-4
+I
+10-5
+(r)p
+10-6
+ref. BothXY
+10-7
+ref. XThenY
+ref. YThenX
+CDM
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+T := K/2efin [-]Fig. 10 also demonstrates that the damage for the X-
+ThenYSim and YThenXSim are very similar to each other.
+However, the eigenvalues are flipped, which is the expected
+behaviour. During the first half of the loading (i.e. τ <
+0.5), the prediction of the dominant eigenvalue matches
+well with the reference value for both loading paths. At
+the same time, the non-dominant eigenvalue, i.e. the one
+belonging to the transverse direction, is captured with less
+but still acceptable accuracy.
+The mismatch is entirely
+due to the rather crude choice of the η coupling parameter
+(see Eq. (12)). However, it indicates that the proposed
+coupling is indeed working.
+Nevertheless, for the second half of the loading (i.e. τ >
+0.5) the CDM is unable to capture the evolution to a satis-
+factory degree. In case of XThenYSim (orange lines), we see
+that the CDM approximation of the x-eigenvalue d(x) re-
+mains constant, since κx is not affected by a loading along
+the y-axis. However, we see that in the reference system
+d(x) continuously increase (see Fig. 11 for more). The y-
+eigenvalue d(y) predicted by the CDM remains initially
+constant due to the coupling. Once �dy(κy) has become
+larger than �
+dx(εfin)/η �dy(κy) starts to increase. However,
+as it can be seen form Fig. 10b, the reference d(y) starts
+to increase almost immediately.
+A different case is the BothXYSim loading path. From
+Fig. 10, it seems that for τ < 0.5 its damage grows slower
+than the dominant damage observed for the other two
+paths. This is because BothXYSim only has half the num-
+bers of loading steps the other two have. If this is corrected
+for then it would actually grow faster. This indicates that
+there is some form of coupling between the two directions
+that is not considered correctly.
+In Fig. 11, we can see how the final damage, i.e. val-
+ues of d(x) and d(y) at εxx = εyy = εfin, depend on the
+control parameter εfin, using either the reference (solid
+lines), the CDM (dash-dotted lines) or the reconstruction
+(dashed lines). The colours distinguish the three different
+load paths that were tested (see Sec. 2.6.2). The collapse
+of the lines indicate that the damage is indeed path in-
+dependent, regardless of the final strain εfin. However,
+the final value depends on the particular method that was
+used. In in Fig. 10, we observe a gap between the final
+damage attained by the reference and the one predicted
+by the CDM. We can now see that this gap is systematic
+and actually increases with larger εfin. This is again in-
+dicating that there is some form of coupling between the
+directions that is not take into account yet.
+3.4. Estimation of the Transfer Function #–�r (#–κ)
+Our procedure to reconstruct a discrete lattice repre-
+sentation based on a damaged continuum state (presented
+in Sec. 2.5) requires two unknown quantities: First, the
+transfer function #–�r (#–κ), which maps the continuum state
+#–κ to the corresponding discrete surrogate state #–r . Sec-
+ond, the directional weight parameter k, which balances
+the orientation and the strength of a beam during the re-
+Figure 11: Final value of the d(x) and d(y) damage eigenvalues, com-
+puted using the reference (solid), CDM (dash doted) and reconstruction
+(dashed) method, plotted against εfin. The colours indicate the three
+different loading cases from Sec. 2.6.2. All lattices have a distortion of
+a = 0.3.
+construction process (see Eq. (9)).
+Analogously to the
+determination of the damage function, the data from the
+UniformSim is used.
+Functional Form of �rx(κx) and �ry(κy).
+As mentioned be-
+fore, it is impossible to measure the components of #–r di-
+rectly. However, as outlined in Sec. 2.6.1 #–r is connected
+to the ratio of failed beam as �r = ∥#–r ∥1 = Nf/NT
+!= rα.
+Thus, we can estimate #–r indirectly. Fig. 12 shows �r for
+the pull direction ϕ = 0° at various distortion levels. As we
+can see, the distortion level has only minor influence. Dif-
+ferent pull directions do not lead to a qualitative change
+(data not shown). For that reason, we approximate the
+mean rfb as
+�r ≈
+���#–�r (#–κ)
+���
+1 := �r(κ; a, ϕ) := α(r)
+a,ϕ · κβ(r)
+a,ϕ,
+(15)
+with the two fit parameters α(r)
+a,ϕ and β(r)
+a,ϕ. Both depend
+on the distortion a and the pull direction ϕ. Due to our
+12
+
+0.007
+0.006
+0.005
+0.004
+(c)p
+0.003
+0.002
+0.001
+0.000
+b
+0.007
+k = 6, a= 0.3
+王
+ref. BothXY
+0.006
+ref. XThenY
+ref. YThenX
+0.005
+CDM
+-王-
+PD, k = 6
+I
+0.004
+(r)p
+0.003
+0.002
+0.001
+0.000
+0.0005
+0.0010
+0.0015
+0.0020
+0.0025
+ fin [-]Figure 12: ⟨˜r⟩ := �
+∥#–r ∥1
+�
+for pull direction ϕ = 0°, at different values
+of the distortion parameter a.
+No qualitative change is observed for
+different pull directions ϕ.
+previous assumptions, we can identify its argument κ di-
+rectly with �κ. To eliminate the dependency on ϕ we use
+the same method as for the damage function (see Sec. 3.2).
+However, parameters associated to pull directions in X are
+used to determine �rx(κx), while the ones belonging to Y
+determine �ry(κy). This will transform the approximation
+of the scalar quantity
+���#–�r (#–κ)
+���
+1 into the one for #–�r (#–κ).
+For the discussion about the estimated α- and β-para-
+meters please see Appendix A.
+Directional Weight Parameter k.
+The empirical tuning
+parameter k influences the selection of beams during the
+reconstruction process. It balances a beam’s strength, i.e.
+its elongation threshold εth, against how well it aligns with
+the damage basis #–t α (see Sec. 2.5). We determine k such
+that the reconstructed damage variable D resembles the
+reference damage D most closely. To this end, we define
+Υk:=
+��D11−D11
+��
+ℓ2+
+��D22−D22
+��
+ℓ2+2
+��D12−D12
+��
+ℓ2 (16)
+as a measure of separation between the two damages. For
+minimising Υk, we select a heuristic approach, in which
+the reconstruction process (see Sec. 2.5) is run for differ-
+ent values of k.
+The k that minimises Υk will then be
+used for the remaining part of this paper. However, for
+this particular reconstruction process, the used α(r)- and
+β(r)-parameters still depended on ϕ. Further, zoning was
+ignored and as damage basis the pull direction ϕ was used.
+The underlying lattice was not distorted. From Fig. 13 we
+see that k = 6 minimises Υk independent of the pull di-
+rection. We also see that pull directions { 30°, 90° } seem
+to be almost unaffected by k, however, their values match
+Υk=6. This is an artefact caused by the regular structure
+of the underlying lattice and the scalar product used in the
+definition of the selection probability (see Eq. (9)). How-
+ever, this artefact is indicating that k = 6 is indeed a good
+Figure 13: Υk, Eq. (16), for some values of k and various pull directions.
+The reconstruction process was done on regular grids, without zoning.
+Further the pull direction and its orthogonal was used as damage basis.
+choice.
+3.5. Tests of the Reconstruction Process
+Now we evaluate the performance of the proposed
+reconstruction scheme to create a mechanically equiva-
+lent lattice, based solely on the continuum state #–κ (see
+Sec. 2.5). For verification, we use the MultiLoadSim sim-
+ulations (see Sec. 2.6.2).
+In addition, we use the Re-
+constrSim simulations to simulate a refinement step (see
+Sec. 2.6.3).
+The MultiLoadSim Results.
+In Fig. 14, we see the re-
+sults from the MultiLoadSim setup with εfin = 0.002
+(see Sec. 2.6.2).
+They impose the bi-axial strain state
+εxx = εyy = εfin, but with loading applied via three dif-
+ferent paths. We used this setup before to assess the CDM
+(see Sec. 3.3). An important note concerning the recon-
+structed states is, that in each loading step the lattice and
+hence the damage is constructed anew. Thus, although it
+looks like a damage evolution, the damage at any loading
+step has no connection to the previous one. However, each
+time the same undamaged but distorted lattice was used.
+If we now compare the damage from the references
+(solid lines) with the one from the reconstructed lattices
+(dashed lines) in Fig. 14, we see that the overall dam-
+age values are very similar to the ones obtained by the
+CDM. As before, we observe that for τ < 0.5 the dom-
+inant eigenvalues, i.e.
+d(x) for XThenYSim and d(y) for
+YThenXSim, are captured well. Then, for τ > 0.5, these
+reconstructed eigenvalues stop growing and thus deviate
+from the references (solid lines).
+An effect we observed
+for CDM (dashed lines), too. But if we look at the other
+eigenvalues, i.e. d(y) for XThenYSim (orange) and d(x) for
+the YThenXSim (green), we see that they start to increase
+almost immediately like the reference. This was not the
+case for the CDM (dash-dotted lines shown in Fig. 10).
+13
+
+10-2
+0=0.0°
+00=D
+10-3
+a = 0.1
+a = 0.2
+a = 0.3
+10-4
+a = 0.5
+10-5
+10-6
+10-7
+10-4
+10-3
+i [-]a = 0.0
+β = 0.0°
+§ = 15.0°
+§ = 30.0°
+§ = 45.0°
+P = 60.0°
+Φ = 75.0°
+= 90.0°
+6
+10-4
+1
+4
+6
+8
+k [-]Figure 14: Damage eigenvalues for the three different loading paths, de-
+scribed in Sec. 2.6.2, with final strain εfin = 0.002, plotted against
+τ := ˜κ/2 εfin. Using fully discrete simulations (solid) as reference and the
+reconstructed damage (dashed). The colours indicate the three different
+loading paths that were taken. The distortion of the lattices was a = 0.3.
+See Fig. 10 for the damage evolution predicted by the CDM.
+The reconstruction process is affected by the ignored cou-
+pling between the directions as well. However, the damage
+eigenvalues generated by it follow the reference much bet-
+ter than the ones computed by the CDM.
+The ReconstrSim Results.
+Now we are using the Recon-
+strSim simulation setup, described in Sec. 2.6.3. The lat-
+tices that are used here were reconstructed for the con-
+tinuum state #–κ = (�ε, 0)T. The system is loaded under
+uni-axial strain along the x-axis, starting at �ε. This setup
+simulates how a discrete region that was loaded up to �ε, as
+continuum and then refined behaves upon further loading.
+In Fig. 15, we see how the damage eigenvalues d(x) and
+d(y) (dashed and dotted lines, respectively) and the rfb �r
+(solid lines), evolve for different reconstruction strains �ε,
+indicated by different colours. The grey lines correspond
+to the reference without any reconstruction.
+The circles in Fig. 15 indicate the values for d(x), d(y)
+and �r have in the reconstructed lattice before any loading
+was applied to them. The circles associated to ∥#–r ∥1 show,
+Figure 15: Behaviour of the d(x) and d(y) damage eigenvalues and �r.
+Colours indicate different reconstruction parameters �ε. Grey is the ref-
+erence, i.e. no reconstruction. Circles indicate damage/rfb values of the
+lattices directly after reconstruction. Squares indicate damage/rfb values
+of the lattices for an applied strain of �ε.
+that the reconstructed lattices have a matching rfb value �r
+which is a consequence of its construction. It is, however,
+much more important, that the reconstructed d(x) eigen-
+value (circles), matches the one predicted by the reference.
+Thus the process is able to reconstruct the dominant eigen-
+value.
+We are also observing that d(y) are not as well re-
+constructed.
+This is a consequence of the assumptions
+that the components of #–r are independent.
+Since Re-
+constrSim only impose strains along the x-axis, we have
+κy ≡ 0 ⇒ ry ≡ 0. Thus, the reconstructed y-eigenvalues
+we are seeing are caused by a directional sampling effect
+created during the reconstruction of the damage. How-
+ever, since d(y) is the non-dominant eigenvalue, we expect
+and allow that it is less well reconstructed.
+The squares in Fig. 15 indicate the state of the lat-
+tices after an uni-axial strain of �ε along the x-axis was
+applied to them. The difference between a square and its
+associated circle proves that this strain causes the failure
+of additional beams. If the reconstruction process would
+work perfectly, any strain below or equal �ε should not lead
+to the failure of any beam. Therefore, we might have re-
+moved the right number of beams and these were more or
+less correctly oriented, the selection of some of them was
+not fully optimal.
+Furthermore, we see that for subsequent loading steps,
+the damage and rfb continue to increase (Fig. 15). While
+the observed values for the restored systems remain above
+the reference, the restored lattices slowly converge towards
+them. This is because the damage created by the subse-
+quent loading steps starts to dominate the artificial one,
+14
+
+e
+10-3
+10-4
+(a)p
+10-5
+10-6
+10-7
+(b)
+10-3
+10-4
+工
+(6)p
+10-5
+a = 0.3,  fin = 0.002
+10-6
+ref. BothXY
+ref. XThenY
+ref. YThenX
+PD. k = 6
+10-7
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+T := K/2efin [-]= 0.003
+= 0.002
+= 0.001
+=0.0
+d(c)
+10-2
+d(y)
+10-3
+10-4
+k = 6, a = 0.3
+10-5
+= 0.001
+= 0.003
+10-6
+= 0.002
+restored: after loading
+restored: before loading
+0.000
+0.001
+0.002
+0.003
+0.004
+0.005
+i [-]that we created through the restoration process.
+4. Summary and Conclusion
+In this study, we presented a generic approach for the
+creation of a discrete twin of a continuum representation
+containing an initial damage. The discrete twin’s damage
+is created in such a way, that it is mechanically consistent
+to the original’s continuum damage. This is a step towards
+adaptive multi-scale simulations, which take the state of
+the coarse description of a region into account upon its
+refinement.
+While the method is general and has no restrictions
+concerning the used numerical representations, we pre-
+sented it in form of a concrete example. As continuum
+representation, we have used FEM with CDM as damage
+measure.
+For the discrete representation, we have used
+a lattice based on a triangular grid consisting of brittle
+beam-truss elements.
+One part of our method is the damage measure used
+inside the continuum representation. This measure is used
+during the initial continuum phase to track the evolving
+continuum damage. Unlike classical CDM, that are cali-
+brated to match the degradation of a particular material,
+we calibrated the CDM against the degeneration of the
+discrete numerical representation. Thus, it measures the
+degeneration that would occur on a hypothetical fine scale
+representation.
+We saw that the determined CDM is indeed able to
+capture the damage caused by uni-axial strains to a sat-
+isfying degree. However, for bi-axial loading, the CDM is
+unable to achieve the same. This is explained by the as-
+sumption that the directions are independent. Directional
+coupling must be used to further improve the CDM’s ac-
+curacy
+The second part of our method is the ability to con-
+struct a discrete damage that is mechanically consistent
+to a given continuum state #–κ. Since this problem is obvi-
+ously not unique, we devised a stochastic scheme to gen-
+erate representations containing such a particular discrete
+damage.
+We have seen, that the reconstruction process is indeed
+able to create discrete lattices, whose initial degeneration
+is consistent with the given continuum state #–κ. A draw-
+back is, that imposing a strain that corresponds to #–κ it-
+self, leads to the failing of some additional beams. This
+is indicating that the selection process needs to be further
+refined. Furthermore, as for the CDM, we observed prob-
+lems for bi-axial strains, which are again caused by the
+independence assumed between the directions.
+Nevertheless, our data is indicating that our approach
+works well for the case of uni-axial loading and is in prin-
+cipal able to work for bi-axial loading. The next step is to
+integrate our method into an adaptive multi-scale simula-
+tion scheme.
+5. CRediT
+Philip Müller: Conceptualisation, Methodology, Soft-
+ware, Validation, Writing - Original Draft, Visualisation;
+Falk Wittel: Conceptualisation, Writing - Review & Edit-
+ing, Supervision; David Kammer: Conceptualisation, Writ-
+ing - Review & Editing, Supervision.
+6. Declaration of Competing Interest
+The authors declare that they have no known compet-
+ing financial interests or personal relationships that could
+have appeared to influence the work reported in this paper.
+7. Data Availability
+The simulation data generated in this study have been
+deposited in the ETH Research collection database avail-
+able at TBA.
+15
+
+Appendix A. Parameters of �rx(κx) and �ry(κy)
+The fitting parameters (see Eq. (15)) of the ratio of
+failed beams (rfb) �r, denoted as α(r)
+a,ϕ and β(r)
+a,ϕ were esti-
+mated in the same way as the ones for the two damage
+functions �dx(κx) and �dy(κy). However, they behave much
+more stable and thus show less variations.
+Figs. A.16a
+Figure A.16:
+Dependence of the two fitting parameters of Eq. (15),
+lg α(r)
+a, ϕ (a) and β(r)
+a, ϕ (b), on the pull direction ϕ.
+Colours indicating
+different distortion levels a. The error bars is given by the 95% confi-
+dence interval.
+show the values for the α- and A.16b for the β-parameters.
+Compared with the parameters we obtained for the dam-
+age law (see Fig. 8), we see much less variability here. This
+is because it is far easier to measure this quantity than the
+damage.
+16
+
+4.80
+4.75
+4.70
+4.65
+4.60
+4.55
+4.50
+4.45
+3.100
+3.075
+3.050
+3.025
+3.000
+2.975
+a= 0.0
+2.950
+a = 0.1
+a= 0.2
+2.925
+a= 0.3
+a = 0.5
+2.900
+0
+20
+40
+60
+80
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+18
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+Entropy of different phases formed by soft rods
+Jayeeta Chattopadhyay,1 Shiang-Tai Lin,2 and Prabal K. Maiti1, a)
+1)Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012,
+India
+2)Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan
+(Dated: 12 January 2023)
+Computation of entropy in liquids and liquid crystal phases is a big challenge in statistical physics. In this work, we
+extend the two-phase thermodynamic model (2PT) to shape anisotropic soft repulsive spherocylinders (SRSs) and
+report the absolute values of entropy for different liquid crystal (LC) phases for a range of aspect ratios L/D = 2 − 5.
+We calculate the density of states (DoS) for different LC phases and decompose it into contributions arising from
+translational and rotational degrees of freedom.
+The translational and rotational modes are further partitioned into
+diffusive, gas-like, and non-diffusive, solid-like components using a fluidicity factor. In the dilute limit, the entropy
+values obtained from the 2PT method match exactly those of an ideal rigid rotor. We find that, for a given packing
+fraction, the magnitude of the total entropy is roughly equal regardless of the different LC phases associated with
+different aspect ratios. We also compute the excess entropy (for L/D = 5) and compare those with the values obtained
+using the standard integration approach of molecular dynamics (MD) or Monte Carlo (MC) equation of state (EOS) of
+SRS. The values obtained using both approaches match very well. The rotational and translational fluidicity factors are
+further used to determine the phase boundaries of different liquid crystal phases for the respective aspect ratios.
+I.
+INTRODUCTION
+The phase behavior of shape anisotropic particles is an
+emerging field of research that gives rise to various liquid
+crystal (LC) phases1–3. Examples span from living organisms
+like tobacco mosaic virus4–6, fd virus7 to synthetic systems
+of rod-like particles like boehmite8, silica9 etc. Different liq-
+uid crystal phases can be identified based on their microscopic
+arrangements, as well as positional and orientational order.
+Onsager, in his seminal work10, showed that a system of
+thin and hard rods could undergo a phase transition from
+disordered isotropic to orientationally ordered nematic phase
+above a critical aspect ratio (L/D > 3.7) that is mainly driven
+by entropy. The loss of orientational entropy in the nematic
+phase is compensated by the increase of translational entropy
+due to the ordered structure.
+Similarly, for the other LC
+phases, entropy plays an important role in studying the stabil-
+ity of the phases. Entropy of a fluid can be expressed as a mul-
+tiparticle correlation expansion of statistical entropy devel-
+oped by Green and Nettleton11,12 and generalized by Lazaridis
+and co-workers13,14 for the non-spherical bodies. Costa et al.
+first used this method to calculate the entropy of a system of
+hard spherocylinders (HSCs)15,16 and later, by Cuetos et al.17
+in a system of soft repulsive spherocylinders (SRSs). It is also
+worth mentioning several interesting works by Dhar et al.18,19
+where they have calculated entropy of hard rods and rigid rect-
+angles in 3D and 2D using analytically solvable lattice model
+and MC simulations.
+In 2003, Lin et al.20 developed the two-phase thermody-
+namic (2PT) model to calculate the entropy, free energy, and
+other thermodynamic properties of liquids from a short MD
+trajectory (20 picoseconds (ps)).
+2PT model has emerged
+as an efficient and accurate method in calculating various
+a)Electronic mail: [email protected]
+thermodynamic properties of Lennard-Jones fluids for the di-
+verse setting of state points both in 2D21 and 3D20, water
+in bulk22 and under different confinement, carbon dioxide23
+and other organic and inorganic molecules24.
+The results
+match very well with those of the experimental studies. In
+the 2PT method, the density of state (DoS) of a liquid, which
+is calculated from the Fourier transform of the velocity auto-
+correlation function (VACF), is decomposed into vibrational
+(solid) and diffusive (gas) components. The thermodynamic
+quantities, including entropy, are then calculated using har-
+monic oscillator approximation to the solid component and
+hard sphere approximation to the gas component. For the ro-
+tational mode, the diffusive part is calculated from the rigid
+rotor approximation20,22. In 2PT method, the entropy of a
+definite state point is calculated from a single MD trajectory.
+Thus, it is far more efficient than the conventional integration
+approach of MD or MC equation of state of the SRS, which
+entails several discrete MD/MC trajectories along the integra-
+tion path. This is advantageous for the systems for which the
+analytical form of the equation of state is unknown (such as
+SRS).
+In this work, we extend the 2PT method to calculate en-
+tropy of various liquid crystal phases formed by a system
+of soft repulsive spherocylinders of different aspect ratios
+(length/diameter) L/D = 2,3,3.5,4 and 5. We validate our
+method by comparing the entropy values obtained using the
+standard integration approach of equation of state of the SRS
+of L/D = 5 at T ∗ = 516,17.
+We find that the entropy val-
+ues do not have any strong dependence on the aspect ratio
+but strongly depend on the packing fraction (η)of the system.
+We also find that LC phase transitions are governed by the
+change of pair entropy. The loss of orientational pair entropy
+in the nematic phase is compensated by the increase of trans-
+lational pair entropy. Similarly, in case of the smectic phase,
+the loss of translational pair entropy is compensated by the
+residual entropy arising from the multi-particle contribution.
+In addition, we present an alternative way to identify the phase
+boundaries of different liquid crystal phases from the fluidic-
+arXiv:2301.04621v1  [cond-mat.soft]  11 Jan 2023
+
+2
+ity factor that is directly related to the diffusivity of the sys-
+tem: the packing fraction at which the translational fluidicity
+ftrans saturates but rotational fluidicity frot decreases sharply
+indicates the phase boundary of the isotropic to nematic (I-
+N) phase transition. Similarly, the nematic to smectic (N-Sm)
+transition is located where frot saturates but ftrans keeps de-
+creasing.
+The rest of the paper is organized as follows: In section
+II, we briefly describe the theoretical background of the 2PT
+method and summarize the multiparticle correlation expan-
+sions method and the integration approach of equation of state
+to calculate the entropy of SRS; in section III, we describe the
+SRS model and the simulation protocol. We present the results
+and analysis in section IV. Finally, in section V, we conclude
+with the discussion on the major benefits of the 2PT method
+and possible applications.
+II.
+MODEL AND COMPUTATIONAL DETAILS
+We model the system as a collection of spherocylinders
+(cylinder with hemispherical caps) of aspect ratios L/D =
+2,3,3.5,4,5. The interacting potential is only due to the ex-
+cluded volume interaction described by the Weeks-Chandler-
+Andersen (WCA) potential given as follows25:
+USRS = 4ε[( D
+dm
+)12 −( D
+dm
+)6]+ε
+if
+dm < 2
+1
+6 D
+= 0
+if
+dm ≥ 2
+1
+6 D
+(1)
+Here, dm is the shortest distance between two SRS that
+determines their relative orientation2,17,26,27,34.
+For conve-
+nience, thermodynamic quantities are expressed in terms of
+interaction strength ε, diameter of the SRS D and mass m:
+temperature T ∗ = kBT
+ε , pressure P∗ = Pvhsc
+kBT , packing fraction
+η = vhscρ, where ρ is the number density of the system
+defined as, ρ = N
+V and vhsc = πD2( D
+6 + L
+4) is the volume of the
+spherocylinder; energy E∗ = E
+ε , entropy S∗ = S
+kB , Helmholtz
+free energy A∗ = A
+ε , Gibbs free energy G∗ = G
+ε , diffusivity
+d∗ = d( m
+ε )1/2/D and the time t∗ = t
+�
+ε/m/D. To compute
+entropy using 2PT method, we convert all the thermody-
+namic quantities in real units using the parameters of argon
+(ε = 0.238 kcal/mol, σ = 3.405Å and mass m = 39.948
+g/mol) and then again convert them into the reduced units.
+We build the system in a hexagonal-closed-packed (HCP)
+crystal structure. As the particles are inherently anisotropic
+in shape, we choose the number of particles in the x, y and
+z directions such that the simulation box can be built in a
+near-cubic geometry. If nx, ny, nz are the number of particles
+in the x, y, z direction respectively and nu is the number of
+particles in one unit cell, then the total number of particles
+in one simulation box N = nu × nx × ny × nz. In our case,
+number of SRSs is chosen to be N = 1024.
+The periodic
+boundary condition in all three directions are used.
+We have carried out a series of MD simulations for a wide
+range of state points spanning the melting transition from solid
+(crystal) to gas (isotropic) for all the aspect ratios. We melt
+the initial crystal structure slowly by reducing the pressure in
+NPT ensemble (Constant particle number, pressure and tem-
+perature) at T ∗ = 5 for each L/D. The positions and velocities
+of the SRSs are updated using Verlet algorithm28 and the rota-
+tional motion by quaternion-based rigid-body dynamics29–33.
+The temperature and pressure of the system are controlled
+using Berendsen thermostat and barostat35 with a tempera-
+ture relaxation time τT = 0.05 and pressure relaxation time
+τP = 2 respectively. We perform 1×105 to 2×105 MD steps
+(with an integration time step δt = 0.001 in reduced unit) to
+reach equilibrium condition and another 2−5×103 steps (5-
+30 ps in real unit using the above-mentioned parameters) with
+δt = 5×10−4(1 fs)in real unit for the 2PT method.
+III.
+THEORY
+A.
+Two phase thermodynamic method
+1.
+Density of State Function
+The density of state (DoS) function G(ν) is defined as the
+mass weighted sum of the atomic spectral densities.
+This
+can be obtained from Fourier transform of velocity auto-
+correlation function (VACF) obtained from MD trajectory20.
+G(ν) =
+1
+kBT
+Natom
+∑
+l=1
+3
+∑
+k=1
+lim
+τ→−∞
+ml
+τ
+����
+� τ
+−τ vk
+l (t)e−i2πνtdt
+����
+2
+(2)
+Here, Natom is the total number of atoms in the system. ml is
+mass of the lth atom and vk
+l is the velocity of the lth atom in kth
+direction ( k indicates spatial coordinates x,y,z respectively).
+G(ν) represents distribution of normal modes in the system i.e
+G(ν) dν represents number of normal modes in the frequency
+range ν to ν + dν. So, total number of modes in the system
+i.e degrees of freedom of the system 3N
+� ∞
+0 G(ν)dν = 3N
+(3)
+The diffusion constant (D) of the system is directly related
+to the zero-frequency density of state of the system G(0):
+D = kBT
+12mN G(0)
+(4)
+For a rigid SRS, there is no vibrational motion. So, total
+number of degrees of freedom for a rigid SRS is 5 compris-
+ing 3 translational and 2 rotational motion. Therefore, total
+number of modes in the system is:
+� ∞
+0 G(ν)dν = 5N
+(5)
+Density of state G(ν) is decomposed into translational and
+rotational part:
+G(ν) = Gtrans(ν)+Grot(ν)
+(6)
+
+3
+where, Gtrans(ν) is obtained from the translational component
+of the center of mass velocity of the SRS:
+Gtrans(ν) =
+1
+kBT
+N
+∑
+j=1
+3
+∑
+k=1
+lim
+τ→−∞
+m j
+τ
+����
+� τ
+−τ vktrans
+j
+(t)e−i2πνtdt
+����
+2
+(7)
+here, N is the total number of SRS in the system and m j is
+the mass of the SRS. vktrans
+j
+is the translational velocity of jth
+SRS in kth direction.
+Grot(ν) =
+1
+kBT
+N
+∑
+j=1
+2
+∑
+k=1
+lim
+τ→−∞
+Ik
+j
+τ
+����
+� τ
+−τ ωk
+j (t)e−i2πνtdt
+����
+2
+(8)
+here, Ik
+j is the moment of inertia of jth SRS along kth the
+principal axis. As SRS is linear, the moment of inertia along
+its director is 0. Therefore, k runs from 1 to 2. ωk
+j represents
+the angular velocity.
+2.
+Thermodynamic properties from 2PT method
+Various thermodynamic quantities like energy, entropy of
+a system can be expressed as a summation over the contribu-
+tions from translational and rotational motion of SRS)22,23:
+E = E0 +Etrans +Erot,
+(9)
+S = Strans +Srot.
+(10)
+Here, E0 is the reference energy. In 2PT method, the density
+of states corresponding to translational or rotational motion is
+partitioned as:
+Gk(ν) = Gs
+k(ν)+Gg
+k(ν)
+(11)
+where, the subscript k stands for translational, or rotational
+motion. The 1st term in Eq. 11 refers to the solid-like and the
+2nd term in Eq. 11 refers to the gas-like contributions. For a
+solid-like system, the DoS can be exactly determined by that
+of harmonic oscillator. But for a liquid, harmonic approxima-
+tion is no longer valid at the low frequency regime due to the
+strong effect of anharmonicity. Also, the diffusive model at
+the zero frequency can lead to singularity. In the 2PT model,
+the anharmonicity effect at the low frequency is treated by de-
+composing the DoS into gas-like and solid-like components
+as mentioned in Eq. 11. The gas-like component is evaluated
+from the DoS at the zero frequency and the fluidicity factor fk
+using the following equation :
+Gg
+k(ν) =
+Gk(0)
+1+
+�
+πνGk(0)
+6fkN
+�2 .
+(12)
+The fluidicity factor fk is calculated using the equation below:
+2∆−9/2
+k
+f 15/2
+k
+−6∆−3
+k
+f 5
+k −∆−3/2
+k
+f 7/2
+k
++6∆−3/2
+k
+f 5/2
+k
++2fk−2 = 0,
+(13)
+where, ∆k is the diffusivity constant in reduced unit that is
+defined as:
+∆k(T,V,N,k,Gk(0)) = 2Gk(0)
+9N
+�πkBT
+k
+�1/2 �N
+V
+�1/3 � 6
+π
+�2/3
+.
+(14)
+The above equation Eq.
+14 indicates ∆k only depends
+on the thermodynamic state points (T,V,N) and Gk(0) that
+can uniquely determines the fluidicity factor fk for different
+modes. Once we calculate Gg
+k(ν) from Eq. 12, the solid-like
+component can be determined by subtracting it from the total
+DoS Gk(ν) (Eq. 11) obtained from velocity auto-correlation.
+Once we calculate Gg
+k(ν) and Gs
+k(ν), each component
+(translational, rotational) of the thermodynamic quantities
+(energy from Eq. 9 and entropy from Eq. 10) can be de-
+termined by integrating the DoS using appropriate weighting
+functions for the respective thermodynamic quantities:
+Ek = β −1
+�� ∞
+0 dνGs
+k(ν)W s
+E,k(ν)+
+� ∞
+0 dνGg
+k(ν)W g
+E,k(ν)
+�
+,
+(15)
+Sk = kB
+�� ∞
+0 dνGs
+k(ν)W s
+S,k(ν)+
+� ∞
+0 dνGg
+k(ν)W g
+S,k(ν)
+�
+,
+(16)
+Ak = β −1
+�� ∞
+0 dνGs
+k(ν)W s
+A,k(ν)+
+� ∞
+0 dνGg
+k(ν)W g
+A,k(ν)
+�
+,
+(17)
+where, β = (kBT)−1 and W g/s
+l,k
+is the weighting function for
+thermodynamic quantity l (E/S/A) for each mode k (transla-
+tion/rotation) partitioned into gas-like (g) or solid-like (s) con-
+tribution. Here,
+W s
+E = βhν
+2
++
+βhν
+exp(βhν)−1,
+(18)
+W s
+S =
+βhν
+exp(βhν)−1 −ln[1−exp(−βhν)],
+(19)
+W g
+E,trans(ν) = W g
+E,rot(ν) = 0.5,
+(20)
+W g
+S,trans(ν) = 1
+3
+SHS
+kB
+,
+(21)
+W g
+S,rot(ν) = 1
+3
+SR
+kB
+(22)
+where, SHS is the hard-sphere entropy and SR is the rotational
+entropy of ideal gas modelled as rigid rotor:
+SHS
+kB
+= 5
+2 +ln
+��2πmkBT
+h2
+�3/2 V
+ftrN z(y)
+�
++ y(3y−4)
+(1−y)2 , (23)
+SR
+kB
+= 1+ln
+� T
+σΘr
+�
+,
+(24)
+
+4
+here, y = f 5/2
+trans/∆3/2
+trans and z(y) is the compressibility factor of
+hard sphere from the Carnahan-Starling equation of state36.
+Θr is the rotational temperature defined as Θr =
+h2
+8π2IrkB and
+σ is the rotational symmetry. The reference energy now be-
+comes,
+E0 = EMD −β −13N(1−0.5 ftrans −0.5 frot),
+(25)
+where, EMD is the total energy calculated from the MD simu-
+lation.
+B.
+Entropy using multiparticle correlation expansion method
+and integration approach on the SRS equation of state
+The configurational entropy Scon is defined as:13–15,17.
+Scon
+tot = Sid +
+∞
+∑
+n=2
+Sn,
+(26)
+where, Sid denotes the entropy of an ideal gas and Sn denotes
+the entropy due to n-particle spatial correlation. Therefore,
+the excess entropy can be calculated from well-known multi-
+particle correlation expansion of the configurational entropy
+(ME) Sex can be written as:
+Sex =
+∞
+∑
+n=2
+Sn = Scon
+tot −Sid,
+(27)
+If S2 represents the entropy due to pair interaction, then the
+residual entropy ∆s that includes the spatial correlation for n ≥
+3 becomes:
+∆s = Sex −S2.
+(28)
+Pair entropy S2 can be expressed as:
+S2 = Strans
+2
++Srot
+2 ,
+(29)
+Strans
+2
+= −2πρ
+�
+[g(r)lng(r)−g(r)+1]r2dr,
+(30)
+Srot
+2 = 4πρ
+�
+g(r)qrot(r)r2dr,
+(31)
+qrot(r) = −1
+4
+� π
+0 g(θ|r)sinθdθ.
+(32)
+In a system of linear molecules, the probability distribution
+function g(r,θ) can be factorized as15, g(r,θ) = g(r)g(θ|r),
+where, g(r) denotes the radial distribution and g(θ|r) denotes
+the conditional probability distribution function between two
+rods at a r distance with a relative angle between θ to θ +dθ.
+The excess entropy can be exactly calculated using the
+equation of state (EOS) of the SRS defined below17:
+SEOS
+ex
+(ρ) = Uex
+T −
+� ρ
+0
+�
+P
+kBTρ′ −1
+� dρ′
+ρ′ ,
+(33)
+where, Uex represents the excess energy, which is the potential
+energy per particle in the units of kB.
+IV.
+RESULTS AND DISCUSSION
+We present equilibrium phase diagram of SRS of aspect
+ratios L/D = 2 − 5 at the temperature T ∗ = 5 (Fig.
+1
+and Fig.7(a)).
+The magnitude of the pressures and densi-
+ties corresponding to different phases for different aspect ra-
+tios are listed in table IV. We obtain 4 stable phases for
+L/D ≥ 3.5 :17,37–40 crystal (K), smectic (Sm), nematic (N) and
+isotropic (I); 3 stable phases for L/D = 3: crystal, smectic,
+and isotropic and two stable phases for L/D = 2: crystal and
+isotropic. For further details of these phases and their charac-
+terization, we refer the reader to our earlier work39,40. Here
+we are interested in entropy computations of these phases.
+I
+I
+I
+N
+Sm
+N
+N
+Sm
+Sm
+K
+K
+K
+(a)
+(c)
+(b)
+FIG. 1. (a) Equation of state (b) nematic order parameter S (c) po-
+tential energy per particle U∗/N are plotted with packing fraction
+η for the system of soft repulsive spherocylinders of aspect ratio
+L/D = 5. Thermodynamic quantities are defined in the reduced unit:
+pressure P∗ = Pvhsc/kBT and packing fraction, η = ρvhsc where vhsc
+is the volume of the spherocylinder. We observe four stable phases:
+isotropic (I), nematic (N), smectic (Sm) and crystal (K). The vertical
+gray lines indicate boundaries between two phases.
+A.
+Validation of 2PT method
+In the dilute limit, the entropy and Helmholtz free energy of
+SRS, calculated using 2PT method, can be compared with the
+values obtained for an ideal diatomic gas modeled as a rigid
+rotor. The analytical expressions for the partition function Z,
+
+15
+10
+N/n
+5
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+n20
+15
+10
+5
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+n0.8
+0.6
+S
+0.4
+0.2
+0
+0.1
+0.2
+0.3
+0.40.5
+0.6
+0.7
+0.8
+0.95
+entropy S, and Helmholtz free energy A of an ideal rigid rotor
+are as follows:
+Z(V,T) =
+�2πmkBT
+h2
+�3/2
+V 8π2IkBT
+σh2
+,
+(34)
+S
+NkB
+= ln
+�2π(m1 +m2)kBT
+h2
+�3/2 Ve5/2
+N
++ln8π2IkBTe
+σh2
+.
+(35)
+A
+NkBT = −
+�
+ln
+�2π(m1 +m2)kBT
+h2
+�3/2 V
+N +ln8π2IkBT
+σh2
++1
+�
+.
+(36)
+The 1st term in Eq. 35 is due to the translational motion,
+and the 2nd term is due to the rotational motion (for an ideal
+rigid rotor, there is no vibrational motion). In Table I and
+II, we compare the entropy of the SRS system in a dilute limit
+calculated from the 2pt method with that of an ideal rigid rotor
+at the same state point calculated using the above equations for
+different aspect ratios which are found to be in a very good
+agreement.
+TABLE I. Comparison of the total Stot, translational Strans and ro-
+tational Srot entropy of SRS of different aspect ratios from the 2PT
+method at the temperature T ∗ = 5 and number density ρ∗ = 0.01
+with that of a rigid rotor at the same state points calculated using Eq.
+35. Here, entropy is calculated in kB/particle unit.
+L/D
+ρ∗
+Sid
+trans
+Sidrot
+Sid
+tot
+S2PT
+trans S2PT
+rot
+S2PT
+tot
+5
+0.01 18.36 12.18 30.54 18.26 12.30 30.56
+3
+0.01 18.36 11.15 29.51 18.36 11.25 29.61
+2
+0.01 18.36 10.34 28.70 18.36 10.44 28.80
+TABLE II. Comparison of the Helmholtz free energy of SRS of dif-
+ferent aspect ratios from the 2PT method with that of the ideal rigid
+rotor using Eq.36 at the dilute limit, temperature T ∗ = 5 and number
+density ρ∗ = 0.01. A∗tot designates the total Helmholtz free energy
+and A∗trans, A∗rot designate the translational and rotational components
+respectively.
+L/D
+ρ∗
+Aid
+trans
+Aidrot
+Aid
+tot
+A2PT
+trans
+A2PT
+rot
+A2PT
+tot
+5
+0.01 -84.24 -55.81 -139.95 -84.91 -55.95 -140.86
+3
+0.01 -84.24 -50.71 -134.98 -85.63 -50.65 -136.28
+2
+0.01 -84.24 -46.66 -130.88 -83.74 -48.33 -132.07
+B.
+Density of states of liquid crystal phases
+We calculate the density of state G(ν) of different liquid
+crystal phases using 2PT method as shown in Fig. 2. For
+each phase, we show the total DoS and its decomposition
+into translational, rotational modes.
+The translational and
+rotational modes are further decomposed into gas-like and
+solid-like components, as mentioned in the 2PT method
+section.
+In Fig.2(a), we plot DoS for the state point P∗ = 1.78,η =
+0.29 which corresponds to the isotropic phase as shown in
+the equilibrium phase diagram [Fig.1]. We find that both the
+translational Gtrans and rotational Grot DoS are dominated by
+the gas like contribution and decay exponentially. At the zero
+frequency ν = 0, both of Gtrans and Grot have large finite val-
+ues, indicating that the system possesses high translational
+and rotational diffusivity. Similarly, in Fig.2(b), we plot DoS
+of nematic phase for the state point P∗ = 6.23,η = 0.50. We
+see that Gtrans decays exponentially and have a fine value at
+ν = 0 indicating gas-like behaviour. However, Grot is domi-
+nated by solid-like behaviour with a low rotational diffusivity.
+In the case of the smectic phase ( P∗ = 8,η = 0.6), both of
+the Gtrans and Grot are dominated by solid-like contribution.
+However, Gtrans has a very low value at zero-frequency indi-
+cating a low-diffusivity which is due to the in-layer fluid-like
+motion. In crystal phase (state point P∗ = 15.13,η = 0.78),
+G(ν) is roughly zero at ν = 0 indicating absence of diffusive
+mode in the system. Both translational and rotational DoS
+exhibit solid-like behaviour.
+C.
+Fluidicity factor of liquid crystal phases
+The decomposition of the translational and rotational DoS
+into gas-like and solid-like components is carried out by cal-
+culating the fluidicity factor f as discussed in Section III-A.
+We find that both of translational and rotational fluidicity fac-
+tors are very high in the isotropic phase, very low in the crystal
+phase and intermediate in the LC phases as mentioned in the
+Table III and in Fig. 3. We also calculate the phase bound-
+aries of different LC phases from the change of ftrans and frot.
+In Fig.3, we find that, both of the ftrans and frot decrease
+with packing fraction η in the isotropic phase. In the ne-
+matic phase, ftrans remains almost constant at its value in the
+isotropic phase, while frot keeps decreasing. This is also con-
+sistent with the DoS calculation showing that rotational dif-
+fusivity is much lower in the nematic phase than translational
+diffusivity. The I-N phase boundary is therefore defined as
+the packing fraction where frot keeps decreasing but ftrans be-
+comes constant (η∗
+I−N ≈ 0.41−0.44 for L/D = 5). Similarly,
+in the Smectic phase, frot remains nearly constant at its value
+in the nematic phase while ftrans drops sharply. Hence, the
+N-Sm phase boundary can be located at the packing fraction
+where ftrans continues to decrease but frot remains almost con-
+stant (η∗
+N−Sm ≈ 0.54 − 0.57 for L/D = 5). Both of ftrans and
+frot acquire a very low value in the crystal phase. These anal-
+yses suggest another method of quantifying the phase bound-
+
+6
+aries using the fluidicity factor.
+TABLE III. Translational and rotational fluidicity factors for different
+liquid crystal phases for the aspect ratio L/D = 5.
+P∗
+η
+ftrans
+frot Phase
+1.78 0.29 0.62 0.53
+I
+6.23 0.50 0.41 0.18
+N
+8.01 0.60 0.26 0.07
+Sm
+15.13 0.78 0.09 0.06
+K
+D.
+Entropy calculation from 2PT method
+In Table IV and in Fig.4, Fig.7, we mention the total en-
+tropy Stot and its decomposition into the translational Strans
+and rotational Srot modes for different liquid crystal phases
+associated to different aspect ratios. We find that entropy de-
+creases as a function of packing fraction for the given aspect
+ratios. We also find that, at a certain packing fraction, to-
+tal entropy is close by for the given aspect ratios, irrespec-
+tive of the different liquid crystal phases they exhibit. As for
+example, at η = 0.60, the magnitude of the total entropy is
+Stot = 18.54 − 19.03 kB/particle; however, it shows smectic
+structure for L/D ≥ 3 and isotropic structure for L/D = 2.
+Similarly, at η = 0.54, Stot = 19.40−19.92 kB/particle while
+it shows nematic structure for L/D ≥ 3.5 and isotropic struc-
+ture for L/D = 3,2. These results indicate that, total entropy
+depends on the thermodynamic state points only, not on the
+different liquid crystal phases corresponding to different L/D
+s. However, the entropy of different L/D s differs at the higher
+packing fractions, as mentioned in Fig. 7(b).
+In Fig.6, we calculate the pair entropy S2 of different LC
+phases using Eq. 29 and its decomposition into translational
+Str
+2 and rotational Srot
+2
+parts. We observe that, Srot
+2
+decreases
+sharply at the I-N phase boundary, while Str
+2 decreases slowly.
+For N-Sm transition, Str
+2 decreases more rapidly than that of
+I-N phase boundary. Our results are consistent with those of
+Cuetos et al.17. These analyses indicate that the change of
+entropy at the LC phase transition points are mainly driven by
+the translational or rotational pair entropy. The sharp decrease
+of rotational pair entropy at the I-N phase boundary is com-
+pensated by residual entropy ∆s arising from the multi particle
+correlation (Eq. 28). Similarly, the N-Sm phase transition is
+driven by the sharp decrease of translational pair entropy that
+is also compensated by residual entropy.
+E.
+Comparison of excess entropy from 2PT method and
+integrating on the SRS equation of state
+We calculate the excess entropy, Sex which is defined as the
+amount of entropy arises due to the particles’ interaction us-
+ing Eq.27. It is calculated from the difference between the
+absolute entropy calculated from the 2PT method or integrat-
+ing over MD/MC equation of state and the entropy of an ideal
+rigid rotor at the same state point. We mention the magni-
+tude of Sex for different liquid crystal phases in Table V for
+L/D = 5 at T ∗ = 5. In Fig. 8, we compare the excess entropy
+of SRS at different packing fractions from the 2PT method
+with those of the standard integration approach on the (a) MD
+equation of state of SRS from our simulation and (b) MC
+equation of state of SRS employed by Cuetos et al.17 We ob-
+serve that the magnitude of Sex are in good agreement at the
+lower densities for the given methods. At the higher densi-
+ties, Sex calculated from 2PT method matches well with the
+MD equation of state, but it differs from the MC equation of
+state17.
+V.
+CONCLUSION AND OUTLOOK
+We describe a technique based on the two-phase thermody-
+namic model (2PT) for computing the entropy of liquid crystal
+phases of SRS with a range of aspect ratios L/D = 2−5. For
+various liquid crystal phases, we compute the density of state
+(DoS) functions and its decomposition into translational and
+rotational motions. In the dilute limit, the entropy calculated
+using the 2PT method matches exactly with that of an ideal
+rigid rotor. We find that, at a definite packing fraction, the
+magnitude of the total entropy is roughly equal regardless of
+the different LC phases associated to different aspect ratios.
+We compare the excess entropy with that of the conventional
+integration approach on equation of state of SRS, that matches
+well. The phase boundaries of different liquid crystal phases
+are also calculated using the rotational and translational flu-
+idicity factors. Our future study will involve to utilise this
+method in calculating absolute value of entropy and other ther-
+modynamic quantities of various liquid crystal molecules and
+compare it with experiments.
+ACKNOWLEDGMENTS
+We thank SERB, India for financial support through provid-
+ing computational facility. JC acknowledges support through
+an INSPIRE fellowship. JC thanks S. Siva Nasarayya Chari
+for insightful discussions.
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+7
+0
+100
+200
+300
+400
+ν
+0
+20
+40
+60
+G( ν)
+η = 0.78 (Crystal)
+0
+100
+200
+300
+400
+ν
+0
+20
+40
+60
+80
+100
+G( ν)
+η = 0.60 (SmecticA)
+0
+100
+200
+300
+400
+ν
+0
+25
+50
+75
+100
+125
+G( ν)
+η = 0.50 (Nematic)
+0
+25
+50
+75
+100
+125
+150
+ν
+0
+100
+200
+300
+400
+G( ν)
+η = 0.29 (Isotropic)
+Gtrans
+g
+Gtrans
+s
+Grot
+g
+Grot
+s
+Gtrans
+Grot
+Gor
+Total
+(a)
+(b)
+(c)
+(d)
+FIG. 2. Density of state (DoS) G(ν) for (a) isotropic (b) nematic (c) smectic and (d) crystal phase. The components of the entropy are
+mentioned in the legend. The snapshots of the configurations are shown for the respective phases. Here, we see that DoS of nematic phase
+[Fig. (b)] comprises both solid and gas like components, whereas for smectic phase [Fig.(c)], it is dominated by solid like components only.
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+
+8
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+η
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+fluidicity (f)
+ftrans
+frot
+I
+N
+Sm
+K
+FIG. 3. Phase diagram of the SRS with aspect ratio L/D = 5 at
+T ∗ = 5 in fluidicity, packing fraction (f − η) space. ftrans and frot
+represent the translational and rotational components respectively.
+The black dotted lines denote phase boundaries of different phases.
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+η
+0
+5
+10
+15
+20
+25
+30
+S(kB/particle)
+Strans
+Srot
+Stot
+I
+N
+Sm
+K
+FIG. 4. Total entropy and its translational and rotational components
+of different liquid crystal phases for the aspect ratio L/D = 5 at T ∗ =
+5. The black dotted lines denote the phase boundaries.
+37A. Cuetos and B. Martínez-Haya, Molecular Physics 113, 1137 (2015).
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+40J. Chattopadhyay, S. Ramaswamy, C. Dasgupta, and P. K. Maiti, arXiv
+preprint arXiv:2205.00667 (2022).
+
+9
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+η
+-200
+-150
+-100
+-50
+0
+A* 
+A*trans
+A*rot
+A*tot
+I
+N
+Sm
+K
+FIG. 5. Helmholtz free energy A∗tot and its translational A∗trans and
+rotational A∗rot components of different liquid crystal phases for the
+aspect ratio L/D = 5 at T ∗ = 5. The black dotted lines denote the
+phase boundaries.
+-6
+-5
+-4
+-3
+-2
+-1
+0
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+η
+S2tr (kB/particle)
+I
+N
+Sm
+FIG. 6. Translational pair entropy per particle of different liquid crys-
+tal phases for the aspect ratio L/D = 5 at T ∗ = 5. The black dotted
+lines denote the phase boundaries.
+
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+--
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+--
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-10
+2
+4
+6
+8
+10
+12
+14
+16
+18
+0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
+P*
+�
+L/D = 2.0
+L/D = 3.0
+L/D = 3.5
+L/D = 4.0
+L/D = 5.0
+15
+16
+17
+18
+19
+20
+21
+22
+23
+0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
+Stot (kB/particle)
+�
+L/D = 2.0
+L/D = 3.0
+L/D = 3.5
+L/D = 4.0
+L/D = 5.0
+9.5
+10
+10.5
+11
+11.5
+12
+12.5
+13
+0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
+Strans (kB/particle)
+�
+5
+5.5
+6
+6.5
+7
+7.5
+8
+8.5
+9
+9.5
+10
+0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
+Srot (kB/particle)
+�
+(a)
+(c)
+(b)
+(d)
+FIG. 7. (a) Equation of state, (b) total entropy Stot and its decomposition into (c) translational Strans and (d) rotational Srot motion as a function
+of packing fraction η for different L/Ds at the temperature T ∗ = 5. Here we see that, at a certain packing fraction, total entropy is roughly
+same irrespective of the different LC phases corresponding to different L/Ds.
+
+11
+0
+0.2
+0.4
+0.6
+0.8
+η
+-10
+-8
+-6
+-4
+-2
+0
+Sex(kB/particle)
+MD_EOS
+MC EOS [Ref. 17]
+2PT
+I
+N
+Sm
+FIG. 8. Excess entropy Sex = S2PT/EOS
+tot
+− Sid
+tot vs packing fraction η for L/D = 5 calculated using 2PT method and equation of state (EOS)
+of SRS. Here, we compare the excess entropy calculated using MD equation of state and 2PT method from our simulation with those of the
+Monte Carlo (MC) EOS from Cuetos et al.17. The black dotted lines denote the phase boundaries.
+
+12
+TABLE IV. Total entropy S2PT
+tot
+and its decomposition into translational S2PT
+trans and rotational S2PT
+rot
+degrees of freedom for different liquid
+crystal phases associated to different aspect ratios L/D s at T ∗ = 5. Here, P∗,ρ∗ and η indicate pressure, number density and packing fraction
+respectively.
+P∗
+ρ∗
+η
+S2PT
+trans
+S2PT
+rot
+S2PT
+tot
+Phase
+L/D = 5
+4.45
+0.093
+0.413
+12.574
+9.230
+21.804
+I
+4.90
+0.098
+0.434
+12.509
+8.985
+21.494
+N
+5.34
+0.103
+0.457
+12.483
+8.667
+21.150
+N
+5.79
+0.107
+0.477
+12.435
+8.391
+20.827
+N
+6.23
+0.111
+0.496
+12.320
+8.354
+20.674
+N
+6.68
+0.116
+0.517
+12.228
+7.928
+20.157
+N
+7.12
+0.121
+0.539
+12.170
+7.751
+19.921
+N
+7.57
+0.130
+0.580
+11.924
+7.344
+19.268
+SmA
+8.01
+0.135
+0.599
+11.780
+7.231
+19.011
+SmA
+8.46
+0.137
+0.611
+11.730
+7.039
+18.768
+SmA
+9.35
+0.144
+0.639
+11.355
+6.841
+18.197
+SmA
+9.79
+0.147
+0.652
+11.298
+6.761
+18.059
+SmA
+10.68
+0.151
+0.674
+11.221
+6.951
+18.172
+SmA
+12.46
+0.160
+0.710
+10.909
+6.553
+17.462
+SmA
+14.24
+0.171
+0.762
+10.161
+5.950
+16.111
+SmA
+16.02
+0.177
+0.788
+9.879
+5.944
+15.823
+K
+17.80
+0.184
+0.818
+9.656
+5.563
+15.219
+K
+L/D = 4
+5.13
+0.121
+0.444
+12.287
+9.081
+21.368
+I
+5.86
+0.127
+0.466
+11.999
+8.783
+20.782
+I
+6.60
+0.134
+0.490
+11.845
+8.493
+20.338
+I
+6.96
+0.137
+0.502
+11.812
+8.438
+20.251
+N
+7.33
+0.141
+0.516
+11.706
+8.290
+19.996
+N
+7.70
+0.146
+0.537
+11.740
+8.008
+19.748
+N
+8.06
+0.155
+0.568
+11.720
+7.787
+19.507
+SmA
+8.43
+0.165
+0.606
+11.630
+7.401
+19.031
+SmA
+8.80
+0.169
+0.620
+11.446
+7.346
+18.792
+SmA
+11.00
+0.187
+0.685
+11.019
+6.625
+17.644
+K
+
+13
+P∗
+ρ∗
+η
+S2PT
+trans
+S2PT
+rot
+S2PT
+tot
+Phase
+L/D = 3.5
+7.20
+0.155
+0.508
+11.708
+8.369
+20.077
+I
+7.85
+0.160
+0.523
+11.608
+8.203
+19.810
+I
+8.18
+0.162
+0.531
+11.549
+8.135
+19.684
+N
+8.51
+0.166
+0.543
+11.385
+8.041
+19.427
+N
+8.84
+0.171
+0.559
+11.412
+7.873
+19.285
+N
+9.16
+0.191
+0.624
+11.410
+7.134
+18.544
+SmA
+9.49
+0.196
+0.643
+11.233
+6.931
+18.164
+SmA
+9.82
+0.198
+0.647
+11.306
+6.870
+18.176
+SmA
+10.14
+0.203
+0.663
+11.130
+6.714
+17.844
+K
+10.47
+0.205
+0.672
+11.082
+6.607
+17.688
+K
+13.09
+0.224
+0.732
+10.282
+6.142
+16.423
+K
+L/D = 3
+2.30
+0.122
+0.352
+13.438
+9.802
+23.239
+I
+6.91
+0.176
+0.506
+11.708
+8.453
+20.161
+I
+8.06
+0.185
+0.534
+11.387
+8.116
+19.504
+I
+8.35
+0.187
+0.539
+11.345
+8.058
+19.403
+I
+9.50
+0.195
+0.562
+11.142
+7.878
+19.020
+I
+9.79
+0.198
+0.570
+11.026
+7.736
+18.762
+SmA
+10.08
+0.211
+0.608
+11.161
+7.460
+18.621
+SmA
+10.37
+0.227
+0.653
+11.052
+6.920
+17.972
+SmA
+10.66
+0.233
+0.670
+10.963
+6.614
+17.577
+K
+12.67
+0.249
+0.717
+10.507
+6.486
+16.992
+K
+L/D = 2
+10.47
+0.282
+0.590
+10.955
+7.584
+18.539
+I
+11.10
+0.287
+0.600
+10.776
+7.510
+18.286
+I
+11.94
+0.293
+0.613
+10.614
+7.404
+18.017
+I
+12.15
+0.326
+0.683
+10.457
+6.351
+16.808
+K
+12.57
+0.344
+0.721
+9.830
+5.992
+15.822
+K
+12.99
+0.348
+0.729
+9.786
+5.862
+15.647
+K
+
+14
+TABLE V. Total entropy S2PT
+tot
+and the excess entropy S2PT
+ex
+calculated from the 2PT method, entropy of ideal rigid rotor Sid
+tot calculated using
+Eq. 35 and excess entropy using Monte Carlo equation of state from Cuetos et al.17 SEOS
+ex
+for different liquid crystal phases of L/D = 5 at
+T ∗ = 5:
+P∗
+ρ∗
+S2PT
+tot
+Sid
+tot
+S2PT
+ex
+SEOS
+ex
+Ref17
+Phase
+4.45
+0.093
+21.803
+28.316
+-6.512
+-4.023
+I
+4.90
+0.098
+21.494
+28.267
+-6.772
+-4.454
+N
+5.34
+0.103
+21.150
+28.215
+-7.064
+-4.598
+N
+5.79
+0.107
+20.827
+28.172
+-7.345
+-4.885
+N
+6.23
+0.112
+20.674
+28.134
+-7.459
+-5.316
+N
+6.68
+0.116
+20.156
+28.093
+-7.936
+-5.891
+N
+7.12
+0.121
+19.921
+28.051
+-8.131
+-6.322
+N
+7.57
+0.130
+19.268
+27.977
+-8.709
+-6.753
+SmA
+8.01
+0.135
+19.011
+27.946
+-8.935
+-7.328
+SmA
+8.46
+0.137
+18.768
+27.926
+-9.158
+-7.615
+SmA
+8.90
+0.141
+18.425
+27.898
+-9.473
+-
+SmA
+9.35
+0.144
+18.197
+27.880
+-9.683
+-8.046
+SmA
+9.79
+0.147
+18.059
+27.860
+-9.801
+-8.333
+SmA
+

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+CAPoW: Context-Aware AI-Assisted Proof of Work
+based DDoS Defense
+Trisha Chakraborty∗, Shaswata Mitra†, Sudip Mittal‡
+Department of Computer Science & Engineering, Mississippi State University
+{tc2006∗, sm3843†}@msstate.edu, mittal‡@cse.msstate.edu
+Abstract—Critical servers can be secured against distributed
+denial of service (DDoS) attacks using proof of work (PoW)
+systems assisted by an Artificial Intelligence (AI) that learns
+contextual network request patterns. In this work, we introduce
+CAPOW, a context-aware anti-DDoS framework that injects la-
+tency adaptively during communication by utilizing context-aware
+PoW puzzles. In CAPOW, a security professional can define
+relevant request context attributes which can be learned by the AI
+system. These contextual attributes can include information about
+the user request, such as IP address, time, flow-level information,
+etc., and are utilized to generate a contextual score for incoming
+requests that influence the hardness of a PoW puzzle. These
+puzzles need to be solved by a user before the server begins
+to process their request. Solving puzzles slow down the volume
+of incoming adversarial requests. Additionally, the framework
+compels the adversary to incur a cost per request, hence making
+it expensive for an adversary to prolong a DDoS attack. We
+include the theoretical foundations of the CAPOW framework
+along with a description of its implementation and evaluation.
+I. INTRODUCTION
+An organization protects its critical servers from distributed
+denial of service (DDoS), which may contain valuable infor-
+mation, such as intellectual property, trade secrets, employee
+personally identifiable information (PII), etc. To launch a
+DDoS attack, the malicious users send a flood of requests to
+these servers. As a result, requests from legitimate users either
+experience delays or their requests are dropped. For more than
+two decades, DDoS attacks have been a prominent issue and
+even today it is far from being solved as these attacks are
+cheaper to launch than to defend, especially with the rise of
+DoS-as-a-Service [25].
+PoW system works by requiring incoming requests to ex-
+pend resources solving an computational puzzles to prove ones
+legitimacy. The general system consists of two parts: prover
+and verifier. The prover finds the solution to the computational
+puzzles, when solved, sends the solution to the verifier. In
+a simple networked client-server environment, the user-side
+contains the prover component, and the server-side contains
+the verifier components. Researchers have proposed PoW-
+based solutions for DDoS which makes the attack expensive to
+launch [4], [21], [34]. Although, these solutions suffer from a
+lack of intuition on how to set puzzle difficulty and adaptability
+in different settings.
+In this paper, we develop a defensive tool that emphasizes
+on learning the normal activity patterns of legitimate users.
+The idea behind the tool is to penalize the users that deviates
+from normal activity patterns by issuing them hard puzzles
+and at the same time issuing easy puzzles to users who
+follow the pattern. We leverage a context-aware AI model
+that can learn these normal activity patterns by contextual
+information. The term context within the scope of legitimate
+activity patterns can be defined as request attributes, such
+as, IP address, time, flow-level information, etc. When the
+context is IP address, network activity is considered deviated
+if the source IP address is part of a known blocked IP
+list. Whereas, when the context is time, network activity is
+considered deviated if it arrives at an unusual time compared
+to the normal activity pattern. Security professionals can select
+relevant request context attributes which can be learned by the
+AI models. The concept of context-aware AI models is derived
+from context-aware computing introduced by Dey et. al [9].
+We introduce CAPOW tool, a context-aware AI-assisted
+PoW system that helps to secure critical servers against DDoS
+attacks. Our framework utilizes context-aware AI models that
+learn the expected context pattern from server-side activity-
+logs. The activity-logs are stored and managed by the server
+which contains user activity (IP address, timestamp, flow-
+level data, etc). The deviation from the learned pattern is then
+leveraged to generate a contextual score for incoming requests
+which tunes the difficulty level of the PoW puzzle to be solved.
+The underlying defensive strategy curtails the ability of a
+malicious user to prolong the attack by adaptively introducing
+latency through PoW puzzles and compelling malicious users
+to expend more resources to complete an attack. The main
+contributions of this paper are as follows.
+Contribution 1: We introduce CAPOW, an anti-DDoS frame-
+work that injects latency adaptively, i.e., the framework en-
+sures that malicious users incur higher latency than legitimate
+users based on the deviation in context pattern. We discuss
+the process of context score calculation from deviation in
+Section III-B.
+Contribution 2: We propose a policy component that is
+created by security personnel to incorporate server-specific
+security demands. We provide intuition for policy construction
+in Section III-C.
+Contribution 3: We discuss an instance of CAPOW imple-
+mentation and perform evaluation to illustrate the effective-
+ness of CAPOW. The implementation details are discussed
+Section IV. The code is released on GitHuB [3].
+The rest of the paper is structured as follows. In Section II
+we discuss the threat model and attack definitions. We discuss
+the theoretical foundation of CAPOW in Section III and
+arXiv:2301.11767v1  [cs.CR]  27 Jan 2023
+
+CAPOW implementation in Section IV. We discuss related
+works of the PoW system and DoS defense in Section V,
+followed by the conclusion in Section VI.
+II. THREAT MODEL
+In this section, we present a series of assumptions associated
+with the adversary’s abilities. An adversary A initiates a DDoS
+attack by sending a flood of requests to the server. The ad-
+versary’s intention is to overwhelm the server’s computational
+resources and disrupt legitimate user communication with the
+server. Although the attack described is a variant of DDoS,
+the usefulness of CAPOW can be extended to other variants.
+These assumptions described below are similar to previous
+literature on DDoS defense using proof of work [17] and in
+some sense, we consider a stronger adversary.
+Assumption 1. Adversary A can eavesdrop on the communi-
+cation channel of the server. A cannot modify any user request
+and cannot read any request payload data.
+Assume a secure network communication channel is used
+by the user to send request packets to the server. The user
+performs encryption on the payload data, including the puzzle
+solution, and sends the packet to the server. When an adversary
+eavesdrops on the channel, they can read the source and
+destination IP of the packet, but they cannot read the encrypted
+payload consisting of the puzzle parameters. Additionally, the
+adversary cannot flip bits of the packet and pollute the puzzle
+solution included in the payload. Hence, we assume that the
+adversary has no knowledge of the puzzle parameters solved
+by a user nor can it deny service to a user who has correctly
+solved the puzzle. In Section IV, we utilize assumption 1 to
+claim that the adversary cannot reverse engineer the base AI
+models to receive easier PoW puzzles.
+Assumption 2 Adversary A can spoof user identifiers, such as
+IP addresses, and deceive a subset of underlying AI models.
+CAPOW uses AI models to learn legitimate network ac-
+tivity patterns and the deviation from the pattern is directly
+proportional to the difficulty of PoW puzzles to be solved by
+the user. A can spoof a legitimate user IP address and send
+requests to the server. An intelligent adversary would send
+probe packets to the server using a set of spoofed IP addresses
+and only utilize IPs that require puzzles to be solved. This way,
+the adversary is able to deceive the AI model and reduce the
+latency introduced. In Section IV, we discuss that sending
+probe packets becomes costly for an adversary to deceive
+multiple base AI models.
+Assumption 3 Adversary A cannot pollute the training data
+of the AI models.
+The AI model used by CAPOW learns normal activity
+patterns and calculates a deviation which directly influences
+the hardness of the puzzle. Hence, it is essential that the AI
+learns normal activity patterns from an unpolluted activity-log
+to maximize the effectiveness of CAPOW. In Section IV-B,
+we describe the training process of a context-aware AI model
+where a security professional is deployed to select secure data
+to train the base AI models.
+III. CAPOW ARCHITECTURAL DESIGN AND
+THEORETICAL FOUNDATIONS
+In this section, we describe the high-level architecture of
+the core components and their inner workings that molds the
+CAPOW framework. As shown in Figure 1, CAPOW consists
+of four core components: request context extractor, context-
+aware AI models, policy, and proof-of-work.
+The AI models learn the normal activity pattern from
+previous activity-logs. When an incoming request packet is
+seen, first the context attributes are extracted from the new
+request packet (see Section III-A). Then, the deviation between
+the learned normal context pattern and new request contexts is
+computed to calculate context score. We elaborate on AI model
+training and score calculation in Section III-B. The policy
+component of CAPOW provides security professionals with
+certain abilities that strengthen the effectiveness of CAPOW
+in various security settings (see Section III-C). The context
+score influences the difficulty of PoW puzzle. In Section III-D,
+we discuss the proof-of-work component and how the PoW
+puzzles can curtails the ability of a malicious user to prolong
+the attack by adaptively introducing latency.
+Data Flow. From Figure 1, the flow of data between different
+components of CAPOW is described below. (1) When a new
+incoming packet is seen, the request packet is forwarded to
+the request context extractor. (2) The extracted request context
+attributes are passed to context-aware AI models which learned
+expected context patterns from activity logs. The context
+score generated by individual AI models is combined using
+a function f to produce the final context score (Φ). (3) The
+context score is forwarded to the policy component which sets
+certain parameters, such as, it maps the context score to a
+puzzle difficulty level. (4) The difficulty level is passed to the
+puzzle solver which solves a puzzle of the defined difficulty
+level using a function func. (5) The computed solution is sent
+to the verifier. (6) When the solution is correct, the request
+packet is placed on the server queue for processing.
+A. Context Extraction from Request Packet
+The concept of context-aware computing was introduced
+by Dey et. al [9], where the proposed mechanism improved
+human-to-computer interaction by delivering contextually rel-
+evant data. In the paper, the author proposed an abstract defini-
+tion of context, which is a piece of information that clarifies the
+characteristics of an entity. When a system contains contextual
+data about a situation or entity, the system can take context-
+aware decisions which improve the overall quality of any
+general decision-making.
+In a security setting, a certain request is deemed suspicious
+if the associated request attributes deviate from the usual
+network activity pattern. For instance, a request packet of
+payload size 65500 bytes is considered suspicious due to
+deviation when the expected normal payload size pattern
+is in the order of a few hundred bytes. To this end, we
+2
+
+Src IP
+Dst IP
+Payload
+Context-Aware AI Model
+f
+Incoming Packet
+Activity  
+Logs
+Context Score 
+Range
+Difficulty  
+Range
+Policy File(s)
+Solution = func (puzzle parameter)
+Security 
+Professional
+Policy
+d-constrained 
+solution found?
+func (solution parameter)
+Proof-of-work
+Server Queue
+Packet n
+Packet n-1
+Packet n-2
+Context C1  
+Model
+Context Score (Φ) 
+W1
+W2
+Wn
+d1
+d2
+d3
+d4
+…
+Φ1
+Φ2
+Φ3
+Φ4
+…
+Calculated 
+context 
+score 
+forwarded 
+to policy 
+module
+Puzzle Verifier
+Puzzle Solver
+Mapped 
+puzzle 
+difficulty 
+level 
+forwarded 
+to puzzle 
+solver
+Packets with correct puzzle 
+solution placed in server 
+queue to process.
+Send solution
+1
+2
+3
+4
+5
+Request Context Extraction
+C1
+C2
+Context C2  
+Model
+Context Ck  
+Model
+Puzzle Parameters 
+Model Parameters 
+…
+Ck
+6
+Fig. 1. The figure illustrates the architecture of CAPOW framework. CAPOW consists of four core components: request context extractor, context-aware AI model, policy, and
+proof of work. The AI model learns context patterns from previous activity-logs selected by security personnel and calculates a context score based on the deviation of the incoming
+packet. The calculated score is mapped to the PoW puzzle difficulty level as defined by the security professional in policy files. The proof of work component performs evaluations
+to find the constrained solution. The request with a correct solution is placed on the server queue to process.
+define context of a request packet as request attributes, such
+as source IP address, time of arrival, port address, time
+to live (TTL), and other flow-level attributes. The contexts
+attributes to be extracted are selected by security personnel
+via policy component. The list of selected context attributes
+are reformed periodically to update the defensive posture of
+the organization deployed. When a new request packet is seen,
+the request context extractor component extracts the selected
+context attributes from the request packet and feeds it to the
+context-aware AI models.
+B. Context-Aware AI Model
+The framework component consumes activity-logs supplied
+by security personnel as input to generate a context-aware AI
+model. The model is generated by considering a set of request
+packets from the activity-log λ = {λ0, λ1, λ2, ..., λi}. Each
+request packet λi consists of a set of request context attributes,
+Cλi = {C0λi, C1λi, C2λi, ..., Ckλi}
+(1)
+where k is the number of request context attributes. Ck is
+represented as n-dimensional vector. When an n-dimensional
+vector of a single context for λ requests is projected in
+Euclidean space, such relative positioning produces a cluster.
+For k context attributes, k clusters are generated. The clusters
+represent the normal activity pattern. To evaluate a new incom-
+ing request, request context extractor from Section III-A, feeds
+the context attributes which are then projected in Euclidean
+space. The deviation ∆(p, q) of context Ck is calculated as the
+Euclidean distance between the corresponding normal activity
+cluster and the new request projection,
+∆(p, q) =
+�
+�
+�
+�
+n
+�
+j=1
+(qj − pj)2
+(2)
+where p is projected single context attribute of the new request
+and q is center of a normal cluster of the same context.
+Consequently, the context score Φ for Ck is calculated as,
+Φ(Ck) =
+�∆(p, q)
+∆max
+�
+× I
+(3)
+where ∆max is the maximum possible deviation for Ck. The
+score is in the range of [0, I], where I ∈ Z+. In Section IV-B,
+we discuss the implementation of context-aware AI models.
+C. Policy
+The policy component is a rule-based strategy that facilitates
+the adaptive security guarantees of CAPOW. The rules are set
+in policy files that determine certain CAPOW characteristics.
+These characteristics include context-aware AI model specifi-
+cations, such as, which activity-logs are supplied to train the
+AI models, which context attributes hold more significance
+over the others, etc. Additionally, these parameters include
+proof-of-work components specifications, such as, what is
+the rule to translate context score to puzzle difficulty, which
+variant of PoW puzzle to be used, etc. Hence, it is evident
+that policy construction is a non-trivial task and requires
+consideration of various facets of the deployed server to bolster
+the effectiveness of CAPOW in different security settings.
+To perform the convoluted task of policy designing, security
+professionals are deployed to design server-specific policies.
+Intuition for AI model parameters. From Section III-A,
+a request packet consists of several context attributes. The
+significance of some contexts holds more importance over
+others depending on the type of attack defense. For instance,
+payload size is an important context attribute to protect against
+large payload DDoS attacks [37], but less important to de-
+fend volumetric DDoS attacks. Policy includes the weight
+associated with context attributes to provide an attack-specific
+defense. Additionally, a policy includes the source of data
+3
+
+to train the AI models to avoid model data pollution attacks
+(Assumption 3).
+Intuition for proof-of-work parameters. The context score
+produced by the context-aware AI model is translated to
+the PoW difficulty level. The policy includes the rules to
+translate context scores to puzzle difficulty. In Section IV-C,
+we implemented three rules to show that the translation leads
+to adaptive latency injected. As stated by Green et. al [13],
+amongst groups of users, the CPU capacity of each device can
+vary 10x times, whereas memory capacity may only vary 4x
+times. Hence, when a memory-bound PoW puzzle is used, it is
+less likely for the adversary to have an edge over a legitimate
+user as the discrepancy in memory power as the resource is
+less compared to CPU-bound puzzles. The policy includes the
+means to set variants of puzzles depending on the expected
+user base.
+D. Proof of Work
+Classical proof of work systems [4], [10], [34] consists
+of two main components – prover and verifier. The prover
+provides verifiable evidence of expanding computational re-
+sources by solving puzzles as assigned by the server. On the
+other hand, the verifier validates whether the solved puzzle
+yielded the desired solution. When PoW systems are used as
+DoS defense [4], [26], [35], a user commits some computation
+resources (CPU cycle, bandwidth, etc.) and burns one of these
+resources for solving the PoW puzzle to prove their legitimacy.
+In CAPOW, when a user deviates from a normal activity
+pattern, the PoW component issues a PoW puzzle to request
+proof of legitimacy. The difficulty level of PoW puzzle is
+a function of context score. The rule to translate to context
+score to difficulty level is defined under policy component
+(Section III-C). PoW solver uses a function func to solve the
+assigned difficulty puzzle (see Figure 1). In general terms, this
+function injects two types of cost: (1) direct cost of resource
+burning [14], and (2) indirect cost of latency. The notion
+of resource burning cost represents the resource consumption
+of a user, where the resource could be computational power,
+memory, network bandwidth, or human capital [14]. This cost
+directly impacts the ability of the adversary to conduct a DDoS
+attack as every request requires the adversary to spend real-
+life resources. The notion of latency cost captures the delay
+in time introduced in communication due to the act of puzzle
+solving. This cost indirectly impacts the adversarial intent by
+throttling the rate of adversarial requests reaching the server
+queue. Both costs ultimately cripple the adversarial capability
+to prolong an ongoing DDoS attack.
+IV. CAPOW IMPLEMENTATION, TOOL INSTANCE
+DEPLOYMENT, AND EVALUATION
+In this section, we present a deployment of CAPOW frame-
+work by implementing a single instance of each core compo-
+nent: context extractor, context-aware AI models, policy, and
+proof-of-work. First, the context extractor instance extracts se-
+lected request context attributes. Second, the extracted contexts
+are relayed to context-aware AI model instances where each
+base AI model is generated using server-side activity-logs.
+Then, the trained AI models calculate the deviation of selected
+contexts to produce a context score. Third, we provide three
+policy designs that maps context score to difficulty of PoW
+puzzle. Finally, we implemented a hash-based PoW puzzle
+instance which, over repeated trials, finds the constrained
+solution of assigned difficulty level. The costs inflicted due
+to the our puzzle instance are CPU-cycles (resource burning)
+and time spent (latency). For the purposes of validating our
+contribution via evaluation, we consider that the main cost
+injected is latency which, when injected, throttles the rate of
+adversarial requests.
+Now, we will describe our evaluation setup. We split the
+CIC-IDS2017 dataset [24] into test and train files where day
+1 to day 5 (Monday - Thursday) is used to train the models
+and day 6 (Friday) is used to evaluate CAPOW. From day
+1 to day 5, we deleted the attack traffic to learn normal
+activity pattern. Consider five users sending requests to the
+server U1, U2, U3, U4, and U5. We fixed four user identifiers
+from day 5 to map the four above-mentioned users. Let the
+fifth user U5, be mapped to the user identifier that performs
+DoS on day 6. Since, the user identifier in CIC-IDS2017
+is IP address, let the mapped IP of user U1, U2, U3, U4, and
+U5 is represented by 104.20.30.120, 83.66.160.22,
+37.59.195.0, 104.16.84.55, and 205.174.165.73
+respectively. Through our evaluation scenario, we provided
+evidence that CAPOW injects latency adaptively based on
+the calculated context score of user U5 which throttles the
+adversarial requests and make it expensive for an adversary to
+prolong a DDoS attack.
+A. Context Extraction Instance
+The context extraction instance consumes the request packet
+and extracts context attributes from the request packet. For
+our implementation, we select three context attributes: (1) IP
+address, (2) temporal activity, and (3) flow-level data. For
+evaluation, we used feature attributes of CIC-IDS2017 dataset
+to serve as context attributes. The source IP feature becomes
+the IP address context, the timestamp feature becomes the
+temporal activity context, and the remaining features become
+flow-level context.
+B. Context-Aware AI Model Instance
+We propose an ensemble learner that consists of dedicated
+base AI models to learn individual contextual patterns. The
+base AI model receives the context attributes from the context
+extractor as inputs. The model that (1) learns the IP address
+pattern is called dynamic attribute-based reputation (DAbR),
+(2) learns the temporal activity pattern is called temporal ac-
+tivity model (TAM), and (3) learns the flow-level data pattern
+is called flow-level model (FLOW). Each model computes a
+context score in the range between [0, 10]. Context scores
+from three AI models are combined using the argmax function.
+Next, we discuss three base models where the subsections are
+divided into model generation, context score calculation, and
+evaluation.
+4
+
+User
+Temporal Activity
+User 1
+[[500, 501, …,600], [800, 801, 
+…,900]]
+User 2
+[[770, 771, …,800], [850, 851, 
+…,860], …]
+User 3
+[[100, 101, …,110], [300, 301, …, 
+315]]
+User 4
+[[550, 551, …,560]]
+User
+Temporal Activity
+User 1
+[[100, 102, …,220], [500,501,…,510]]
+User 2
+[[200, 201,…, 230],]
+User 3
+[[190, 191, …, 200], [630, …690]]
+User 4
+[[100, 101, …,250], [260, 261, … 410]]
+User 1
+Time (seconds)
+100
+200
+400
+300
+500
+600
+700
+User 2
+User 3
+User 4
+User
+Temporal Activity
+User 1
+[[650, 651, …, 700], [760, 761, …, 800]]
+User 2
+[[175, 176, …,190], [790, 791, …,800]]
+User 3
+[[530, 531, …,602], [740, 741, …, 750]]
+User 4
+[[350, 351, …,440], [690, 691, …, 701]]
+User
+Temporal Activity
+User 1
+[[300, 301, …,405], [500,501,…,510]]
+User 2
+[[505, 540], [640, 641, …680]]
+User 3
+[[190,…200], [410, 530]]
+User 4
+[[100, 101, …, 250], [260, 261, …, 
+410], [500, 501, …, 515]]
+Activity log on t-3 day
+Activity log t-1 day
+Activity log t day
+Activity log on t-2 day
+Aged Activity Logs
+User Activity Cluster
+Current Activity Logs
+Fig. 2. The figure shows that selected activity-logs (left) are used to generate a temporal activity model (TAM) (right). The illustration shows that out of four activity logs, currently
+only two activity logs are used to form the model (blue box). The remaining activity-logs are aged in an attempt to keep the model up-to-date.
+Dynamic Attribute-based Reputation (DAbR): We utilize
+DAbR [29] as the base AI model that learns context patterns
+for IP attributes. The AI model is generated by projecting
+malicious IP attributes from Cisco Talos dataset [31] into
+Euclidean space. The dataset contains a list of malicious
+IP addresses and IP-related attributes [29]. The red dots in
+Figure 3(A) represent the projected malicious IP attributes that
+form a cluster in Euclidean space. When a new request is
+evaluated, the IP attributes of the new request are projected
+in Euclidean space and a deviation is calculated as Euclidean
+distance to the malicious cluster center. The distance calculated
+produces the context score for DAbR (α). The multi-colored
+stars represent U1, U2, U3, U4, and U5. User U1, U2, U3, U4, and
+U5 receives 2.87, 1.16, 3.15, 2.18, and 2.98 reputation score
+respectively.
+Temporal Activity Model (TAM): We propose a temporal
+activity model (TAM) that learns the pattern of user request
+activity based on time of arrival from activity-logs. The model
+is generated using previous t-days server activity-logs. The
+selected activity-logs can be either previous t consecutive days,
+or t specific days (as defined in the policy). The temporal
+model can be updated by aging the older activity models
+(see Figure 2). The red rectangular blocks in Figure 3(B)
+represent an activity cluster per user. The term active in
+practice can represent a user session or concurrent requests.
+When a user request U arrives at the server, the server finds
+the corresponding user activity cluster (UCLS) formed by the
+temporal activity model. The user activity cluster (UCLS) is a
+list of time intervals that represents the user’s historical activity
+times. The deviation in time is calculated as the distance
+between the two nearest clusters. From CIC-IDS2017 dataset,
+the cluster formed for user U1 shows that the user was active
+between 130 − 140 minutes, 160 − 170 minutes, 600 − 670
+minutes, and 720−760 minutes. When user U1 arrived at time
+700 minutes on day 6, the two nearest clusters are 600 − 670
+and 720−760 (see Figure 3(B)). This deviation is called ∆local
+which is the distance between the two nearest clusters. Finally,
+the context score for TAM is calculated as,
+β = ∆local
+∆max
+× 10
+(4)
+where, ∆max represents the maximum deviation possible
+which in our implementation is 720 minutes. Note that no
+cluster is found for U5, hence the context score calculates is
+the highest in range.
+Flow-level Model (FLOW): Flow-level Model (FLOW) learns
+network flow context patterns from activity-logs. The network
+flow attributes of a request packet are flow-related data, such as
+TTL, flow duration, payload size, protocol, etc. To generate the
+model, the n-dimensional flow attribute vectors are projected
+in Euclidean space. In Figure 3(C), the green dots represent
+projected network flow attributes of legitimate requests, and
+the red dots represent projected network flow attributes of
+malicious requests. When a new request is seen, its flow-
+level attributes are projected and the Euclidean distance to
+malicious and legitimate clusters are computed. The context
+score is calculated as,
+γ = ∆l,m
+∆max
+× 10
+(5)
+where, ∆l,m is the deviation from malicious and legitimate
+clusters and ∆max is the maximum deviation possible in flow-
+level context.
+C. Policy Component Instance
+We constructed three policy instances, policy 1, policy 2,
+and policy 3. These policies only set the mapping function
+between context scores to the PoW puzzle difficulty level.
+Context score is directly proportional to the difficulty of the
+PoW puzzle, such as the increase in contextual deviation leads
+to a higher difficulty puzzle and more latency injected.
+Policies 1 and 2: Linear mapping. Assume a linear map
+function. Policy 1 maps f(Φ) → d, where Φ ∈ [0, 10] is the
+range of context score and d ∈ [0, 10] is the difficulty levels of
+the PoW puzzle. Similar to policy 1, policy 2 maps f(Φ) → d,
+where Φ ∈ [0, 10] and d ∈ [10, 20]. Note that, the error bar
+in Figure 4 shows the discrepancy in time to solve d-level
+PoW puzzle. As discussed in Section III-C, this discrepancy
+in time to solve can be avoided by using memory-bound PoW
+puzzles.
+Policy 3: Error range mapping For policy 3, we incorporated
+the error ϵ of the context-aware AI model. Assume a linear
+map function. Policy 3 maps f(Φ) → d, where Φ ∈ [0, 10] and
+d ∈ [0, 10]. The final difficulty level assigned is a difficulty
+value chosen at random in the interval [⌈di − ϵ⌉, ⌈di + ϵ⌉],
+where ϵ = 0.2. Figure 4 shows that as contextual deviation
+increases, the amount of injected latency increases.
+5
+
+0.2
+0.4
+0.6
+0.8
+1.0
+Malicious cluster centre 
+User 1
+User 2
+User 3
+User 4
+User 5
+Malicious IP
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+2
+4
+6
+8
+10
+Time (minutes)
+Context Score
+100
+200
+400
+300
+500
+600
+700
+800
+0.2
+0.4
+0.6
+0.8
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.0
+Iu87
+Malicious
+User 1
+User 2
+User 3
+User 4
+User 5
+Benign
+User 1
+Context Score
+User 4
+User 3
+User 2
+User 5
+Model B
+Model C
+Model A
+2
+4
+6
+10
+8
+(A)
+(B)
+(C)
+(D)
+Fig. 3. The figure contains four sub-figures. (A) Representation of trained DAbR in the 2-D plot. The red dot cluster represents malicious IP attributes. (B) Representation of
+trained TAM. The stars represent the current time of arrival. (C) Representation of FLOW. The green cluster represents legitimate flow-level attributes and the red cluster represents
+malicious ones. (D) Represents the calculated context score after combining scores from Model A is DAbR, Model B is TAM, and Model C is FLOW.
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Context Score (
+)
+0
+200
+400
+600
+800
+Latency (millisecond)
+Policy 1
+Policy 2
+Policy 3
+Fig. 4. An evaluation of our three implemented policies. The median of 30 trials is
+reported for each reputation score.
+D. PoW Instance – Hash Function
+We discuss two sub-components of CAPOW that mimic
+proof-of-work system: puzzle solver, and puzzle verifier.
+Puzzle Solver. The puzzle solver takes user identifiers as input,
+such as the timestamp of the arrival of the request packet (t),
+and the user IP address (u). Additionally, the solver takes
+a server seed value (ρ) to protect against pre-computational
+attacks. To this, a n-bit string is added, which the client
+modifies upon each hash function evaluation. We call this
+string nonce denoted by η.
+The user evaluates this input until it finds an output string
+Y where Y = H(u||t||ρ||η) with d leading zeroes, where d is
+the difficulty level assigned to the request packet. The puzzle
+solver is a user-end component that is installed either in the
+browser [19] or kernel-level. After solving, the user sends the
+nonce back to the server for verification.
+Puzzle Verifier. Puzzle verification is a server-side compo-
+nent that performs straightforward verification of the puz-
+zle solution by performing one hash evaluation, i.e., Y ′ =
+H(u||t||ρ||η). If the sent η value leads to desired number of
+leading 0’s, then the solution is verified.
+Summary of CAPOW implementation evaluation. The con-
+text scores produced by DAbR, TAM, and FLOW models are
+combined to produce the final context score (Φ). As discussed
+in Section III-C, some contexts might be more relevant than
+others to provide attack specific defense. We denote weight w
+as the significance of each context in the final context score.
+The weights for each AI model are fixed through the policy
+instance as discussed in Section IV-C.
+Φ = arg max(w1α, w2β, w3γ)
+(6)
+where w1, w2, and w3 represent weights associated with
+DAbR, TAM, and FLOW respectively. Figure 3(D) illustrates
+the combined context score where w1, w2, and w3 is set to 1.
+User U1 and U2 show that the final context score is decided
+by FLOW model. Similarly, U3, U4, and U5 the final score
+is decided by TAM model. Using policy 2, user U5 incurs
+≈ 300ms latency for a context score of 8, which is the highest
+latency amongst other users introduced by CAPOW.
+Notably, the evaluation performed using a simulated dataset
+might not reflect the worst case efficiency of CAPOW as in
+practice, user U5 might not be deviate in a temporal activity
+context. In this section, we discuss that the cost of deceiving
+multiple AI models is expensive for the adversary. In our
+implementation, user U5 has to deceive three AI models to
+receive an easy PoW puzzle by receiving lower context scores.
+User U5 can receive a lower context score for DAbR by
+trivially spoofing the IP address (Assumption 2). To deceive
+TAM, the user can engineer the requests around the same time
+as noticed during eavesdropping (Assumption 1). As reading
+or tracking flow-level data embedded in request payload data
+while eavesdropping is not possible (Assumption 1), the only
+way to deceive FLOW is by sending multiple probe packets to
+land on a low context score. This is an extensive approach as
+a security personnel may select new contexts to improve the
+defensive posture of the organization periodically. Therefore,
+deceiving all AI models becomes expensive for the adversary.
+To validate contribution 3, we designed and evaluated an
+implementation instance on CAPOW and provided policy
+designs to validate contribution 2. Finally, CAPOW ensures
+that malicious users incur higher latency than legitimate users
+based on the deviation in context pattern that prevents DDOS.
+Hence, we validate contribution 1 (see Section I).
+V. RELATED WORKS
+In this section, we discuss the overview of proof-of-work
+(PoW) literature in DDoS. Relevant to our work, we will also
+discuss the current advances in AI-assisted cybersecurity.
+6
+
+A. Classical Proof-of-Work
+Dwork et. al [10] coined the term proof-of-work (PoW)
+when they proposed the use of cryptographic hash functions
+(also known as client puzzles) to combat unsolicited bulk
+emails (junk emails). Following that, Franklin et. al [11]
+proposed a lightweight website metering scheme in 1997 to
+prevent fraudulent web server owners from inflating their
+website’s popularity. In 1999, Jakobsson et. al [16] proposed
+MicroMinting (originally proposed by Rivest et. al [30] as a
+digital payment scheme) as a candidate problem that can reuse
+the computational effort of solving the POW puzzle. Later that
+year, Laurie et. al [18] proposed that proof of work does not
+work in a spam setting.
+B. Proof-of-Work as DoS defense
+Similar to spam emails, in DDoS, it is significantly cheaper
+for the attacking party to launch a DDoS attack than to defend
+an infrastructure with the defending party. According to Arbor
+network, launching a DoS attack costs an average of $66 per
+attack and can cause damage to the victim of around $500 per
+minute [20]. Aura et. al [4] proposed the first client puzzle
+authentication protocol for a DoS resilient system. Mankins
+et. al [21] investigated methods for tuning the amount of re-
+source consumption to access server resources based on client
+behavior, where the costs imposed can be either monetary
+or computational. In a similar vein, Wang and Reiter [33]
+investigate how clients can bid on puzzles through auctions.
+Ndibwile et. al [22] proposed web traffic authentication as a
+replacement for CAPTCHA-based defenses. Wu et. al [36]
+proposed a software puzzle framework that disqualifies the
+adversary’s ability to gain an advantage by using a GPU to
+solve puzzles. A framework was put forth by Dean et. al [8] to
+reduce DoS in TLS servers. A DoS variant was introduced by
+Wood et. al [35]. Certain PoW defenses against DoS are layer-
+specific. The network layer of the proof-of-work system used
+by Parno et. al [26] prioritizes users who use more CPU time to
+solve puzzles. The Heimdall architecture, which can detect any
+change in network flow in routers, was introduced by Chen et.
+al [7]. When a change in network flow is identified for any new
+connection, a puzzle is generated and sent to the new user. The
+difficulty of the computational challenges used in the context
+of DoS attacks on the transport layer was recently assessed
+using game theory by Noureddine et. al [23]. Walfish et.
+al [32] propose an alternative resource called communication
+capacity as a defense against application-layer flood attacks.
+Other research has concentrated on incorporating PoW puzzles
+into practical browsing experiences [5], [6], [19].
+C. Automated DoS defense
+In this section, we revisit the literature on ensemble learning
+techniques for network traffic classification problems. En-
+semble learning is a branch of supervised machine learning
+technique that aggregates the learning of multiple base learners
+to improve overall prediction accuracy [28]. Like network
+traffic classification problems, each base learner is trained to
+become an expert in the local area of the total feature space.
+Gaikwad et. al [12] proposed a bagging ensemble approach
+using REPTree base learners to improve classification over
+weaker AI models. Gupta et. al [2] suggested an IDS system
+that uses ensemble learning to address a class imbalance
+problem. The ensemble learner uses three base learners. First,
+the deep neural network classifies normal and suspicious
+traffic. Second, eXtreme Gradient Boosting is used to identify
+major attacks. Third, random forest is used to classify minor
+attacks. Zhou et. al [1] proposed feature selection process
+using ensemble learning in two stages. The first stage involves
+feature reduction using the heuristic method CFS and the
+Bat Algorithm (BA). The second stage involves aggregating
+C4.5 and Random Forest (RF) algorithms. Jabbar et. al [15]
+suggested an ensemble classifier that uses Alternating Decision
+Tree (ADTree) and the k-Nearest Neighbor algorithm (kNN)
+as base AI models. Paulauskas and Auskalnis [27] proposed an
+ensemble learner that employs four base classifiers: J48, C5.0,
+Naive Bayes, and Partial Decision List (PART) to improve
+classification results over individual AI models.
+VI. CONCLUSION AND FUTURE WORK
+In this paper, we design and evaluate CAPOW a context-
+aware AI-assisted PoW framework that protects critical servers
+against DDoS. The underlying defensive strategy involves
+adaptively introducing latency on malicious users. To achieve
+this functionality, our framework employs an AI model that
+takes the context attributes from the incoming user request
+packet as input. The AI model computes deviation from
+normal activity patterns to output a context score. This score
+influences the difficulty level of a PoW puzzle that injects
+latency adaptively during communication. CAPOW ensures
+that the ability of a malicious user to prolong the attack is
+constrained by adaptively introducing latency through PoW
+puzzles and compelling malicious users to expend more re-
+sources to complete an attack.
+For future work, different design variants of CAPOW can
+be configured to combat different DDoS attack types. PoW
+systems suffer from inherent pitfalls of resource wastage which
+can be circumvented by replacing the model with proof of
+stake (PoS) component. Additionally, alternate design can
+include enhanced human in loop strategy which provides
+control of the framework to the security personnel deploying
+the framework.
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+

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+Rock Guitar Tablature Generation via
+Natural Language Processing
+Josue Casco-Rodriguez
+Rice University
+Houston, TX, USA
+[email protected]
+Abstract
+Deep learning has recently empowered and democratized generative modeling of images and
+text [1, 2], with additional concurrent works exploring the possibility of generating more complex
+forms of data, such as audio [3, 4]. However, the high dimensionality, long-range dependencies,
+and lack of standardized datasets currently makes generative modeling of audio and music very
+challenging.
+We propose to model music as a series of discrete notes upon which we can use
+autoregressive natural language processing techniques for successful generative modeling. While
+previous works used similar pipelines on data such as sheet music and MIDI [5, 6], we aim to
+extend such approaches to the under-studied medium of guitar tablature. Specifically, we develop
+the first work to our knowledge that models one specific genre—heavy rock—as guitar tablature.
+Unlike other works in guitar tablature generation, we have a freely available public demo at
+https://huggingface.co/spaces/josuelmet/Metal Music Interpolator.1
+1
+Introduction
+Music, like images and language, is a fundamental form of art and a quintessential piece of the hu-
+man experience. Despite the fact that recent works—such as Stable Diffusion and OpenAI’s DALL-E
+[7, 1]—have produced explosive breakthroughs in the generation and modeling of visual art, such
+breakthroughs for music production have not yet been realized; however, recent works, such as Ope-
+nAI’s Jukebox [4], have made progress towards advanced music generation. A large factor in why such
+breakthroughs have yet to be is that music is challenging to model, requiring sequence-modeling of
+data mediums that are not as well-understood or intuitive as images or words.
+When investigating how a machine can learn to understand or generate music, one can begin by
+understanding how people learn to work with music. Although music exists as a continuous-time au-
+dio signal, people most efficiently understand and analyze music as a pattern of discrete frequencies
+(for example, the note “A” = 440 Hz) that are played for discretely quantized intervals of time (e.g.,
+quarter-, half-, and whole-notes). As such, sequence-modeling techniques for understanding sequences
+of discrete data can be leveraged towards music modeling, given a sufficient dataset of music samples
+represented in discrete forms.
+While sheet music and Musical Instrument Digital Interface (MIDI) files are conventional forms of
+discretely compressed representations of music, one prominent form of music representation that has
+been studied less is guitar tablature (see fig. 1 and fig. 2). While appearing similar to traditional
+sheet music, guitar tablature differs by representing notes as fret and string indices upon which the
+instrument-players must place their fingers so as to produce a specific note, since stringed instruments
+are unique in that most notes that can be played on them have more than one fret and string combi-
+nation that can produce them.
+We develop a new dataset of compressed representations of guitar tablature from one specific genre of
+1Our source code is used to train final demo model is available at https://github.com/Josuelmet/Metal-Music-
+Interpolator.
+1
+arXiv:2301.05295v1  [eess.AS]  12 Jan 2023
+
+Figure 1:
+A guitar tablature snippet of two measures written in 4/4 time. Each note is represented
+not as its pitch, but rather as the specific fret index upon which a player should press upon a certain
+string so as to produce the note. The fret index of a note on the string it is played on is equivalent to
+the number of semitones between the note’s pitch and the lowest pitch that the string can play. Note
+that certain notes contain information about their dynamics: for example, the P.M. symbol indicates
+that certain notes should be played in a semi-muted fashion.
+Figure 2:
+Another guitar tablature snippet in 4/4 time, this time consisting of the iconic first measure
+of Sweet Child O’ Mine by Guns N’ Roses. The six pitches arranged vertically on the left are the lowest
+pitches that each of the six guitar strings can play.
+music (heavy rock), as well as a neural network architecture that can leverage sequence modeling (such
+as in long short-term memory networks or natural language processing models) to produce new guitar
+tablature sequences when conditioned on a brief snippet of an existing guitar tablature. Specifically,
+the proposed autoregressive model aims to estimate the most likely new tablature token xN+1 when
+given the previous tokens x1, x2, ..., xN (i.e., estimate the conditional probability p(xN+1|x1, ..., xN)),
+thus enabling an iterative procedure through which an M-token sequence can be generated from an
+N-token sequence, for M > N.
+2
+Background
+2.1
+Sequence Models
+Recurrent networks. Sequence modeling is a long-standing problem in machine learning and statis-
+tics, with one of its earliest prominent efforts being recurrent neural networks [8]. While recurrent
+neural networks are able to leverage a form of memory to model sequences of theoretically unbounded
+contexts, in practice they and their recent variants [9] struggle to do so, in part due to gradient prop-
+agation issues [10].
+Transformers.
+Meanwhile, transformers [11] circumvent the problems with recurrent neural net-
+works by replacing recurrent operations with one feedforward attention operation that compares every
+element of a sequence with every other element of the sequence; such an approach could initially
+seem disadvantaged due to the memoryless nature, inherently finite bounded context [12], and O(N 2)
+runtime of an attention mechanism on a sequence of length N [13]. However, when combined with
+additional innovations such as positional embeddings and token dimensionality reduction via vector-
+ization, transformer architectures have yielded enormous advances in sequence and image modeling
+[2, 1, 14].
+Self-attention. The key behind any transformer architecture is the self-attention mechanism. Given
+a vector-valued input sequence X = [x1, x2, ..., xN] ∈ RN×Dx such that each element is Dx-dimensional
+and the transformer feedforward dimension is D, a self-attention head transforms X into an output
+sequence ˆV through the following: [13]
+2
+
+P. M.
+P. M.
+P. M.
+4
+5
+18
+仆9
+10
+0
+0
+12
+0
+0
+11
+0
+81
+D#
+15
+14
+4
+A#
+15
+F#
+14
+—12
+14
+14
+C#
+4
+12
+G#
+D#1. Using the weights WQ, WK ∈ RD×Dx and WV ∈ RDv×Dx, project X into three distinct
+matrices—the query, key, and value matrices Q, K, and V—via these linear transformations:
+Q = XWT
+Q
+K = XWT
+K
+V = XWT
+V
+2. Let us express the query, key, and value matrices as Q = [q1, ..., qN]T , K = [k1, ..., kN]T , and
+V = [v1, ..., vN]T , where the vectors qi, ki, vi for i ∈ {1, 2, ..., N} are the query, key, and value
+vectors, respectively.
+Each output sequence vector ˆvi is calculated by multiplying each value vector vj by a score
+determined as the similarity between the query vector qi the key vector kj :
+ˆvi =
+N
+�
+j=1
+softmax
+�qT
+i kj
+√
+D
+�
+vj
+Calculation of ˆV = [ˆv1, ..., ˆvN]T can thus be simply expressed as:
+ˆV = softmax
+�
+QKT
+√
+D
+�
+V = AV,
+where the attention matrix A is computed by applying the softmax operation to each row of the
+matrix QKT /
+√
+D [13].
+2.2
+Related Works
+Music/audio generation. Our key contribution to musical sequence modeling is publicly available
+guitar tablature modeling of heavy rock. Various previous and ongoing works have approached music
+generation both continuous and discrete data modalities. For example, the recent SaShiMi [3] and
+Jukebox [4] architectures approach audio and music generation in the spaces of continuous waveforms
+and discrete notes, respectively. The advent of diffusion models [15] has also found influence in a new
+model combining spectrogram and MIDI music generation [16].
+Guitar tablature literature.
+The field of guitar tablature analysis is small but growing, with
+various works tackling challenges such as graph-based solo analysis [17], transcription [18], dataset
+collection, and sequence modeling [19, 20, 21]. Of particular importance to our work are AnimeTab
+[19] and DadaGP [20], since they also opt for a transformer-based approach to statistically generate
+sequences of guitar tablature. Unlike DadaGP, our model has token representations that are much
+more simple and easy to understand, is trained on one specific genre, and has a publicly available
+demo. While our model may share some similarities with AnimeTab, which was published during the
+development of this work and is supposed to have a demo released soon, our model has a demo already
+available and is trained on the genre of heavy rock music instead of anime/video game music.
+3
+Methods
+3.1
+Data Processing
+Initial preprocessing. The success of statistical inference methods often reflects the quality of data
+used for training—data preprocessing is just as important to a successful model as the model itself.
+Our data preprocessing pipeline begins by first collecting a sizeable volume of songs, in guitar tabla-
+ture format, that accurately represent one subgenre of music2. For each tablature file, every song is
+first converted into 4/4 time for ease of processing, and is then converted into a Python object via
+PyGuitarPro3 for ease of querying. Each track of each song (i.e., each instrument or voice of each
+2Complete list of songs: https://github.com/Josuelmet/Metal-Music-Interpolator/blob/main/songs/README.md
+3https://github.com/Perlence/PyGuitarPro
+3
+
+Figure 3:
+Note embedding scheme illustrated for the example note of a whole note on fret 0 of the
+lowest string on a guitar/bass. The fret value is one-hot encoded as 0, the note length is one-hot
+encoded as a whole note, and none of the flags are set to 1 because the note is neither dotted nor
+palm-muted nor a dead/rest/tied note.
+song, not including drums) is then converted into a one-dimensional list containing each note in the
+song; each note is represented as a tuple containing the note’s pitch (with special designations for tied,
+dead4, and rest notes) , duration, the chordal nature if applicable (with the represented chords being
+4th, diminished 5th, and perfect 5th chords), and two flags indicating whether the note is dotted and
+whether the note is muted. Note that the pitch of each note is represented not as the musical pitch
+of each note (e.g., “A4” or “C3”), but rather as the fret on the guitar (or bass) upon which a player
+should place their finger so as to generate the note. Once all songs’ notes have been represented as
+tuples, each tuple is converted to an integer via an invertible dictionary map.
+Embedding initialization. After initial pre-processing, each song exists as a set of sequences, where
+each sequence represents one voice or instrument and contains integers that represent each note. While
+a na¨ıve sequence model could attempt inference upon these scalar sequences, modern sequence models
+have found success in instead representing the individual tokens or elements of a sequence as vectors,
+allowing for more expressive and informative representations token modalities. Unlike previous works
+[20], we opt for a simple, but effective, initial token vectorization illustrated in fig. 3. Each initial
+vectorized token embedding, before training, has 72 dimensions: the first 59 are reserved for one-hot
+encoding the number of semitones (equivalent to the number of frets on a guitar or bass) between the
+pitch value and the lowest pitch playable by the given instrument; the next 3 dimensions are flags indi-
+cating if the note is a dead, rest, or tied note; the next 8 dimensions one-hot encode the note’s duration
+(e.g., whole-, half-, and quarter-notes); and the last two dimensions are flags indicating if the note’s
+duration is dotted and if the note is played in a muted fashion. While the vectorized token embeddings
+are trained, and thus iteravely refined, during the training process, we found that our hand-crafted
+initialization scheme performed better than default random token initialization. As in any other suc-
+cessful transformer architecture, each token also has a positional embedding that accompanies the
+vectorized token embedding before going into the transformer model.
+3.2
+Model Architecture
+In addition to implementing the entire data preprocessing pipeline with the only starting point being
+PyGuitarPro for querying tablature files as Python objects, we also had to manually implement the
+transformer architecture, since we opted for a mini-GPT model due to the relatively small dataset at
+our disposal compared to the number of sequences and number of parameters used to train conven-
+tional language models [2]. Since we were not using a pre-written transformer model, we also had to
+implement a causal masking mechanism to ensure that the transformer cannot use information from
+any tokens after the token it is trying to predict. After extensive hyperparameter tuning on a 90-10
+testing-validation, split, our final hyperparameter values are located in table 1. The final mini-GPT ar-
+chitecture consists of an embedding layer (as described earlier), three transformer blocks in sequence,
+4Dead notes are noted that are heavily muted such that they lose a distinct sense of pitch.
+4
+
+0
+0
+1
+2
+57
+58
+Extra
+Dead
+Rest
+Tied
+Fret:
+Note
+10
+0
+0
+0
+0
+0
+0
+0
+Types:
+cat
+Dotted
+Palm
+Whole
+Half
+64th
+128th
+Duration
+Mute
+Note
+1
+0
+Length:
+0
+0
+0
+0
+0Hyperparameter
+Value
+N (sequence length)
+100
+Output dimensionality (number of unique tokens)
+1629
+Initial embedding dimensionality
+72
+Number of transformer blocks
+3
+Transformer feedforward dimension
+512
+Transformer dropout rate
+30%
+Attention heads per transformer
+10
+Initial learning rate
+0.003
+β1 (Adam parameter)
+0.96
+Batch size
+512
+Training epochs (determined by early stopping)
+∼ 120
+Table 1:
+Hyperparameters of the final mini-GPT model.
+Figure 4:
+Training and validation accuracy and loss of the final mini-GPT model using a 90-10
+training-validation split.
+and one final feedforward layer that returns a 1629-dimensional vector representing the conditional
+probability of the next token’s value p (xN+1|x1, ..., xN).
+4
+Results
+Performance. Using the hyperparameters specified in table 1 and early stopping based on the valida-
+tion loss, our mini-GPT model achieved over 70% training accuracy and over 60% validation accuracy,
+as shown in fig. 4. While it was possible to improve the training accuracy by reducing the transformer
+dropout, we empirically found that doing so worsened validation generalization due to overfitting.
+Qualititative evaluation of our model is made possible through the interactive demo provided5. Not
+only is our work the first, to our knowledge, that provides a publicly accessible generation demo, but
+our model’s success also further validates the usage of and need for natural language processing tech-
+niques in the realm of audio and music generation.
+Future works. Straightforward and interesting extensions of our model include more explicity incor-
+porating rhythmic information [21] in the embeddings or as a separate embedding; such an approach
+would be better able to capture and understand the differences between notes that land on downbeats,
+upbeats, and backbeats. Another extension would be to model multiple tracks or voices at a time,
+allowing for modeling of rich chords or drum sequences. For note-based chord or drum modeling, some
+form of specialized translational invariance could be key to ensuring that the autoregressive model
+understands that, when multiple notes are placed at the same time, their order in a sequence does not
+matter. Using a more novel attention mechanism [13] could further improve performance. Diffusion
+5https://huggingface.co/spaces/josuelmet/Metal Music Interpolator
+5
+
+Accuracy
+Loss
+0.7
+Train Accuracy
+6 
+Train LosS
+Val Accuracy
+Val Los5
+0.6
+5 
+0.5 
+4 
+0.4 
+3 
+0.3
+0.2 
+2 -
+0.1 
+1 
+0
+20 
+40
+60
+80
+100
+120
+20 
+40 
+60 
+80 
+100
+120models could also play a role in generating sequences of guitar tablature, but whether they can out-
+perform autoregressive language models has yet to be shown; recent work has shown that transformers
+and diffusion models can be combined to produce state-of-the-art results in music generation [16].
+References
+[1] A. Ramesh, P. Dhariwal, A. Nichol, C. Chu, and M. Chen, “Hierarchical text-conditional image
+generation with CLIP latents,” arXiv preprint arXiv:2204.06125, 2022.
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+7
+

69E4T4oBgHgl3EQf2A2F/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf,len=322
+page_content='Rock Guitar Tablature Generation via  Natural Language Processing  Josue Casco-Rodriguez  Rice University  Houston, TX, USA  jc135@rice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='edu  Abstract  Deep learning has recently empowered and democratized generative modeling of images and  text [1, 2], with additional concurrent works exploring the possibility of generating more complex  forms of data, such as audio [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' However, the high dimensionality, long-range dependencies,  and lack of standardized datasets currently makes generative modeling of audio and music very  challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  We propose to model music as a series of discrete notes upon which we can use  autoregressive natural language processing techniques for successful generative modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' While  previous works used similar pipelines on data such as sheet music and MIDI [5, 6], we aim to  extend such approaches to the under-studied medium of guitar tablature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Specifically, we develop  the first work to our knowledge that models one specific genre—heavy rock—as guitar tablature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  Unlike other works in guitar tablature generation, we have a freely available public demo at  https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='co/spaces/josuelmet/Metal Music Interpolator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='1  1  Introduction  Music, like images and language, is a fundamental form of art and a quintessential piece of the hu-  man experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Despite the fact that recent works—such as Stable Diffusion and OpenAI’s DALL-E  [7, 1]—have produced explosive breakthroughs in the generation and modeling of visual art, such  breakthroughs for music production have not yet been realized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' however, recent works, such as Ope-  nAI’s Jukebox [4], have made progress towards advanced music generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' A large factor in why such  breakthroughs have yet to be is that music is challenging to model, requiring sequence-modeling of  data mediums that are not as well-understood or intuitive as images or words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  When investigating how a machine can learn to understand or generate music, one can begin by  understanding how people learn to work with music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Although music exists as a continuous-time au-  dio signal, people most efficiently understand and analyze music as a pattern of discrete frequencies  (for example, the note “A” = 440 Hz) that are played for discretely quantized intervals of time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=',  quarter-, half-, and whole-notes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' As such, sequence-modeling techniques for understanding sequences  of discrete data can be leveraged towards music modeling, given a sufficient dataset of music samples  represented in discrete forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  While sheet music and Musical Instrument Digital Interface (MIDI) files are conventional forms of  discretely compressed representations of music, one prominent form of music representation that has  been studied less is guitar tablature (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' 1 and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' While appearing similar to traditional  sheet music, guitar tablature differs by representing notes as fret and string indices upon which the  instrument-players must place their fingers so as to produce a specific note, since stringed instruments  are unique in that most notes that can be played on them have more than one fret and string combi-  nation that can produce them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  We develop a new dataset of compressed representations of guitar tablature from one specific genre of  1Our source code is used to train final demo model is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='com/Josuelmet/Metal-Music-  Interpolator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='05295v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='AS]  12 Jan 2023  Figure 1:  A guitar tablature snippet of two measures written in 4/4 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Each note is represented  not as its pitch, but rather as the specific fret index upon which a player should press upon a certain  string so as to produce the note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' The fret index of a note on the string it is played on is equivalent to  the number of semitones between the note’s pitch and the lowest pitch that the string can play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Note  that certain notes contain information about their dynamics: for example, the P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' symbol indicates  that certain notes should be played in a semi-muted fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  Figure 2:  Another guitar tablature snippet in 4/4 time, this time consisting of the iconic first measure  of Sweet Child O’ Mine by Guns N’ Roses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' The six pitches arranged vertically on the left are the lowest  pitches that each of the six guitar strings can play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  music (heavy rock), as well as a neural network architecture that can leverage sequence modeling (such  as in long short-term memory networks or natural language processing models) to produce new guitar  tablature sequences when conditioned on a brief snippet of an existing guitar tablature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Specifically,  the proposed autoregressive model aims to estimate the most likely new tablature token xN+1 when  given the previous tokens x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=', xN (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=', estimate the conditional probability p(xN+1|x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=', xN)),  thus enabling an iterative procedure through which an M-token sequence can be generated from an  N-token sequence, for M > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  2  Background  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='1  Sequence Models  Recurrent networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Sequence modeling is a long-standing problem in machine learning and statis-  tics, with one of its earliest prominent efforts being recurrent neural networks [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' While recurrent  neural networks are able to leverage a form of memory to model sequences of theoretically unbounded  contexts, in practice they and their recent variants [9] struggle to do so, in part due to gradient prop-  agation issues [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  Meanwhile, transformers [11] circumvent the problems with recurrent neural net-  works by replacing recurrent operations with one feedforward attention operation that compares every  element of a sequence with every other element of the sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' such an approach could initially  seem disadvantaged due to the memoryless nature, inherently finite bounded context [12], and O(N 2)  runtime of an attention mechanism on a sequence of length N [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' However, when combined with  additional innovations such as positional embeddings and token dimensionality reduction via vector-  ization, transformer architectures have yielded enormous advances in sequence and image modeling  [2, 1, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  Self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' The key behind any transformer architecture is the self-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Given  a vector-valued input sequence X = [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=', xN] ∈ RN×Dx such that each element is Dx-dimensional  and the transformer feedforward dimension is D, a self-attention head transforms X into an output  sequence ˆV through the following: [13]  2  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  4  5  18  仆9  10  0  0  12  0  0  11  0  81  D#  15  14  4  A#  15  F#  14  —12  14  14  C#  4  12  G#  D#1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Using the weights WQ, WK ∈ RD×Dx and WV ∈ RDv×Dx, project X into three distinct  matrices—the query, key, and value matrices Q, K, and V—via these linear transformations:  Q = XWT  Q  K = XWT  K  V = XWT  V  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Let us express the query, key, and value matrices as Q = [q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=', qN]T , K = [k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=', kN]T , and  V = [v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=', vN]T , where the vectors qi, ki, vi for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=', N} are the query, key, and value  vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  Each output sequence vector ˆvi is calculated by multiplying each value vector vj by a score  determined as the similarity between the query vector qi the key vector kj :  ˆvi =  N  �  j=1  softmax  �qT  i kj  √  D  �  vj  Calculation of ˆV = [ˆv1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=', ˆvN]T can thus be simply expressed as:  ˆV = softmax  �  QKT  √  D  �  V = AV,  where the attention matrix A is computed by applying the softmax operation to each row of the  matrix QKT /  √  D [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='2  Related Works  Music/audio generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Our key contribution to musical sequence modeling is publicly available  guitar tablature modeling of heavy rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Various previous and ongoing works have approached music  generation both continuous and discrete data modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' For example, the recent SaShiMi [3] and  Jukebox [4] architectures approach audio and music generation in the spaces of continuous waveforms  and discrete notes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' The advent of diffusion models [15] has also found influence in a new  model combining spectrogram and MIDI music generation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  Guitar tablature literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  The field of guitar tablature analysis is small but growing, with  various works tackling challenges such as graph-based solo analysis [17], transcription [18], dataset  collection, and sequence modeling [19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Of particular importance to our work are AnimeTab  [19] and DadaGP [20], since they also opt for a transformer-based approach to statistically generate  sequences of guitar tablature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Unlike DadaGP, our model has token representations that are much  more simple and easy to understand, is trained on one specific genre, and has a publicly available  demo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' While our model may share some similarities with AnimeTab, which was published during the  development of this work and is supposed to have a demo released soon, our model has a demo already  available and is trained on the genre of heavy rock music instead of anime/video game music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  3  Methods  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='1  Data Processing  Initial preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' The success of statistical inference methods often reflects the quality of data  used for training—data preprocessing is just as important to a successful model as the model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  Our data preprocessing pipeline begins by first collecting a sizeable volume of songs, in guitar tabla-  ture format, that accurately represent one subgenre of music2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' For each tablature file, every song is  first converted into 4/4 time for ease of processing, and is then converted into a Python object via  PyGuitarPro3 for ease of querying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Each track of each song (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=', each instrument or voice of each  2Complete list of songs: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='com/Josuelmet/Metal-Music-Interpolator/blob/main/songs/README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='md  3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='com/Perlence/PyGuitarPro  3  Figure 3:  Note embedding scheme illustrated for the example note of a whole note on fret 0 of the  lowest string on a guitar/bass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' The fret value is one-hot encoded as 0, the note length is one-hot  encoded as a whole note, and none of the flags are set to 1 because the note is neither dotted nor  palm-muted nor a dead/rest/tied note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  song, not including drums) is then converted into a one-dimensional list containing each note in the  song;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' each note is represented as a tuple containing the note’s pitch (with special designations for tied,  dead4, and rest notes) , duration, the chordal nature if applicable (with the represented chords being  4th, diminished 5th, and perfect 5th chords), and two flags indicating whether the note is dotted and  whether the note is muted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Note that the pitch of each note is represented not as the musical pitch  of each note (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=', “A4” or “C3”), but rather as the fret on the guitar (or bass) upon which a player  should place their finger so as to generate the note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Once all songs’ notes have been represented as  tuples, each tuple is converted to an integer via an invertible dictionary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  Embedding initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' After initial pre-processing, each song exists as a set of sequences, where  each sequence represents one voice or instrument and contains integers that represent each note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' While  a na¨ıve sequence model could attempt inference upon these scalar sequences, modern sequence models  have found success in instead representing the individual tokens or elements of a sequence as vectors,  allowing for more expressive and informative representations token modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Unlike previous works  [20], we opt for a simple, but effective, initial token vectorization illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Each initial  vectorized token embedding, before training, has 72 dimensions: the first 59 are reserved for one-hot  encoding the number of semitones (equivalent to the number of frets on a guitar or bass) between the  pitch value and the lowest pitch playable by the given instrument;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' the next 3 dimensions are flags indi-  cating if the note is a dead, rest, or tied note;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' the next 8 dimensions one-hot encode the note’s duration  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=', whole-, half-, and quarter-notes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' and the last two dimensions are flags indicating if the note’s  duration is dotted and if the note is played in a muted fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' While the vectorized token embeddings  are trained, and thus iteravely refined, during the training process, we found that our hand-crafted  initialization scheme performed better than default random token initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' As in any other suc-  cessful transformer architecture, each token also has a positional embedding that accompanies the  vectorized token embedding before going into the transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='2  Model Architecture  In addition to implementing the entire data preprocessing pipeline with the only starting point being  PyGuitarPro for querying tablature files as Python objects, we also had to manually implement the  transformer architecture, since we opted for a mini-GPT model due to the relatively small dataset at  our disposal compared to the number of sequences and number of parameters used to train conven-  tional language models [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Since we were not using a pre-written transformer model, we also had to  implement a causal masking mechanism to ensure that the transformer cannot use information from  any tokens after the token it is trying to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' After extensive hyperparameter tuning on a 90-10  testing-validation, split, our final hyperparameter values are located in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' The final mini-GPT ar-  chitecture consists of an embedding layer (as described earlier), three transformer blocks in sequence,  4Dead notes are noted that are heavily muted such that they lose a distinct sense of pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  4  0  0  1  2  57  58  Extra  Dead  Rest  Tied  Fret:  Note  10  0  0  0  0  0  0  0  Types:  cat  Dotted  Palm  Whole  Half  64th  128th  Duration  Mute  Note  1  0  Length:  0  0  0  0  0Hyperparameter  Value  N (sequence length)  100  Output dimensionality (number of unique tokens)  1629  Initial embedding dimensionality  72  Number of transformer blocks  3  Transformer feedforward dimension  512  Transformer dropout rate  30%  Attention heads per transformer  10  Initial learning rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='003  β1 (Adam parameter)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='96  Batch size  512  Training epochs (determined by early stopping)  ∼ 120  Table 1:  Hyperparameters of the final mini-GPT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  Figure 4:  Training and validation accuracy and loss of the final mini-GPT model using a 90-10  training-validation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  and one final feedforward layer that returns a 1629-dimensional vector representing the conditional  probability of the next token’s value p (xN+1|x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=', xN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  4  Results  Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Using the hyperparameters specified in table 1 and early stopping based on the valida-  tion loss, our mini-GPT model achieved over 70% training accuracy and over 60% validation accuracy,  as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' While it was possible to improve the training accuracy by reducing the transformer  dropout, we empirically found that doing so worsened validation generalization due to overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  Qualititative evaluation of our model is made possible through the interactive demo provided5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Not  only is our work the first, to our knowledge, that provides a publicly accessible generation demo, but  our model’s success also further validates the usage of and need for natural language processing tech-  niques in the realm of audio and music generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  Future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Straightforward and interesting extensions of our model include more explicity incor-  porating rhythmic information [21] in the embeddings or as a separate embedding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' such an approach  would be better able to capture and understand the differences between notes that land on downbeats,  upbeats, and backbeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Another extension would be to model multiple tracks or voices at a time,  allowing for modeling of rich chords or drum sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' For note-based chord or drum modeling, some  form of specialized translational invariance could be key to ensuring that the autoregressive model  understands that, when multiple notes are placed at the same time, their order in a sequence does not  matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Using a more novel attention mechanism [13] could further improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Diffusion  5https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='co/spaces/josuelmet/Metal Music Interpolator  5  Accuracy  Loss  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='7  Train Accuracy  6  Train LosS  Val Accuracy  Val Los5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='6  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='5  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='4  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='2  2 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='1  1  0  20  40  60  80  100  120  20  40  60  80  100  120models could also play a role in generating sequences of guitar tablature, but whether they can out-  perform autoregressive language models has yet to be shown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' recent work has shown that transformers  and diffusion models can be combined to produce state-of-the-art results in music generation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
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+page_content=' Chen, “Hierarchical text-conditional image  generation with CLIP latents,” arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
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+page_content='  [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
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+page_content=' Dhariwal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Neelakantan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Shyam,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Sastry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Askell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
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+page_content='  [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQf2A2F/content/2301.05295v1.pdf'}
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7NE4T4oBgHgl3EQfCQsk/content/tmp_files/2301.04858v1.pdf.txt

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+Advancing carrier transport models for InAs/GaSb type-II superlattice MWIR
+photodetectors
+Rohit Kumar, Anup Kumar Mandia, Anuja Singh and Bhaskaran Muralidharan∗
+Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India
+(Dated: January 13, 2023)
+In order to provide the best possible performance, modern infrared photodetector designs necessitate
+extremely precise modeling of the superlattice absorber region. We advance the Rode’s method for
+the Boltzmann transport equation in conjunction with the k.p band structure and the envelope
+function approximation for a detailed computation of the carrier mobility and conductivity of layered
+type-II superlattice structures, using which, we unravel two crucial insights. First, the significance of
+both elastic and inelastic scattering mechanisms, particularly the influence of the interface roughness
+and polar optical phonon scattering mechanisms in technologically relevant superlattice structures.
+Second, that the structure-specific Hall mobility and Hall scattering factor reveals that temperature
+and carrier concentrations significantly affect the Hall scattering factor, which deviates significantly
+from unity even for small magnetic fields.
+This reinforces the caution that should be exercised
+when employing the Hall scattering factor in experimental estimations of drift mobilities and carrier
+concentrations.
+Our research hence offers a comprehensive microscopic understanding of carrier
+dynamics in such technologically relevant superlattices. Our models also provide highly accurate
+and precise transport parameters beyond the relaxation time approximation and thereby paving the
+way to develop physics-based device modules for mid-wavelength infrared photodetectors.
+I.
+INTRODUCTION
+Modeling state-of-the-art infrared (IR) photodetectors
+[1–6] require highly accurate transport parameters for de-
+veloping dark and photocurrent performance projections
+[5, 7–10]. Current technologically relevant IR photode-
+tectors use III-V materials such as InAs/GaSb [11, 12]
+due to numerous advantages [13, 14]. Type-II superlat-
+tices (T2SLs) based on stacks of InAs/GaSb [1, 2, 14] are
+thus extensively used to design high-performance third-
+generation IR detectors [15, 16]. Despite the fact that the
+mobility of the photogenerated minority carriers has a
+significant impact on the performance of IR photodetec-
+tors, carrier transport in technologically relevant T2SL
+structures has not as extensively been explored. Recent
+explorations in this context [17–23] which include carrier
+mobility calculations [24], do not conclusively bring to
+the fore structure-specific impact of important scatter-
+ing mechanisms such as Piezoelectric (PZ), polar opti-
+cal phonon (POP), acoustic deformation potential (ADP)
+scattering mechanisms and most importantly the inter-
+face roughness scattering (IRS).
+With the necessity to develop a deeper understand-
+ing of carrier transport in technologically relevant T2SLs,
+this work advances an accurate model for transport calcu-
+lations, wherein, we investigate different scattering lim-
+ited transport under low-field in InAs/GaSb superlattices
+(SLs) as a function of free electron carrier concentration,
+temperature, and SL structural parameters. In our cal-
+culations, five primary scattering mechanisms that limit
+carrier mobility are the ionized impurity (II) [25], the PZ
+[26], the ADP [27], the POP and the IRS [28–31].
+∗ corresponding author: [email protected]
+We advance the Rode’s method [32–34] which goes be-
+yond the relaxation time approximation (RTA) [35, 36],
+coupled with band structure calculations via the k.p [37–
+41] technique that also includes the strain effect due to
+lattice mismatch between InAs and GaSb materials [42].
+We demonstrate the effect of both the elastic and the
+inelastic scattering mechanisms [43] on the electron mo-
+bility of the composite structure for a wide range of tem-
+peratures and doping concentrations. Our studies reveal
+that the low-temperature mobility of T2SLs is limited
+by the II, PZ and IRS scattering mechanisms. In con-
+trast, the mobility at higher temperatures is mainly lim-
+ited by the POP scattering mechanism, an inelastic and
+anisotropic process. At intermediate temperatures, how-
+ever, the mobility decreases due to a combined effect of
+ADP and IRS mechanisms. The effects of several struc-
+tural parameters including layer thicknesses, interface
+roughness heights, correlation lengths, and ion densities
+are thoroughly investigated. Our calculations thereby re-
+inforce the superiority of the Rode’s method [32, 34] over
+the conventionally employed RTA, wherein, the former is
+applicable over a wide temperature range in the presence
+of inelastic and anisotropic scattering mechanism.
+In order to experimentally obtain the carrier concen-
+tration and drift mobility in a SL structure, it is also im-
+portant to ascertain the Hall scattering factor, which is
+frequently thought of as being equal to unity, indicating
+that the Hall mobility and the drift mobility are equal.
+However, in many heterostructures, it differs significantly
+from unity, which results in inaccurate estimates of the
+carrier density and drift mobility. We clearly show that
+the temperature and carrier concentrations significantly
+affect the Hall scattering factor, and that it ranges from
+0.3 to about 1.48 even for weak magnetic fields, thereby
+reinforcing that caution should be exercised when em-
+ploying this factor in calculations involving drift mobil-
+arXiv:2301.04858v1  [cond-mat.mtrl-sci]  12 Jan 2023
+
+2
+ity and carrier concentration. The models developed here
+pave the way to develop physics-based device modules for
+mid-wavelength IR (MWIR) photodetectors.
+This paper is structured as follows. In Sec. II we de-
+scribe the k.p model to compute the band structure, elec-
+tron distribution function, Boltzmann transport formal-
+ism, Rode’s approach and various scattering processes.
+In Sec. III we illustrate the simulation methodology. In
+Sec. IV, we explain the findings and finally, in Sec. V,
+we summarize our results.
+II.
+ANALYTICAL FORMALISM
+A.
+Electronic band structure
+The energy band structure of T2SLs can be calcu-
+lated using various theoretical approaches like the den-
+sity functional theory (DFT) [44], the empirical tight-
+binding method [35, 45, 46], the empirical pseudopoten-
+tial method [47, 48], many-body perturbation theory [49]
+and the k.p perturbation method [37]. For this study, we
+use the k.p technique with the envelope function approx-
+imation (EFA) [24, 50, 51] since it overcomes the compu-
+tational limitations of first-principles methods. The k.p
+model is extensively used because of its superiority in
+computing the energy band gap. Unlike ab − initio and
+tight binding methods, the k.p technique requires fewer
+input parameters I, with the related calculation proce-
+dure being straightforward.
+In this work, we solve the 8-band Kane Hamilto-
+nian [52], by perturbatively extending the wave function
+around high-symmetry points of the reciprocal space, em-
+ploying the Lowdin’s perturbation approach [52].
+We
+also consider the spin-orbit coupling [53] in our com-
+putation, which provides additional contributions to the
+spin splitting of the energy bands [3]. The SL wavefunc-
+tions (Φn(z)) in the orbital basis states (u0(z)) along the
+growth direction (z) are articulated in terms of the slowly
+varying envelope functions (F(z)), which are given as
+Φn(z) =
+�
+j
+Fj(z)uj0(z).
+(1)
+Such envelope functions under the periodic boundary
+conditions can be rewritten as
+F i
+j(k, z0) = e−iadF i
+j(k, zM),
+F i
+j(k, zM+1) = eiadF i
+j(k, z1),
+(2)
+where, d denotes the thickness of a period, M represents
+the number of grid points, a denotes the Bloch vector of
+the envelope function that spans the Brillouin zone (BZ)
+and k represents the momentum along the transverse di-
+rection. The final Hamiltonian of the SL in the basis set
+comprises three matrices (H0, HI and HII), given by
+H(k, kz) = H0 + HI �
+−i ∂
+∂z
+�
++
+�
+−i ∂
+∂z
+�
+HII �
+−i ∂
+∂z
+�
+. The
+entire coupled differential equation is then solved using
+a numerical finite difference method [54], as described in
+earlier work [3].
+The interface between the InAs and the GaSb layers
+is very abrupt as depicted in Fig. 1(a). The energy dif-
+ference between the conduction band minimum (CBM)
+and the first heavy hole (HH) maximum at the center of
+the BZ determines the band gap in an InAs/GaSb-based
+T2SL, as shown in Fig. 1(b). Figure 1(b) also demon-
+strates that the InAs conduction band (CB) is lower than
+the GaSb valence band (VB), indicating that the band
+structure is a staggered T2SL [55].
+B.
+Carrier transport model
+1.
+Boltzmann transport equation and its solution
+In order to characterize the behavior of the T2SL sys-
+tem, we solve the Boltzmann transport equation (BTE)
+and compute the probability of finding a carrier with a
+crystal momentum k at a location r at a time t as indi-
+cated by the distribution function f(r, k, t). Solving the
+BTE (3) yields the average distribution of the carriers in
+both the position and the momentum space. The BTE
+can be written as [56–58]
+∂f
+∂t − ∂f
+∂t
+���
+diff
+− ∂f
+∂t
+���
+forces
+= ∂f
+∂t
+���
+coll
++ s(r, p, t) .
+(3)
+The term s(r, p, t), in Eq.
+(3), represents generation-
+recombination processes [59], where p is the classical mo-
+mentum. The term (∂f/∂t)forces, represents the change
+in the distribution function due to applied electric and
+magnetic fields.
+The term (∂f/∂t)forces = −F · ∇pf,
+where, F = (dp/dt) = ℏ(dk/dt) = −e(E + v × B), repre-
+sents the total force equal to the sum of the electric-force
+and the Lorentz-force owing to the magnetic flux density
+B, where e is the electron charge, E is the applied elec-
+tric field and v denotes the group velocity of the carriers.
+The term (∂f/∂t)diff = −v.∇rf, refers to the spatial
+change in the distribution function caused by tempera-
+ture or concentration gradients, which results in carrier
+diffusion in the coordinate space. Here, (∂f/∂t)coll is the
+collision term, which indicates how the distribution func-
+tion changes over time due to collision events, and can
+be described as the difference between the in- and the
+out-scattering processes, i.e.,
+�∂f
+∂t
+�
+coll
+=
+�
+k1
+�
+S(k1, k) f (k1)
+�
+1 − f (k)
+�
+− S(k, k1) f (k)
+�
+1 − f (k1)
+��
+,
+(4)
+where, S(k, k1) and S(k1, k) are the transition rates
+for an electron moving between states k and k1.
+Un-
+der steady-state, ∂f
+∂t = 0, in case of spatial homogeneity,
+∇rf = 0, and assuming that there is no recombination-
+
+3
+(a)
+(b)
+FIG. 1. Preliminaries. (a) Schematic of InAs/GaSb based T2SL structure. The electron wave function in the InAs layer
+extends beyond the interface into the GaSb layer and overlaps with the heavy hole wave function. Here, nML and mML are
+the numbers of monolayers of InAs and GaSb respectively, in a single period d. (b) Band alignment of InAs/GaSb based
+T2SL system showing the optical transition between the heavy-hole valence miniband and the electrons from lowest conduction
+minibands that is employed to detect IR radiation. The periodic potential of the d period emerges in the material due to the
+modulation of the semiconductor layers. The creation of hole (electron) minibands in the valence (conduction) band is caused
+by the overlap of hole (electron) wave functions between adjacent GaSb (InAs) layers. The difference between the first electron
+miniband in the CB and the first heavy hole miniband in the valence band is used to compute the effective bandgap energy Eg
+of the T2SL (highlighted in black).
+generation term, the BTE (3) can be rewritten as
+−eE
+ℏ · ∇kf =
+�
+k1
+�
+S(k1, k) f (k1)
+�
+1 − f (k)
+�
+− S(k, k1) f (k)
+�
+1 − f (k1)
+��
+.
+(5)
+In the low-electric field regime, the distribution func-
+tion can be represented as [60]
+f(k) = f0
+�
+ε(k)
+�
++ g(k) cos θ ,
+(6)
+where, k=|k|, f denotes the actual electron distribution
+function, which includes both the elastic and the inelas-
+tic scattering mechanisms, g(k) is the perturbation term
+to f0[ε(k)] produced by the electric field, θ is the angle
+between applied electric field (along the symmetry axis)
+and the electron wave vector k, and f0 represents the dis-
+tribution function under equilibrium conditions, which is
+taken according to Fermi-Dirac statistics [35, 59].
+By
+solving Eqs.
+(5) and (6), the perturbation term g(k),
+can be calculated as [32, 34, 61, 62]
+gi(k) =
+�Si
+�
+gi(k)
+�
+− (−e)E
+ℏ
+�
+∂f0
+∂k
+�
+So(k) +
+1
+τel(k)
+�
+,
+(7)
+FIG. 2.
+The various dominant scattering mechanisms in-
+volved in a T2SL structure.
+where E = |E|, and gi(k) appears on both sides of Eq.
+(7).
+Hence, we solve Eq.
+(7) iteratively and the con-
+vergence is exponentially fast which takes a few itera-
+tions. Once gi(k) is obtained, we calculate the mobil-
+ity. In Eq. (7), the term i indicates the iteration index,
+and the terms, Si & So are the in-scattering and the
+out-scattering operators, respectively, for inelastic scat-
+tering mechanisms, as explained in Sec. II B. The term
+1
+τel(k), represents the total momentum relaxation rate of
+all the elastic scattering mechanisms, which is calculated
+according to the Matthiessen’s rule (8), and can be writ-
+ten as
+
+Interface scattering
+GaSb
+Growth direction
+Extended Phonons
+Interface
+nMI
+KeD
+InAs
+Temperature gradientInAs.
+GaSb
+InAs
+GaSb
+InAs
+GaSb
+InAs
+z
+GaSb
+GaSb
+CB
+GaSb
+VB
+Ec
+Eg
+HH
+LH
+M
+InAs
+InAs
+InAs
+InAs
+Spatial coordinateDominant scattering mechanisms in T2SL
+Defect scattering
+Lattice scattering
+IRS
+Impurity
+Intravalley
+4
+Ionized
+Acoustic
+Optical
+Deformation potential
+Piezoelectric
+Polar4
+1
+τel(k) =
+1
+τII(k) +
+1
+τP Z(k) +
+1
+τADP (k) +
+1
+τIRS(k) .
+(8)
+The various dominant scattering mechanisms involved
+in an InAs/GaSb-based T2SL structure are shown in Fig.
+2.
+2.
+Ionized impurity scattering
+The II scattering mechanism [25] arises due to the
+Coulomb interactions between electrons and ions, when
+a charged center is introduced inside the bulk material.
+The II scattering mechanism is entirely elastic and dom-
+inates usually at high doping concentrations and low
+temperatures. The II scattering mechanism dominates
+near the CB edge but reduces drastically as the energy
+increases [63].
+The scattering rate for the II increases
+rapidly with decreasing temperature. Here, we use the
+Brooks-Herring approach [64] for the calculation of II
+scattering rate [34, 65], which is given by
+1
+τII(k) =
+e4N
+8π ν(k) (ϵ0 ϵs)2 (ℏ k)2
+�
+P(k) ln
+�
+1 + 4
+� k
+β
+�2�
+− Q(k)
+�
+,
+(9)
+where, ϵ0 is the permittivity of the free space, ϵs is the
+static dielectric constant, ℏ is the reduced Planck’s con-
+stant and N is the ionized impurity concentration, which
+is the sum of the acceptor and donor impurity concen-
+tration i.e., N = NA + ND. Here, β indicates the inverse
+screening length, which is given as
+β =
+�
+e2
+ϵ0 ϵs kB T
+�
+DS(ε)f0(1 − f0)dε ,
+(10)
+where, DS(ε) is the density of states (DOS) at energy ε
+and kB is the Boltzmann constant. P(k) and Q(k) can
+be expressed as follows [34, 62]
+P(k) =
+� 3
+4
+�β c(k)
+k
+�4
++ 2
+�β c(k)
+k
+�2
++ 1
+�
+,
+(11)
+Q(k) =
+�3 β4 + 6 β2 k2 − 8 k4
+(β2 + 4 k2) k2
+�
+c4(k)
++ 8
+� β2 + 2k2
+β2 + 4 k2
+�
+c2(k) +
+�
+4
+�
+k/β
+�2
+1 + 4
+�
+k/β
+�2
+�
+.
+(12)
+The detailed explanation of the P and Q parameters are
+given in the literature [34]. Here, the wave function ad-
+mixture c(k) represents the contribution of the p-orbital
+to the wave function of the band.
+3.
+Piezoelectric scattering
+The PZ effect arises due to the acoustic phonon scat-
+tering in polar semiconductors. Being a weak effect, the
+PZ scattering is elastic and significant only at low doping
+concentrations and low temperatures, where other scat-
+tering mechanisms are weak. The momentum relaxation
+rate for the PZ scattering is given by [32, 65]
+1
+τP Z(k) =
+(eP)2 kB T
+6π ϵ0 ϵs ν(k) ℏ2
+�
+4c4(k) − 6c2(k) + 3
+�
+,
+(13)
+where, P is a piezoelectric coefficient, which is a dimen-
+sionless quantity. For the zincblende structure, it is given
+as [34, 62]
+P 2 = h2
+14 ϵ0 ϵs
+35
+��12
+cl
+�
++
+�16
+ct
+��
+,
+(14)
+where, h14 is an element of the PZ stress tensor, and ct
+and cl represents the spherically averaged elastic con-
+stants for transverse and longitudinal modes, respec-
+tively, and are given by [26, 32, 34]
+cl = 3
+5c11 + 1
+5
+�
+2c12 + 4c44
+�
+,
+ct = 1
+5
+�
+c11 − c12
+�
++ 3
+5c44 ,
+(15)
+where c11, c12, and c44 are three independent elastic con-
+stants.
+4.
+Acoustic deformation potential scattering
+The ADP scattering mechanism is caused by the inter-
+action of electrons with non-polar acoustic phonons. It
+is approximately elastic near room temperature For the
+ADP scattering mechanism, the momentum relaxation
+rate is given by [34, 65]
+1
+τADP (k) =
+kB T
+�
+e ΞD k
+�2
+3π cel ν(k) ℏ2
+�
+6c4(k)−8c2(k)+3
+�
+, (16)
+
+5
+where, cel denotes the spherically averaged elastic con-
+stant and ΞD represents the acoustic deformation poten-
+tial, which is obtained by the CB shift (in eV) per unit
+strain, owing to the acoustic waves(17). To calculate the
+acoustic deformation potential (ΞD), we use the following
+relation (17)
+ΞD = −V ×
+�
+∂ECBM
+∂V
+������
+V =V0
+,
+(17)
+where, V denotes the volume, ECBM represents the en-
+ergy of the CBM and V0 is the zero pressure volume of
+the structure.
+5.
+Interface roughness scattering
+The existence of the interface roughness in a T2SL
+[17, 18, 23, 29, 66] structure leads to endemic variations
+in InAs well widths, causes modulation of the associated
+energy levels and introduces an unstable potential for the
+motion of the confined electrons. The IRS mechanism
+can occur due to the imperfections that arise during the
+growth of the material. The earlier related works [67, 68]
+show that the degree of scattering decreases in propor-
+tion to the well width hence it is important in MWIR
+detectors. The IRS mechanism is an elastic process and
+dominates at low temperatures in thin-film systems for
+a short period of T2SL, and it is significant at high elec-
+tron density. The momentum relaxation rate for the IRS
+mechanism is given as [57, 69, 70]
+1
+τIRS(k) =
+�
+e2 ∆ Λ
+ϵ0 ϵ∞
+�2
+k
+ℏ2 ν(k)
+�
+Nd + Ns
+2
+�2
+×
+1
+�
+1 + (kΛ)2 ε
+�
+kΛ
+�
+1 + (kΛ)2
+�
+,
+(18)
+where, Λ is the lateral correlation length, ∆ is the rough-
+ness height, Ns is the sheet carrier concentration, and Nd
+is the doping carrier density.
+6.
+Polar optical phonon scattering
+The POP scattering results from the interaction of op-
+tical phonons with electrons. The POP scattering mech-
+anism is inelastic and anisotropic, which occurs via the
+emission or the absorption of a phonon hence, RTA is in-
+applicable in such SL structures. The scattering rate due
+to the POP scattering mechanism is approximately con-
+stant at very high energies, and it depends on the POP
+frequencies. The POP scattering dominates in the higher
+temperature domain. Hence, it is significant at both near
+and beyond room temperature. The out-scattering oper-
+ator is given by [34]
+So =
+�
+Npop + 1 − f −�
+λ−
+o +
+�
+Npop + f +�
+λ+
+o ,
+(19)
+λ±
+o = L±�
+(A±)2ln
+���k± + k
+k± − k
+��� − A±cc± − aca±c±�
+, (20)
+L± =
+e2 ωpop k±
+4π ℏ k ν(k±)
+�ϵs − ϵ∞
+ϵs ϵ∞
+�
+,
+(21)
+where, ϵ∞ and ϵs are high and low-frequency dielectric
+constants, respectively.
+A± = aa± + [(k±)2 + k2] cc±/ 2 k±k ,
+(22)
+where c, c±, a and a± are the wave function coefficients,
+k± is the solution of Eq. ε(k) ± ℏωpop. Any quantity
+superfixed by plus/minus is to be evaluated at the en-
+ergy corresponding to k+ or k−. The superscript plus
+denotes scattering by the absorption and is evaluated at
+an energy ε(k) + ℏωpop. Similarly, superscript minus de-
+notes scattering by the emission and is evaluated at en-
+ergy ε(k)−ℏωpop. Emission of phonons is possible only if
+the phonons’ energy is greater than ℏωpop energy. There-
+fore, if the phonon energy is less than ℏωpop, the term λ−
+o
+has to be considered as zero. The term Npop, indicates
+the number of optical phonons and is given by the Bose
+distribution as [32, 34]
+Npop =
+1
+exp (ℏ ωpop / kB T) − 1 .
+(23)
+The in-scattering operator Si, is given by
+Si = (Npop + 1 − f)λ+
+i g+ + (Npop + f)λ−
+i g− ,
+(24)
+where, plus and minus superscripts indicate the absorp-
+tion and emission processes, respectively.
+The term
+λ±
+i (k) can be expressed as
+λ±
+i (k) = L± �(k±)2 + k2
+2 k± k
+(A±)2 ln
+���k± + k
+k± − k
+���
+− (A±)2 − c2(k) (c±(k))2
+3
+�
+.
+(25)
+The mobility can be calculated after calculating the
+rates of all the elastic scattering mechanisms
+1
+τel(k) (8)
+and the influence of inelastic scattering mechanisms on g
+(7) through the terms Si(g) (24) and So (19). The rates
+of various elastic scattering mechanisms are calculated by
+using the expressions given in Eqs. (9), (13), (16), (18).
+C.
+Mobility and conductivity
+The RTA [56] cannot be used if the scattering process
+is inelastic and anisotropic because there is no way to
+define the relaxation time that is independent of the dis-
+tribution function.
+In such instances, Rode’s iterative
+approach can be applied to compute the real distribu-
+tion function under low-field conditions. After calculat-
+ing the perturbation distribution by using Rode’s algo-
+rithm, we finally calculate the low-field carrier mobility,
+
+6
+FIG. 3. Flowchart for the calculation of electronic transport
+parameters.
+µ [32, 34, 65]
+µ =
+1
+3E
+�
+ν(ε) DS(ε) g(ε) dε
+�
+DS(ε) f0(ε) dε
+.
+(26)
+The term g(ε), can be obtained from Eq. (7) and the
+carrier velocity ν(k) can be calculated from the band
+structure as
+ν(k) = 1
+ℏ
+∂ε
+∂k .
+(27)
+Once the mobility is determined, it is pretty easy to
+calculate the electrical conductivity by using
+σ = n e µ ,
+(28)
+where, µ is the electron drift mobility, and n is the elec-
+tron carrier concentration. The entire sequence for cal-
+culating the transport coefficients using Rode’s approach
+is shown in Fig. 3.
+Similarly, in the presence of an arbitrary magnetic
+field, the BTE can be solved. The distribution function
+in such cases can be written as [33, 71]
+f(k) = f0[ε(k)] + xg(k) + yh(k) ,
+(29)
+where, y is the direction, cosine from B × E to k, and
+h(k) is the perturbation distribution function due to the
+magnetic field. Substituting Eq. (29) in (3) gives a pair
+of coupled equations that can be solved iteratively [33]
+gi+1(k) = Si(gi(k) − (−e)E
+ℏ
+( ∂f0
+∂k ) + βSi(hi(k))
+So(k) (1 + β2)
+,
+(30)
+hi+1(k) = Si(hi(k) + β (−e)E
+ℏ
+( ∂f0
+∂k ) − βSi(gi(k))
+So(k) (1 + β2)
+, (31)
+where, β =
+(−e)ν(k)B
+ℏkSo(k) , and B is the applied magnetic
+field. The expression for the Hall mobility and the Hall
+scattering factor can be written as [60]
+µH = 1
+B
+�
+ν(ε) DS(ε) h(ε) dε
+�
+ν(ε) DS(ε) g(ε) dε ,
+(32)
+rH = µH
+µ ,
+(33)
+where, µH and µ are the Hall and the drift mobility,
+respectively, and rH is the Hall scattering factor. This
+solution gives a more accurate result for the Hall scatter-
+ing factor compared with the other expressions based on
+the RTA [71].
+III.
+SIMULATION APPROACH
+First, we calculate the band structure using the k.p
+technique as discussed in Sec. II A and then analytically
+fit it to produce a smooth curve for the calculation of
+group velocity [62]. By using Eq. (34), the Fermi level is
+determined with a smooth band structure obtained after
+the analytical fitting, where V0 represents the volume of
+the cell and εc represents the energy at the bottom of the
+CB.
+n = 1
+V0
+� ∞
+εc
+DS(ε)f(ε)dε .
+(34)
+Equations (9), (13), (16), (18), (19), (24) are used to
+calculate the various scattering rates, and the perturba-
+tion in the distribution function is determined using Eq.
+(7) with Si(k) = 0. The term g(k), is calculated itera-
+tively until g(k) converges and it gives results beyond the
+RTA.
+IV.
+RESULTS AND DISCUSSION
+A.
+Dispersion relation for T2SL
+We calculate the band structure of an InAs/GaSb-
+based T2SL, with layer widths nML/mML, where n, m
+= 8, 8 correspondingly, using the 8 × 8 k.p technique
+as described in Sec. II A, at a temperature of T=77 K,
+and the results are shown in Fig. 4. In a single period
+of 8ML/8ML InAs/GaSb configuration, the thickness of
+
+Start
+Calculation of Input Parameters and
+Band Structure
+Analytical Fitting of Band Structure
+Calculation of Fermi-Level
+Calculation of Various Scattering
+Rates
+Si (g(k))=0
+Perturbation g(k) of the
+distribution function
+(-e)Ecfo
+Si(gi(k) -
+h 
+Cok
+gi(k)
+7
+So(k) +
+Tel(k)
+No
+g(k) Converged ?
+Si (g(k))=0
+Yes
+Transport Coefficients
+Stop7
+TABLE I.
+Material parameters required to calculate the electronic band structure using the k.p technique at T = 77 K
+[37, 72–74]
+Quantity
+Unit
+InAs
+GaSb
+Lattice constant
+˚A
+6.0584
+6.0959
+Effective mass of electron (m∗
+e)
+-
+0.022
+0.0412
+Energy band gap at 0 K
+eV
+0.418
+0.814
+Luttinger parameter γ1
+-
+19.4
+11.84
+Luttinger parameter γ2
+-
+8.545
+4.25
+Luttinger parameter γ3
+-
+9.17
+5.01
+Varshini Parameter α
+meV/K
+0.276
+0.417
+Varshini Parameter β
+K
+93
+140
+Interband mixing parameter Ep
+eV
+21.5
+22.4
+Spin-orbit splitting (SO)
+eV
+0.38
+0.76
+Valence band offset (VBO)
+eV
+-0.56
+0
+(a) (110)
+(b) (001)
+FIG. 4. Calculated band structure in the first BZ using the
+periodic boundary condition of a T2SL based on 8 ML InAs
+/ 8 ML GaSb at T = 77 K using the k.p method (a) The
+in-plane dispersion and (b) the out-of-plane dispersion.
+FIG. 5.
+DOS calculated using the k.p method in an
+InAs/GaSb SL as a function of energy.
+The inset clearly
+shows how the DOS for the carriers in the VB varies as a
+function of energy.
+each layer is roughly 24 ˚A. The dispersion curve along the
+in-plane and the out-of-plane directions are presented in
+Figs. 4(a) and 4(b), respectively and the calculated band
+gap is 270 meV. The band gap of 270 meV corresponds
+to a cut-off wavelength of 4.59 µm which confirms that
+our model is best suited for the MWIR spectrum. In Fig.
+5 we show the DOS of an SL as a function of energy, cal-
+culated using the k.p method. Table I summarizes the
+values of the parameters, utilized in the k.p calculations.
+B.
+Scattering rates
+In Fig. 6, we show the dependence of scattering rates
+with energy for the temperatures of 77 K, 300 K, and
+500 K at doping densities of ND = 1 × 1013 cm−3
+and ND = 2 × 1017 cm−3.
+Here, we show the rela-
+tive importance of each of the scattering mechanisms in
+a T2SL. The IRS mechanism is the strongest scatter-
+ing mechanism for low as well as high doping densities
+at a temperature of 77 K and 300 K as shown in Fig.
+6. At a temperature of 77 K and a doping density of
+ND = 1 × 1013 cm−3, the most dominant contributions
+are due to the IRS followed by the ADP and the POP
+scattering mechanisms. The II scattering mechanism is
+the least significant scattering mechanism at this partic-
+ular temperature and doping density, whereas it has a
+significant contribution at higher doping densities.
+At room temperature, the average energy of the carri-
+ers is 3/2kBT = 0.0388 eV , indicating that the majority
+of the carriers are in the low-energy region. Hence, it is
+clear from Fig. 6(e) that at room temperature, the signif-
+icant contribution comes from the IRS mechanism as well
+as the POP scattering mechanism. Both scattering mech-
+anisms are dominant at this temperature, and the dom-
+inance of the POP scattering mechanism changes with
+respect to temperature and the average energy of the
+carriers, which signifies that the POP scattering mech-
+anism plays a significant role in such a T2SL structure.
+As a result, it is important to note that the POP scatter-
+ing mechanism is the primary factor limiting the carrier’s
+mobility from room temperature to higher temperatures.
+At a temperature of 500 K, the average energy of
+the carriers is 0.0646 eV and, most of the carrier con-
+tributes to the POP scattering mechanism hence, this
+
+Energy (eV)
+Ec
+Eg
+0
+HH
+.1
+-0.5
+0
+0.5
+r /a
+TEnergy (eV)
+0.4
+Ec
+0.2
+HH
+0
+-0.2
+-0.4
+-1
+0
+1
+T /L1020
+(ev-1cm*
+1018
+1020
+DOS
+1019
+@1018
+sO
+-0.02405
+-0.02340
+Energy (eV)
+-0.05 0.25
+0.35
+0.45
+Energy (eV)8
+(a) T=77 K
+(b) T=300 K
+(c) T=500 K
+(d) T=77 K
+(e) T=300 K
+(f) T=500 K
+FIG. 6.
+Scattering rates for 8ML/8ML InAs/GaSb based T2SL with roughness parameters Λ = 3 nm and ∆ = 0.3 nm as
+a function of electron energy at
+(a) T = 77 K and ND = 1 × 1013 cm−3
+(b) T = 300 K and ND = 1 × 1013 cm−3
+(c)
+T = 500 K and ND = 1 × 1013 cm−3 (d) T = 77 K and ND = 2 × 1017 cm−3 (e) T = 300 K and ND = 2 × 1017 cm−3 and (f)
+T = 500 K and ND = 2 × 1017 cm−3 .
+TABLE II. Material parameters required to compute the various scattering rates [32, 73, 75–78].
+Parameter
+Unit
+InAs
+GaSb
+Elastic constant c11
+GPa
+832.9
+884.2
+Elastic constant c12
+GPa
+452.6
+402.6
+Elastic constant c44
+GPa
+395.9
+432.2
+Acoustic deformation potential
+eV
+4.90
+6.70
+Low freq. dielectric constant
+-
+14.55
+15.00
+High freq. dielectric constant
+-
+11.78
+13.80
+Piezoelectric coefficient
+C/m2
+0.045
+0.126
+Optical phonon frequency
+1/cm
+240 (LO)a, 218 (TO)b
+193 (LO)a, 215 (TO)b
+a LO : Longitudinal Optical Phonon Frequency.
+b TO : Transverse Optical Phonon Frequency.
+again demonstrates that the POP scattering mechanism
+is the most dominant scattering mechanism for T2SL at
+and beyond the ambient temperature for both doping
+densities, as shown in Figs. 6(c) and 6(f). Figure 6 shows
+a sudden change in the POP scattering rate after partic-
+ular energy, which is because if the electron energy is
+less than the POP energy, the electron can only scat-
+ter by the absorption of the optical phonons, whereas if
+the energy is greater than the phonon energy, the elec-
+tron can scatter by both the absorption and the emission
+of phonons, where the optical phonon energy is deter-
+mined using ℏωP OP . The PZ scattering is the least domi-
+nant scattering mechanism at higher doping densities, as
+shown in Figs. 6(d), 6(e), 6(f). Table II lists the ma-
+terial parameters that are used to compute the various
+scattering rates.
+It is generally known that the ADP scattering mech-
+anism becomes substantial at temperatures of 77 K and
+above, reducing electron mobility. Therefore, it is also
+important to include the effect of the ADP scattering
+mechanism, which is significant near the room tempera-
+ture for low as well as high doping densities, which was
+
+PZ
+(sec)
+POP
+IRS
+ADP
+Total
+rate (
+Scattering
+1011
+107
+0.03
+0.055
+0.1
+Energy (eV)1015
+PZ
+(sec
+POP
+IRS
+ADP
+Total
+rate 
+Scattering
+1011
+107
+0.03
+0.055
+0.1
+Energy (eV)1015
+PZ
+(sec
+POP
+IRS
+ADP
+Total
+rate (
+Scattering
+1011
+107
+0.03
+0.055
+0.1
+Energy (eV)PZ
+(sec)
+POP
+IRS
+ADP
+Total
+rate (
+Scattering
+1011
+107
+0.03
+0.055
+0.1
+Energy (eV)1015
+PZ
+(sec'
+POP
+IRS
+ADP
+Total
+rate 
+M
+Scattering
+1011
+107
+0.03
+0.055
+0.1
+Energy (eV)1015
+PZ
+(sec
+POP
+IRS
+ADP
+Total
+rate 
+Scattering
+1011
+107
+0.03
+0.055
+0.1
+Energy (eV)9
+FIG. 7.
+Calculated mobility contribution for electrons due
+to the various scattering mechanism involved in (8ML/8ML)
+InAs/GaSb T2SL as a function of temperature for ND = 9 ×
+1016 cm−3.
+FIG. 8.
+Calculated low-field electron drift mobility in
+8ML/8ML InAs/GaSb SL as a function of doping concen-
+tration for temperatures of 77 K, 120 K and 150 K.
+not highlighted in the earlier works for such SL struc-
+tures. At lower temperatures and in the thin-film sys-
+tems, the IRS scattering is considerable, and to compute
+the roughness scattering rate, we utilize a sheet carrier
+density Ns, of 4.6 × 1012 cm−2 and a doping carrier den-
+sity Nd, of 1 × 1011 cm−2 with the roughness height ∆,
+fixed at 0.3 nm, and the correlation length of the fluctu-
+ations Λ kept at 3 nm. The IRS mechanism is temper-
+ature independent, but the carrier distribution function
+depends on the temperature. Therefore, the electron mo-
+bility through the IRS mechanism is somewhat tempera-
+ture sensitive. Except for the IRS scattering rate, which
+is temperature independent, we see that all the scattering
+rates increase as the temperature rises as shown in Figs.
+6(a), 6(b), 6(c). When the temperature is either low or
+intermediate, the II scattering rate increases with an in-
+crease in the doping concentration, which suppress the
+contribution from the PZ scattering, as shown in Figs.
+6(a), 6(d), 6(b), 6(e).
+FIG. 9.
+Comparison of conductivity in a T2SL as a func-
+tion of temperature, calculated using the Rode’s and the RTA
+method for various doping concentrations.
+FIG. 10.
+Calculated temperature dependence of electronic
+mobility with IRS heights for a correlation length of 3 nm
+& ND = 9 × 1016 cm−3. The mobility due to only the IRS
+mechanism is shown.
+C.
+Electron transport parameters
+We calculate the mobility and the conductivity for
+a T2SL at various temperatures and doping concentra-
+tions.
+Figure 7 shows the contribution to the mobil-
+ity due to various scattering mechanisms calculated for
+ND = 9 × 1016 cm−3.
+To the best of our knowledge,
+the combined effect of these scattering mechanisms in a
+T2SL structure has never been shown in earlier works.
+These five types of scattering mechanisms show their sig-
+nificant contribution to the overall mobility calculation.
+From Fig. 7 it turns out that the scattering mechanism
+with the lowest mobility values is the dominant one in
+that temperature range. Therefore, starting at a tem-
+perature of 150 K, the POP scattering mechanism is the
+most dominant scattering mechanism until 700 K; below
+77 K, a significant contribution to the mobility comes
+from the II scattering and the IRS mechanisms as shown
+
+1010
+Overall
+RTAO
+pOp
+ADP
+PZO
+IRS
+ Overall + RTA + Il+ POP + ADP + PZ + IRS
+103
+102
+300
+500
+700
+Temperature (K)
+106
+102
+20
+150
+300
+500
+700
+Temperature (K)6
+10
+△77K口120K*150K
+104
+10°
+1012
+1014
+1016
+1018
+Doping Concentration (cm-3104
+Conductivity (S/cm)
+-. 1×1016 cm-3-*. 1×1016 cm-3
+- - 9×1016 cm-3...*.. 9×1016 cm-3
+101
+....0..0....0
+:::
+10-2
+Rode
+RTA
+10-5
+20
+150
+300
+500
+700
+Temperature (K)Mobility (cm? / V-sec)
+A=0.1nm-0-A=0.3nm-0A=0.5nm
+△=0.7nm
+106
+105
+104
+103
+20
+150
+300
+Temperature (K)10
+FIG. 11.
+Calculated mobility for electrons in an 8ML
+InAs/8ML GaSb SL as a function of temperature and cor-
+relation length for an IRS height of 0.3 nm with ND =
+9 × 1016 cm−3. Here, the mobility due to only the IRS mech-
+anism is shown.
+FIG. 12. Temperature dependence of electron Hall mobility
+in a T2SL calculated using the Rode’s and the RTA method
+at B = 0.69 T for various doping concentrations.
+in Fig. 7.
+In case of II scattering mechanism, with increasing
+temperature, the electron density increases exponentially
+and causes growth in the screening length.
+As a re-
+sult, the mobility at low temperatures increases sharply
+with rising temperatures because the scattering rates are
+inversely related to the square of the screening length.
+Since the POP scattering mechanism is more prominent
+above 150 K; hence the overall mobility is reduced as
+shown in Fig. 7. In Fig. 7, we also compare the mo-
+bility computed using the RTA approach to the overall
+mobility calculated using Rode’s method and it is found
+that in the RTA approach, the mobility is underestimated
+because the POP scattering mechanism is inelastic and
+nonrandomizing, making it impossible to characterize the
+perturbation in the distribution function using the relax-
+FIG. 13. Hall scattering factor versus temperature at B= 0.69
+T for ND = 9 × 1017cm−3.
+FIG. 14. Hall scattering factor as a function of temperature
+and carrier concentration at B= 0.69 T.
+ation time. The POP scattering mechanism becomes in-
+significant at low temperatures, resulting in nearly com-
+parable mobilities determined using the RTA and Rode’s
+iterative technique.
+In Fig. 8, we demonstrate the overall mobility versus
+doping concentration at different temperatures and em-
+phasize on the mobility at 77K, which is the usual operat-
+ing temperature of most high-performance IR detectors.
+The graph illustrates a decrease in mobility as the doping
+concentration increases due to a rise in the number of ion-
+ized centers. As we raise the temperature, the mobility
+diminishes as expected because at higher temperatures
+the phonon scattering increases. The mobility values do
+not differ significantly for low carrier concentrations be-
+cause the II scattering mechanism is less significant at
+this range and the primary contributions for lower doping
+concentration at low temperatures come from the PZ and
+the ADP scattering mechanisms, while at greater doping
+concentrations, the II scattering mechanism is compara-
+ble to the ADP and the PZ scattering mechanisms. The
+mobility owing to the II scattering mechanism is a de-
+
+A=30nm
+A=20nm
+^=12nm
+Correlation lengthN
+106
+A=8nm
+Λ=6nm
+Λ=3nm
+Λ=2nm
+103
+A=1nm
+20
+150
+300
+Temperature
+(K)Hall mobility (cm? / V-sec)
+105
+D—1×1013 cm-3——1×1013 cm-3
+-0- 1×1016 cm-3-★- 1×1016 cm-3
+.O..9×1016 cm-3...*...9×1016 cm-3
+104
+Rode
+RTA
+20
+150
+300
+Temperature (K)Hall scattering factor r
+.o..Rode ..o..RTA
+1.00
+0.75
+20
+100
+200
+Temperature (K)3
+Carrier conc. (cm*
+1016
+H
+1014
+2
+0.5
+12
+10
+30
+77
+120
+150
+200
+Temperature(K)11
+creasing function of ND, the mobility begins to decrease
+as ND exceeds 1 × 1016 cm−3.
+In Fig. 9, we show the conductivity versus temperature
+for the doping concentrations of ND = 1 × 1013 cm−3,
+ND = 1 × 1016 cm−3 and ND = 9 × 1016 cm−3, re-
+spectively, and to demonstrate the supremacy of our
+approach, we compare the results obtained using both
+the Rode’s and the RTA method. At higher tempera-
+tures, the difference in the result of Rode’s method and
+the RTA is due to the POP scattering mechanism, the
+POP scattering is weaker at lower temperatures hence
+both the RTA and the Rode exhibit the same conduc-
+tivity. We demonstrate that the conductivity in a T2SL
+increases with an increase in the carrier concentration
+but decreases as we increase the temperature.
+In Figs. 10 and 11, we show the mobility due to only
+the IRS mechanism.
+The calculated mobilities are vi-
+tal functions of the roughness parameters and the carrier
+scattering. The existing mobility calculations reveal that,
+up to temperatures where the POP scattering mechanism
+takes over, the IRS is the dominating scattering mecha-
+nism in T2SL. The screening is included in our calcu-
+lation using Thomas-Fermi screening which lowers the
+scattering rates and increases the mobility. As illustrated
+in Fig. 10, the mobility is shown to be strongly reliant
+on the roughness height ∆, and decreases monotonically
+with increasing ∆, and is proportional to ∆−2.
+Figures 10 and 11 show that at low temperatures, the
+mobility rises since the value of ∂f
+∂ε is an ascending func-
+tion of temperature and the denominator of Eq. (26) is
+virtually constant at lower temperatures. Also, the elec-
+tron density increases at higher temperatures and hence
+the mobility drop smoothly. Figure 11 shows that the
+mobility is high for smaller values of correlation length
+Λ, and drops rapidly as the correlation length of rough-
+ness increases until it reaches a saturation point. The
+mobility reaches its maximum value at roughly 50 K for
+smaller values of Λ, and this maximum point moves to-
+ward the higher temperatures for greater values of Λ.
+The Hall mobility in InAs/GaSb T2SLs is depicted in
+Fig. 12. At temperatures above 50 K, the mobility re-
+duces as expected from a combination of the ADP and
+the POP scattering mechanisms. In T2SL, the mobility
+increases with decreasing temperature, preferable to the
+T −3/2 dependency associated with the phonon scattering.
+The greater temperature dependency of the electron mo-
+bility in InAs/GaSb-based T2SL may indicate stronger
+electron-phonon coupling than in the bulk material. The
+increased mobility near 50 K could be attributed to a
+longer scattering time or a lower electron-effective mass
+at the CB edge.
+When the Hall scattering factor rH, deviates signifi-
+cantly from unity, it indicates that to derive the elec-
+tron drift mobility from the experimentally calculated
+Hall mobility data, the Hall scattering factor must be
+precisely determined. Figure 13 shows the predicted val-
+ues of the Hall scattering factor against the temperature
+at B = 0.69 T for ND = 9 × 1017 cm−3, while Fig. 14
+depicts the Hall scattering factor as a function of tem-
+perature and the carrier concentration at B = 0.69 T.
+To the best of our knowledge, calculations of the Hall
+scattering factor in such SLs have not been performed
+yet in earlier works. The contribution of various scat-
+tering mechanisms decides the Hall scattering factor’s
+value. Figures 13 and 14 indicate that the value of rH at
+low temperatures deviates significantly from unity, while
+many researchers use one as an ideal value for a variety
+of calculations and studies, which is not accurate. The
+carrier concentration and the drift mobility may both be
+overestimated and underestimated when the Hall scatter-
+ing factor is used as unity. The Hall scattering factor, in
+our calculation, fluctuates between the values as low as
+0.3 at low temperature and electron concentration, and
+as high as 1.48 and even more at high temperature and
+electron concentration as shown in Fig. 14. Therefore,
+it is worth pointing out that, while evaluating the car-
+rier concentration and the drift mobility in such SLs, one
+must use caution.
+In this work, we calculate the precise values of the
+Hall scattering factor and show that for a doping value
+of ND = 9 × 1017 cm−3, the computed values of rH are
+0.914, 0.952 and 1.01 at temperatures of 77 K, 150 K
+and 190 K, respectively, as also depicted in Fig.
+13.
+At higher temperatures, the value of the Hall scatter-
+ing factor is more than unity, indicating that the drift
+mobility is lower than the Hall mobility, implying that
+the phonon-assisted scattering mechanisms are substan-
+tial and diminish the drift mobility. As shown in Fig.
+14, at temperatures of 30 K and 77 K, the Hall scat-
+tering factor is equal to 0.335 & 0.638 for lower doping
+concentrations of ND = 1 × 1012 cm−3 and it is equal
+to 0.369 & 0.691 with slightly higher doping concentra-
+tions of ND = 5×1015 cm−3 which signifies that the Hall
+scattering factor increases as the temperature and elec-
+tron concentrations rise, but as we increase the carrier
+concentration beyond 3 × 1017 cm−3, the Hall scattering
+factor starts decreasing. The higher electron concentra-
+tion causes a rapid variation in the Hall factor.
+V.
+CONCLUSION
+In this paper, we developed the Rode algorithm on the
+BTE in conjunction with the k.p band structure and the
+EFA for a detailed computation of the carrier mobility
+and conductivity, in order to primarily unravel two cru-
+cial insights. First, the significance of both elastic and in-
+elastic scattering mechanisms, particularly the influence
+of the IRS and POP scattering mechanisms in techno-
+logically relevant SL structures. Second, the structure
+specific Hall mobility and Hall scattering factor, which
+reveals that temperature and carrier concentrations sig-
+nificantly affect the Hall scattering factor, which devi-
+ates significantly from unity, i.e., from 0.3 to about 1.48,
+even for small magnetic fields. This reinforces the cau-
+tion that should be exercised when employing the Hall
+
+12
+scattering factor in experimental estimations of drift mo-
+bilities and carrier concentrations. Our research offers a
+comprehensive microscopic understanding of carrier dy-
+namics in such technologically relevant SLs. Our model
+also provides highly accurate and precise transport pa-
+rameters beyond the RTA and hence paves the way to
+develop physics based device modules for MWIR pho-
+todetectors.
+ACKNOWLEDGMENTS
+The authors acknowledge funding from ISRO under
+the ISRO-IIT Bombay Space Technology Cell.
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+

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+IMPROVED FRONT STEEPEST DESCENT
+FOR MULTI-OBJECTIVE OPTIMIZATION
+Matteo Lapucci
+Global Optimization Laboratory (GOL)
+Department of Information Engineering
+University of Florence
+Via di Santa Marta, 3, 50139, Florence, Italy
+[email protected]
+Pierluigi Mansueto
+Global Optimization Laboratory (GOL)
+Department of Information Engineering
+University of Florence
+Via di Santa Marta, 3, 50139, Florence, Italy
+[email protected]
+ABSTRACT
+In this paper, we deal with the Front Steepest Descent algorithm for multi-objective optimization.
+We point out that the algorithm from the literature is often incapable, by design, of spanning large
+portions of the Pareto front. We thus introduce some modifications within the algorithm aimed to
+overcome this significant limitation. We prove that the asymptotic convergence properties of the
+algorithm are preserved and numerically show that the proposed method significantly outperforms
+the original one.
+Keywords Multi-objective optimization · Steepest descent · Pareto front
+Mathematics Subject Classification (2020) 90C29 · 90C30
+1
+Introduction
+In this paper, we are interested in optimization problems of the form
+min
+x∈Rn F(x) = (f1(x), . . . , fm(x))T ,
+(1)
+where F : Rn → Rm is a vector-valued continuously differentiable function. We are thus dealing with smooth,
+unconstrained multi-objective optimization problems, where many functions have to be simultaneously minimized and
+Pareto’s efficiency concepts have to be considered to establish optimality. We refer the reader to [8] for an introduction
+to multi-objective optimization.
+Multi-objective descent methods [9–11, 16] constitute a class of algorithmic approaches designed to tackle these
+problems; these approaches basically extend classical iterative optimization algorithms for scalar optimization to the
+multi-objective setting. Descent methods are receiving increasing attention and have consistently become significant
+alternatives to scalarization methods [6, 7, 15] and evolutionary algorithms [4]. This is particularly true for recent
+versions of descent approaches that are specifically designed to handle sets of points and to construct an approximation
+of the entire Pareto front, rather than a single solution.
+In this short manuscript, we focus on the Front Steepest Descent (FSD) algorithm proposed in [2]. In particular, we
+argue that, although being far superior than the original single point steepest descent algorithm for multi-objective
+optimization [10], FSD as defined in [2] has limited exploration capabilities and it is quite frequently unable to span
+large portions of the Pareto front.
+We thus propose small but crucial modifications to the algorithm, that allow to turn it tremendously effective at spanning
+the entire Pareto front, regardless of the starting set of points. We show that the proposed approach still enjoys the nice
+convergence guarantees of the original FSD.
+The rest of the paper is organized as follows: in Section 2, we summarize the FSD algorithm, recalling its convergence
+properties; we then point out in Section 2.1 that in certain, common situations the algorithm is unable to span large
+arXiv:2301.03310v1  [math.OC]  9 Jan 2023
+
+Improved Front Steepest Descent for MOO
+MATTEO LAPUCCI AND PIERLUIGI MANSUETO
+portions of the Pareto front. In Section 3 we introduce the novel strategy for generating nondominated solutions within
+FSD and we provide the convergence analysis for the resulting algorithm in Section 3.1. In Section 4, we present the
+results of numerical experiments showing that the proposed modification significantly improves effectiveness and
+consistency of the FSD algorithm. We finally give some concluding remarks in Section 5.
+2
+The Front Steepest Descent algorithm
+The Front Steepest Descent algorithm [2] was designed to solve problem (1) according to Pareto’s optimality concepts.
+Given the standard partial ordering in Rm, i.e.,
+u ≤ v ⇐⇒ uj ≤ vj, ∀ j = 1, . . . , m,
+u < v ⇐⇒ uj < vj, ∀ j = 1, . . . , m,
+u ≨ v ⇐⇒ u ≤ v ∧ u ̸= v,
+the aim is to find solutions ¯x ∈ Rn that satisfy the following properties, listed in decreasing order of strength:
+• Pareto optimality: ∄ y ∈ Rn s.t. F(y) ≨ F(¯x);
+• Weak Pareto optimality: ∄ y ∈ Rn s.t. F(y) < F(¯x);
+• Pareto stationarity: min
+d∈Rn
+max
+j=1,...,m ∇fj(¯x)T d = 0.
+In fact, there typically exist many Pareto optimal solutions (the Pareto set) that account for different trade-offs between
+the contrasting objectives; these trade-offs, that constitute in the objectives space the Pareto front, can a posteriori be
+evaluated by the decision makers, who are thus willing to have the broadest possible range of available options.
+FSD method specifically aims to construct an approximation of the entire Pareto front; the algorithm works in an iterative
+fashion, maintaining at each iteration a set Xk of solutions that are mutually nondominated, i.e., for any x ∈ Xk there
+is no y ∈ Xk such that F(y) ≨ F(x).
+The points for the set Xk+1 are computed carrying out search steps starting from the points ˆx ∈ Xk along:
+• the steepest common descent direction [10]:
+v(ˆx) = arg min
+d∈Rn
+max
+j=1,...,m ∇fj(ˆx)T d + 1
+2∥d∥2;
+(2)
+• the steepest partial descent directions [1,2]: given I ⊂ {1, . . . , m},
+vI(ˆx) = arg min
+d∈Rn
+max
+j∈I ∇fj(ˆx)T d + 1
+2∥d∥2.
+(3)
+The use of equality notation in the definition of steepest descent directions is justified by the uniqueness of the solution
+set for the above optimization problems (the objective is strongly convex and continuous). Given any subset of objectives
+I, a partial descent direction exists if
+θI(ˆx) = min
+d∈Rn max
+j∈I ∇fj(ˆx)T d + 1
+2∥d∥2 < 0;
+of course, the steepest common descent direction v(ˆx) and the corresponding θ (ˆx) are considered when I = {1, . . . , m}.
+Both mappings vI(ˆx) and θI(ˆx) are continuous [10].
+The instructions of the FSD procedure are summarized in Algorithm 1. In brief, at each iteration k, all points in the
+current set of nondominated solutions, Xk, are considered; for each one of these points, xc, a line search along the
+steepest partial descent direction is carried out for any subset of objectives I ⊆ {1, . . . , m} such that θI(xc) < 0; in
+addition, a subset I is only considered for xc if the point is nondominated with respect to that subset of objectives.
+The line search is an Armijo-type procedure whose scheme is reported in Algorithm 2. Given a nondominated point and
+a search direction w.r.t. the objectives in I, the algorithm returns a new point such that it is “sufficiently nondominated”.
+The obtained point is added to the set of nondominated points, while all the points that are now dominated by it are
+filtered out.
+Algorithm 2 enjoys the following finite termination properties.
+2
+
+Improved Front Steepest Descent for MOO
+MATTEO LAPUCCI AND PIERLUIGI MANSUETO
+Algorithm 1: FrontSteepestDescent
+1 Input: F : Rn → Rm, X0 set of mutually nondominated points w.r.t. F.
+2 k = 0
+3 while a stopping criterion is not satisfied do
+4
+ˆXk = Xk
+5
+forall xc ∈ Xk do
+6
+forall I ⊆ {1, . . . , m} such that
+• ∄y ∈ ˆXk s.t. FI(y) ≨ FI(xc) and
+• θI(xc) < 0
+7
+do
+8
+α = ArmijoLS(F(·), I, ˆXk, xc, vI(xc), θI(xc))
+9
+ˆXk = ˆXk \ {y ∈ ˆXk | F(xc + αvI(xc)) ≨ F(y)} ∪ {xc + αvI(xc)}
+10
+Xk+1 = ˆXk
+11
+k = k + 1
+12 return Xk
+Algorithm 2: ArmijoLS
+1 Input: F : Rn → Rm, I ⊆ {1, . . . , m}, ˆX set of mutually nondominated points w.r.t. F, xc ∈ ˆX, vI(xc) ∈ Rn,
+θI(xc) ∈ R, α0 > 0, δ ∈ (0, 1), γ ∈ (0, 1).
+2 α = α0
+3 Let ˆXI be the set of points in ˆX that are mutually nondominated w.r.t. FI
+4 while ∃ y ∈ ˆXI s.t. FI(y) + 1γαθI(xc) < FI(xc + αvI(xc)) do
+5
+α = δα
+6 return α
+Proposition 1 ( [2, Proposition 4]). Let I ⊆ {1, . . . , m}, ˆX be a set of mutually nondominated solutions containing
+xc; xc is also nondominated w.r.t. FI and it is such that θI(xc) < 0. Then, ∃ α > 0, sufficiently small, such that
+FI(y) + 1γαθI(xc) ≮ FI(xc + αvI(xc)),
+∀ y ∈ ˆXI,
+with ˆXI being the set of points in ˆX that are mutually nondominated w.r.t. FI. Furthermore, the produced point
+xc + αvI(xc) is nondominated by any point in ˆX with respect to F.
+Remark 1. An improved version of Algorithm 2 was also proposed in [2], which is based on an extrapolation strategy
+and allows to possibly obtain many nondominated solutions along the search direction. When used within Algorithm 1,
+the extrapolation technique does not alter theoretical convergence results, but the resulting algorithm is reported to be
+significantly more effective.
+Now, we shall recall the convergence properties of Algorithm 1, which are based on the concept of linked sequence [14].
+Definition 1. Let {Xk} be the sequence of sets of nondominated points produced by Algorithm 1. We define a linked
+sequence as a sequence {xjk} such that, for any k = 1, 2, . . ., the point xjk ∈ Xk is generated at iteration k − 1 of
+Algorithm 1 by the point xjk−1 ∈ Xk−1.
+Proposition 2 ( [2, Proposition 5]). Let us assume that there exists x0 ∈ X0 such that
+• x0 is not Pareto stationary;
+• the set L(x0) = �m
+j=1{x ∈ Rn | fj(x) ≤ fj(x0)} is compact.
+Let {Xk} be the sequence of sets of nondominated points produced by Algorithm 1. Let {xjk} be a linked sequence,
+then it admits limit points and every limit point is Pareto-stationary for problem (1).
+3
+
+Improved Front Steepest Descent for MOO
+MATTEO LAPUCCI AND PIERLUIGI MANSUETO
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+f1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+f2
+(a)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+f1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+f2
+(b)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+f1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+f2
+(c)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+f1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+f2
+(d)
+Figure 1: Pareto fronts obtained by the FSD algorithm on the convex JOS problem (n = 5). (a) FSD starts from 1 Pareto
+point; (b) FSD starts from 2 Pareto points; (c) 3 independent FSD runs, started from 3 different random points; (d) 3
+independent runs of FSD with the extrapolation strategy, started from the same 3 random points as in (c).
+2.1
+FSD may not span the Pareto front
+The FSD algorithm constitutes, in practice, a significant improvement w.r.t. the simple multi-start steepest descent
+strategy for multi-objective optimization. However, in experimental settings, it is not uncommon to observe situations
+where FSD is unable to retrieve large portions of the Pareto front.
+Here, we highlight this shortcoming and argue that it is the direct result of algorithmic design. In particular, the first
+condition at step 6 of Algorithm 1 makes the outcome of the algorithm very strongly dependent on the starting point(s).
+When a point xc is considered for exploration in Algorithm 1, a partial descent direction obtained according to the
+subset of objectives I ⊆ {1, . . . , m} is only considered if xc is nondominated within Xk w.r.t. FI; in other words, there
+is no y ∈ Xk such that FI(y) ≨ FI(xc). This condition was required by the authors of [2] in order to establish finite
+termination properties for the line search (Algorithm 2).
+Unfortunately, that same condition results in a limited fraction of points in Xk to be used for starting a partial descent
+search. This fact can be visualized, with very extreme outcomes, in the bi-objective case; indeed, when m = 2, for
+each of the two proper subsets of indices, I1 = {1} and I2 = {2} there is only one point that satisfies the (partial)
+nondominance condition: xI1 = arg minx∈Xk f1(x) and xI2 = arg minx∈Xk f2(x).
+Thus, partial descent is only carried out starting from the two current extreme points in the Pareto front. Moreover,
+these partial descent steps will only allow to explore, outwards, the extreme parts of the current front approximation,
+whereas the other descent step will mainly drive points to Pareto stationarity; as a result, even large holes within the
+current solutions set cannot be filled.
+Taking the reasoning to the extreme, let us assume that the starting set of solutions already lies on the Pareto front; if the
+set contains only one point, then by repeated partial descent w.r.t. I1 and I2 the entire Pareto front can be spanned quite
+uniformly; this situation is depicted in Figure 1a. If, on the other hand, there are two starting solutions, possibly far
+away from each other in the objective space, then only the extreme parts of the front will be spanned, while the gap
+4
+
+Improved Front Steepest Descent for MOO
+MATTEO LAPUCCI AND PIERLUIGI MANSUETO
+between the two points is not tackled (Figure 1b). Of course, the same reasoning applies with more than two starting
+points.
+The paradoxical behavior of the algorithm is such that it might be convenient to start far away from the Pareto front.
+In this way, FSD may have many iterations at its disposal to increase the size of the set Xk and uniformly span the
+objectives space; points are then driven to Pareto stationarity thanks to steps carried out considering I = {1, 2}.
+Anyhow, the results are still influenced, somewhat randomly, by the starting solutions, as shown in Figure 1c. Moreover,
+the extreme parts of the front are always spanned much more densely than the central one. We shall remark that, as the
+intermediate regions of the front often provide the most interesting trade-offs to users, this is a very significant issue in
+practice.
+The extrapolation technique proposed in [2] might allow to partly alleviate the issue discussed here, as much more
+nondominated solutions are obtained at each iteration; however, it is again the exploration of the extreme regions that is
+mainly enhanced and sped up, with possibly overall counterproductive results (Figure 1d).
+3
+Improved Front Steepest Descent
+In Algorithm 3, we report the scheme of a modified Front Steepest Descent (IFSD) algorithm that overcomes the
+limitations of Algorithm 1 discussed in Section 2.1.
+Algorithm 3: ImprovedFrontSteepestDescent
+1 Input: F : Rn → Rm, X0 set of mutually nondominated points w.r.t. F, α0 > 0, δ ∈ (0, 1), γ ∈ (0, 1).
+2 k = 0
+3 while a stopping criterion is not satisfied do
+4
+ˆXk = Xk
+5
+forall xc ∈ Xk do
+6
+if xc ∈ ˆXk then
+7
+if θ(xc) < 0 then
+8
+αk
+c = maxh=0,1,...{α0δh | F(xc + α0δhv(xc)) ≤ F(xc) + 1γα0δhθ(xc)}
+9
+else
+10
+αk
+c = 0
+11
+zk
+c = xc + αk
+cv(xc)
+12
+ˆXk = ˆXk \ {y ∈ ˆXk | F(zk
+c ) ≨ F(y)} ∪ {zk
+c }
+13
+forall I ⊆ {1, . . . , m} s.t. θI(zk
+c ) < 0 do
+14
+if zk
+c ∈ ˆXk then
+15
+αI
+c = maxh=0,1,...{α0δh | ∀ y ∈ ˆXk ∃j ∈ {1, . . . , m} s.t. fj(zk
+c + α0δhvI(zk
+c )) < fj(y)}
+16
+ˆXk = ˆXk \ {y ∈ ˆXk | F(zk
+c + αI
+cvI(zk
+c )) ≨ F(y)} ∪ {zk
+c + αI
+cvI(zk
+c )}
+17
+Xk+1 = ˆXk
+18
+k = k + 1
+19 return Xk
+Algorithm 3 includes a bunch of modifications w.r.t. the original FSD approach:
+• for any point in Xk that is still nondominated when it is considered for exploration, a preliminary steepest
+descent step is carried out; this step exploits a classical single point Armijo line search [10];
+• further searches w.r.t. subsets of objectives start at the obtained point, as long as it is not dominated;
+• for partial descent searches, we require the obtained point to be nondominated by all other points in ˆXk.
+The idea is that, with these modifications, all points may be used to start exploration based on partial descent;
+convergence of all the produced points towards stationarity is then forced by means of the “preliminary” steepest
+descent step, that ensures the sufficient decrease. In the next section we prove that the algorithm is well defined and
+actually produces convergent sequences of points.
+5
+
+Improved Front Steepest Descent for MOO
+MATTEO LAPUCCI AND PIERLUIGI MANSUETO
+3.1
+Convergence analysis
+In this section, we provide the formal convergence analysis for Algorithm 3.
+Proposition 3. The line search at step 8 of Algorithm 3 is well defined.
+Proof. The result follows from [10, Lemma 4] and by the if condition at step 7 that ensures that θ(xc) < 0.
+Proposition 4. Step 15 of Algorithm 3 is well defined if zk
+c is nondominated with respect to points in ˆXk.
+Proof. Let y be any point in ˆXk; if F(y) = F(zk
+c ), then by [10, Lemma 4] and the condition θI(zk
+c ) < 0, there exists
+¯α > 0 such that FI(zk
+c + αvI(zk
+c )) < FI(zk
+c ) = FI(y) for all α < ¯α; thus there exists h sufficiently large such that
+fj(zk
+c +α0δhvI(zk
+c )) < fj(y) for all j ∈ I. If, on the other hand, there exists j ∈ {1, . . . , m} such that fj(zk
+c ) < fj(y),
+then by the continuity of F there exists α = α0δh sufficiently small such that fj(zk
+c + αvI(zk
+c )) < fj(y). Thus, the
+condition can be satisfied for all y ∈ ˆXk and αI
+c is the minimum of the corresponding values of α0δh.
+Proposition 5. If Xk contains mutually nondominated points with respect to F, then ˆXk contains nondominated points
+at any time during iteration k; thus step 15 is always well defined and Xk+1 is finally a set of nondominated solutions.
+Proof. At iteration k, the set ˆXk is initialized with the nondominated points Xk; then, it is only updated at steps 12
+and 16. At step 12, either zk
+c = xc, and the set is not modified, or, by the definition of αk
+c, zk
+c dominates xc, which in
+turn was nondominated. Thus, the added point zk
+c is nondominated, while all the newly dominated points are removed.
+At step 16, the added point zk
+c + αI
+cvI(zk
+c ) is nondominated by the definition of αI
+c; all the newly dominated points are
+removed. Thus, ˆXk always contains mutually nondominated solutions. By Proposition 4 step 15 is therefore always
+well defined. Moreover, since Xk+1 = ˆXk at the end of the iteration, Xk+1 inherits the nondominance property from
+ˆXk.
+Lemma 1. After step 12 of Algorithm 3, zk
+c belongs to ˆXk. Moreover, for all ˜k > k, there exists y ∈ X˜k such that
+F(y) ≤ F(zk
+c ).
+Proof. The first assertion of the proposition trivially follows from the update rule of ˆXk, at step 12. Now, either
+zk
+c ∈ X˜k or zk
+c /∈ X˜k; in the former case, we trivially have y = zk
+c ; otherwise, we can notice that, by the instructions of
+the algorithm, any set X˜k, ˜k > k, is the result of repeated application of steps 12 and 16, starting from ˆXk at some point
+when zk
+c ∈ ˆXk. When zk
+c was removed from the set, a point y1 was certainly inserted such that F(y1) ≤ F(zk
+c ). Then,
+either y1 ∈ X˜k, or y1 was removed when a point y2 such that F(y2) ≤ F(y1) was added. By recursively applying
+the reasoning, we have that there is certainly a point yt ∈ X˜k such that F(yt) ≤ F(yt−1) ≤ . . . ≤ F(y2) ≤ F(y1) ≤
+F(zk
+c ). This completes the proof.
+Proposition 6. Let X0 be a set of mutually nondominated points and x0 ∈ X0 be a point such that the set L(x0) =
+�m
+j=1{x ∈ Rn | fj(x) ≤ fj(x0)} is compact. Let {Xk} be the sequence of sets of nondominated points produced by
+Algorithm 3. Let {xjk} be a linked sequence, then it admits limit points and every limit point is Pareto-stationary for
+problem (1).
+Proof. For any k, either x0 ∈ Xk or x0 /∈ Xk. In the former case, since all points in Xk are mutually nondominated,
+we certainly have xjk ∈ L(x0). Otherwise, by a similar reasoning as in the proof of Lemma 1, we have that there is a
+point yk ∈ Xk such that F(yk) ≤ F(x0); since yk does not dominate xjk, we have that there exists h ∈ {1, . . . , m}
+such that fh(xjk) ≤ fh(yk) ≤ fh(x0); thus, again, xjk ∈ L(x0). Therefore the entire sequence {xjk} belongs to the
+compact set L(x0), and thus admits limit points.
+Now, let us consider a limit point ¯x of a linked sequence {xjk}, i.e., there exists K ⊆ {1, 2, . . .} such that
+lim
+k→∞
+k∈K
+xjk = ¯x.
+We assume by contradiction that θ(¯x) < 0 and thus there exists ε > 0 such that for all k ∈ K sufficiently large we have
+θ(xjk) ≤ −ε < 0. Let zjk = xjk + αjkv(xjk) the point obtained at step 11 of the algorithm starting from xjk. Now,
+αjk ∈ [0, α0], which is a compact set, thus there exists a further subsequence K1 ⊆ K such that αjk → ¯α ∈ [0, α0].
+6
+
+Improved Front Steepest Descent for MOO
+MATTEO LAPUCCI AND PIERLUIGI MANSUETO
+Moreover, function v(·) is continuous, thus v(xjk) → v(¯x) for k → ∞, k ∈ K1. Hence, taking the limits along K1 we
+also get that zjk → ¯x + ¯αv(¯x) = ¯z.
+By the definition of αjk and zjk (steps 8-11) we have that
+F(zjk) ≤ F(xjk) + 1γαjkθ(xjk).
+Taking the limits for k ∈ K1, k → ∞, recalling the continuity of θ(·), we get
+F(¯z) ≤ F(¯x) + 1γ ¯αθ(¯x) ≤ F(¯x) − 1γ ¯αε.
+(4)
+Now, given k ∈ K1, let k1(k) be the smallest index in K1 such that k1(k) > k. By Lemma 1, there exists yjk1(k) ∈
+Xk1(k) such that F(yjk1(k)) ≤ F(zjk); moreover, xjk1(k) ∈ Xk1(k); by Proposition 5, the points in Xk1(k) are mutually
+nondominated, hence there exists h(k) ∈ {1, . . . , m} such that
+fh(k)(xjk1(k)) ≤ fh(k)(yjk1(k)) ≤ fh(k)(zjk).
+Considering a further subsequence K2 ⊆ K1 such that h(k) = h for all k ∈ K2 and taking the limits, we obtain
+fh(¯x) ≤ fh(¯z).
+Putting this last result together with (4), we get
+fh(¯x) ≤ fh(¯z) ≤ fh(¯x) − γ ¯αε.
+Since ¯α ∈ [0, α0], ε > 0 and γ > 0, the above chain of inequalities can only hold if ¯α = limk→∞,k∈K2 αjk = 0.
+For all k ∈ K2 sufficiently large, we have θ(xjk) < 0 and, thus, αjk is defined at step 8. Since αjk → 0, for any
+q ∈ N, for all k ∈ K2 large enough we certainly have αjk < α0δq; thus, the Armijo condition F(xjk + αv(xjk)) ≤
+F(xjk) + 1γαθ(xjk) is not satisfied by α = α0δq, i.e., there exists ˜h(k) such that
+f˜h(k)(xjk + α0δqv(xjk)) > f˜h(k)(xjk) + γα0δqθ(xjk).
+Taking the limits along a suitable subsequence such that ˜h(k) = ˜h, we get
+f˜h(¯x + α0δqv(¯x)) ≥ f˜h(¯x) + γα0δqθ(¯x).
+Now, since q is arbitrary and θ(¯x) < 0, this is absurd by [10, Lemma 4]. The proof is thus complete.
+4
+Numerical results
+In this section, we show the results of computational experiments, supporting the discussion in Sections 2-3. The code,
+which was written in Python3, was executed on a computer with the following characteristics: Ubuntu 22.04, Intel
+Xeon Processor E5-2430 v2 6 cores 2.50 GHz, 16 GB RAM. In order to solve instances of problems (2)-(3), the Gurobi
+optimizer (version 9.5) was employed.
+We compared our approach (IFSD) to the original FSD, Algorithm 1, equipped with the base line search (Algorithm 2)
+or the extrapolation strategy (EFSD). The following parameters setting was used for line searches: α0 = 1, δ = 0.5, γ =
+10−4.
+With respect to the conceptual scheme in Algorithm 3, we employed within IFSD a strategy to limit the number of points
+used for partial descent searches, in order to improve the efficiency of the overall procedure and avoid the production
+of too many, very close solutions. In particular, we added a condition based on the crowding distance [4] to decide
+whether a point should be considered for further exploration after the steepest descent step or not.
+The benchmark used for the comparisons consists of the following unconstrained problems: CEC09_2, CEC09_3 [17],
+JOS_1 [12], MAN [13] (m = 2) and CEC09_10 (m = 3) [17]. For all the problems, we considered instances with
+values of n in {5, 10, 20, 30, 40, 50, 100, 200}. Moreover, each problem was tested twice, with different strategies for
+the initial points: a) n points are uniformly sampled from the hyper-diagonal of a suitable box; b) only the midpoint of
+the hyper-diagonal is selected. The hyper-diagonal refers to the box constituting the constraints in the bounded version
+of CEC and MAN problems, whereas it is [−100, 100]n for the JOS problem.
+In order to appreciate the relative performance and robustness of the approaches, we employed the performance
+profiles [5]. In brief, this tool shows the probability that a metric value achieved by a method in a problem is within
+a factor τ ∈ R of the best value obtained by any of the algorithms in that problem. We employed classical metrics
+for multi-objective optimization: purity, Γ–spread, ∆–spread [3] and hyper-volume [18]. Purity and hyper-volume
+7
+
+Improved Front Steepest Descent for MOO
+MATTEO LAPUCCI AND PIERLUIGI MANSUETO
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+f1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+f2
+(a)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+f1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+f2
+(b)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+f1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+f2
+(c)
+Figure 2: Pareto fronts obtained by the IFSD algorithm on the convex JOS problem (n = 5) starting from different
+initial points: (a) 1 Pareto point as in Figure 1a; (b) 2 Pareto points as in Figure 1b; (c) 3 independent runs from the
+same random points as those of Figure 1(c)-(d).
+have increasing values for better solutions: then, the corresponding profiles are produced considering the inverse of the
+obtained values.
+In Figure 2, the behavior of the proposed approach in the same setting as in Figure 1 is shown. In this example we can
+observe that now, regardless, of the starting point(s), the entire Pareto front is effectively spanned, with not even tiny
+holes.
+For a more consistent assessment of algorithms performance, we report in Figure 3 the performance profiles for the
+IFSD, FSD and EFSD algorithms on the entire benchmark of 80 problem instances.
+We observe a remarkable superiority of the proposed approach w.r.t. the original variants of the algorithm, especially in
+terms of the spread metrics, which points out that the Pareto front is indeed spanned more widely and uniformly. The
+strong hypervolume performance also supports this result. As for purity metric, the three algorithms appear to be closer,
+but we still observe a slight advantage of IFSD.
+5
+Conclusions
+In this paper, we introduced an improved Front Steepest Descent algorithm with asymptotic convergence guarantees
+similar as those of the original method. The novel algorithm is designed so as to overcome some empirically evident
+limitation of FSD, that is often unable to span large portions of the Pareto front. Numerical evidence suggests that the
+proposed procedure effectively achieves this goal.
+Future work should be focused on the integration of the proposed approach and the extrapolation strategy proposed
+in [2]. Moreover, the employment of the proposed approach within memetic procedures for global multi-objective
+optimization [13] might be considered. Finally, the algorithm defined in this work could be extended to deal with
+constrained optimization problems.
+8
+
+Improved Front Steepest Descent for MOO
+MATTEO LAPUCCI AND PIERLUIGI MANSUETO
+1
+2
+3
+4
+5
+6
+7
+8
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Cumulative
+Purity
+IFSD
+FSD
+EFSD
+(a) Purity profile
+1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Cumulative
+Hypervolume
+(b) Hypervolume profile
+1
+2
+3
+4
+5
+6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Cumulative
+-spread
+(c) Γ-spread profile
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Cumulative
+-spread
+(d) ∆-spread profile
+Figure 3: Performance profiles for the IFSD, FSD and EFSD algorithms on a benchmark of 80 multi-objective problems.
+Conflict of interest
+The authors declare that they have no conflict of interest.
+References
+[1] G. Cocchi, M. Lapucci, and P. Mansueto. Pareto front approximation through a multi-objective augmented
+lagrangian method. EURO Journal on Computational Optimization, page 100008, 2021.
+[2] G. Cocchi, G. Liuzzi, S. Lucidi, and M. Sciandrone. On the convergence of steepest descent methods for
+multiobjective optimization. Computational Optimization and Applications, pages 1–27, 2020.
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+[8] G. Eichfelder. Twenty years of continuous multiobjective optimization in the twenty-first century. EURO Journal
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+
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+MATTEO LAPUCCI AND PIERLUIGI MANSUETO
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+[10] J. Fliege and B. F. Svaiter. Steepest descent methods for multicriteria optimization. Mathematical Methods of
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+10
+

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+DRAFT VERSION JANUARY 16, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+A Simultaneous Dual-Frequency Scintillation Arc Survey of Six Bright Canonical Pulsars Using the Upgraded Giant
+Metrewave Radio Telescope
+JACOB E. TURNER,1, 2 BHAL CHANDRA JOSHI,3 MAURA A. MCLAUGHLIN,1, 2 AND DANIEL R. STINEBRING4
+1Department of Physics and Astronomy, West Virginia University, P.O. Box 6315, Morgantown, WV 26506, USA
+2Center for Gravitational Waves and Cosmology, West Virginia University, Chestnut Ridge Research Building, Morgantown, WV 26505, USA
+3National Centre for Radio Astrophysics, Tata Institute of Fundamental Research, Post Bag 3, Ganeshkhind, Pune - 411007, India
+4Department of Physics and Astronomy, Oberlin College, Oberlin, OH 44074, USA
+ABSTRACT
+We use the Upgraded Giant Metrewave Radio Telescope to measure scintillation arc properties in six bright
+canonical pulsars with simultaneous dual frequency coverage. These observations at frequencies from 300 to
+750 MHz allowed for detailed analysis of arc evolution across frequency and epoch. We perform more robust
+determinations of arc curvature and scattering delay frequency-dependence than allowed by single-frequency-
+band-per-epoch measurements, which we find to agree with theory and previous literature. We report the dis-
+covery of a strong correlation between arc asymmetry and arc curvature, potentially indicating a link between
+scattering screen distance and refraction strength or the effect of asymmetric distribution of scattering material
+on a scattering screen. The inclusion of a 155 minute observation allowed us to resolve the scale of scintilla-
+tion variations on short timescales, which we find to be directly tied to the amount of ISM sampled over the
+observation. Some of our pulsars showed either consistent or emerging asymmetries in arc curvature, indicating
+instances of refraction across their lines of sight. The presence of significant features in various pulsars, such
+as multiple scintillation arcs in PSR J1136+1551 and flat arclets in PSR J1509+5531, that have been found in
+previous works, were also sufficiently detected. Possible evidence for a timescale over which a given scatter-
+ing screen dominates signal propagation was found by tracking visible scintillation arcs in each epoch in PSR
+J1136+1551. The interesting pulsar science accomplished with this upgraded telescope shows strong promise
+for important future work in pulsar astronomy.
+Keywords: methods: data analysis – stars: pulsars – ISM: general – ISM: structure
+1. INTRODUCTION
+The scintillation of pulsar emission occurs as the result of
+the propagation of this emission through non-uniform distri-
+butions of free electrons in the ionized interstellar medium
+(IISM). This interaction results in frequency-dependent and
+time-evolving variations in the flux density of the pulsar sig-
+nal as measured at a detector.
+When these variations are
+examined across observing frequency and time in so-called
+dynamic spectra, representations of the change in the pulsar
+signal’s intensity across frequency and time, for a given ob-
+servation, they can provide valuable insight into the structure
+of these electron density variations along our line of sight
+(LOS) to a given pulsar. For observations where the scin-
+tles, bright patches in a dynamic spectrum resulting from
+constructive interference between photons as the result of
+propagation through the free electrons in the ISM, are fully
+resolved in frequency, have sufficient coverage in time, and
+exhibit structures over the course of observations including,
+but not limited to, non-zero slopes across frequency and time,
+known as scintillation drift, as well as “crisscrossing” scin-
+tillation patterns, then additional information can be gained
+about the ISM structure along the LOS by examining the
+parabolic arcs, known as scintillation arcs, that can emerge
+by examining the Fourier transform of the dynamic spec-
+trum (Stinebring et al. 2001). Some current hypotheses on
+the physical origins of these arcs postulate that they originate
+from compressed plasma along the boundaries of 50−100 pc
+size bubbles in the ISM (Stinebring et al. 2022).
+Traditional measurements of scintillation arcs have typ-
+ically been limited to either one observing band over all
+epochs (i.e., Trang & Rickett (2007)), or alternated between
+observing bands from epoch to epoch (i.e., Stinebring et al.
+(2019)). While generally sufficient for most analyses, this
+band limit results in a bottleneck for examining the evo-
+lution of various frequency-dependent effects over shorter
+timescales, including scintillation arc curvature, structures
+within individual arcs, and asymmetries in both arc bright-
+ness and power as a function of differential time delay. By
+making use of the subarray capabilities of the Upgraded Gi-
+ant Metrewave Radio Telescope (uGMRT), we can effec-
+arXiv:2301.05306v1  [astro-ph.HE]  12 Jan 2023
+
+2
+J. E. TURNER ET AL.
+tively create an ultra wideband receiver by setting multiple
+groups of dishes to simultaneously observe at different fre-
+quencies. This work is primarily data-focused and aims to
+highlight the results of some multi-frequency analyses per-
+formed on a small survey of six strong canonical pulsars us-
+ing this approach. In Section 2 we discuss the data taken
+as part of our survey. Section 3 describes the analyses per-
+formed and the physical parameters extracted. Section 4 de-
+tails the results of these analyses. Finally, Section 5 summa-
+rizes our results and discusses possible next steps.
+2. DATA
+Our data were taken across across eight epochs span-
+ning MJD 58987−59497 using 22 dishes split into subarrays
+for simultaneous multi-frequency observations at uGMRT’s
+Band 3 and Band 4, centered at 400 MHz and 650 MHz,
+respectively, each with 200 MHz of bandwidth. This simul-
+taneous low-frequency accessibility is comparable to instru-
+ments like CHIME that can observe continuously between
+400-800 MHz and better than instruments such as the Green
+Bank Telescope, which, while having a wide range of low
+frequency coverage, can only observe below one GHz with
+at most 240 MHz of bandwidth at frequencies close to one
+GHz and less than 200 MHz of bandwidth in lower frequency
+ranges. Observations were also made at Band 5 centered
+at 1360 MHz, although due to a combination of RFI and
+low S/N no scintles were detectable in the dynamic spec-
+tra. The observing bands were split into 4096 49 kHz wide
+frequency channels and observed with 10 second subintegra-
+tions. These data were flux calibrated using observations of
+either 3C147 or 3C286 taken at the beginning of every ob-
+serving session and every pulsar was phase calibrated with
+a nearby source for five minutes once every 40 minutes of
+observing time on the pulsar. Two to three pulsars were ob-
+served at each epoch for 40 minutes each, except for MJD
+59497, where three pulsars were observed for 155 minutes
+each. As a result of the phase calibration, each of those obser-
+vations were comprised of three 40 minute sub-observations
+plus an additional 20 minute sub-observation.
+3. ANALYSIS
+All observations were processed to extract their dynamic
+spectra by calculating the intensity, S, of the pulsar’s signal
+at each observing frequency, ν, and time, t, via
+S(ν, t) = Pon(ν, t) − Poff(ν, t)
+Pbandpass(ν, t)
+,
+(1)
+where Pbandpass is the total power of the observation as a
+function of observing frequency and time, and Pon and Poff
+are the power in all on- and off-pulse components, respec-
+tively, as a function of frequency and time. Each dynamic
+spectrum was then broken up into four 50 MHz spectra to
+allow for more in-depth frequency-dependent analyses and
+manually zapped by examining dynamic spectra data arrays
+and removing pixels that were brighter than the brightest
+scintle maxima.
+Secondary spectra were then created by
+taking the absolute square of the Fourier transform (i.e., the
+power spectrum) of the corresponding dynamic spectrum and
+converting it to units of dB. The primary (brightest) scintilla-
+tion arcs on the positive and negative side of each secondary
+spectrum’s fringe frequency axis were then found via sepa-
+rate fν = ηf 2
+t fits, where fν is the differential time delay, η
+is the arc curvature, and ft is the fringe frequency.
+We also determined scintillation parameters by using the
+python package Pypulse (Lam 2017) to create the 2D au-
+tocorrelation functions (ACFs) of each dynamic spectrum
+and fit 2D Gaussians to these ACFs to determine their scin-
+tillation bandwidth, ∆νd, defined as the half-width at half-
+maximum (HWHM) along the frequency axis of the ACF at
+lag 0, scintillation timescale, ∆td, defined as the half-width
+at e−1 along the time axis at ACF lag 0, and scintillation
+drift rate, dν/dt, defined as the rotation of the 2D Gaussian
+fit to the 2D ACF in the plane of the frequency and time lags.
+For our scattering delay scaling index analysis, our measured
+scintillation bandwidths were converted to scattering delays
+using
+2π∆νdτ = C1,
+(2)
+where C1 is a dimensionless quantity between 0.6 − 1.5 that
+depends on the spectrum of the electron density fluctuations
+and geometry of the medium (Cordes & Rickett 1998). For
+this work we use C1 = 1.
+4. RESULTS & DISCUSSION
+Measured arc curvatures and their corresponding dynamic
+spectrum scintillation drift rates can be found in Table 1. All
+curvatures and their uncertainties have been scaled to their
+corresponding value at 1 GHz assuming a ν−2 frequency de-
+pendence (Hill et al. 2003). Here errors on the arc curva-
+tures represent fitting errors from the linear least squares fit.
+Some pulsars on MJD 59497 have multiple curvature mea-
+surements at a given frequency, which is the result of this
+epoch being 155 minutes instead of the 40 minutes of the
+other observations. As a result, a new η was measured after
+every 40 minutes since these observations were broken up
+into 40 minute chunks separated by five minute phase cali-
+brations. On days where measurements are given for the left
+or right arm only, arcs on the other side of the fringe fre-
+quency axis may have been present, but were unmeasureable
+due to either having insufficient extension along the differ-
+ential delay axis, insufficient flux relative to the background
+noise, being too diffuse, being too close to central spike in
+flux that commonly occurs around a fringe frequency of 0
+mHz, or some combination of these factors.
+
+SIMULTANEOUS DUAL-FREQUENCY SCINTILLATION ARC SURVEY
+3
+Table 1. Pulsar Scintillation Arc Curvatures and Drift Rates
+Pulsar
+MJD
+Frequency
+ηL
+σηL
+ηR
+σηR
+dν/dt
+σdν/dt
+(MHz)
+(s3)
+(s3)
+(s3)
+(s3)
+(MHz/min)
+(MHz/min)
+J0630–2834
+58987
+325
+—
+—
+0.2071
+0.0069
+–0.0164
+0.0009
+J0630–2834
+58987
+375
+—
+—
+0.2331
+0.0160
+–0.0676
+0.0097
+J0630–2834
+58987
+425
+—
+—
+0.2932
+0.0223
+–0.0343
+0.0027
+J0630–2834
+58987
+475
+—
+—
+0.2037
+0.0207
+–0.0574
+0.0059
+J0630–2834
+58987
+575
+—
+—
+0.1220
+0.0061
+0.0076
+0.0025
+J0630–2834
+58987
+625
+—
+—
+0.1233
+0.0049
+–0.0779
+0.0097
+J0630–2834
+58987
+675
+—
+—
+0.1553
+0.0058
+0.0818
+0.0118
+J0630–2834
+58987
+725
+—
+—
+0.2566
+0.0162
+0.2685
+0.0196
+J1136+1551
+58987
+325
+0.0154
+0.0007
+0.0238
+0.0006
+—
+—
+J1136+1551
+58987
+375
+0.0198
+0.0009
+0.0231
+0.0005
+0.0202
+0.0067
+J1136+1551
+58987
+425
+0.0195
+0.0010
+0.0225
+0.0004
+–0.0287
+0.0074
+J1136+1551
+58987
+575
+0.0214
+0.0005
+0.0218
+0.0003
+–0.0173
+0.0067
+J1136+1551
+58987
+625
+0.0202
+0.0007
+0.0229
+0.0002
+—
+—
+J1136+1551
+58987
+675
+0.0199
+0.0005
+0.0249
+0.0003
+–0.0322
+0.0250
+J1136+1551
+58987
+725
+0.0203
+0.0007
+0.0282
+0.0006
+0.00004
+0.7446
+J1136+1551
+58991
+325
+0.0126
+0.0004
+0.0234
+0.0005
+0.0048
+0.0093
+J1136+1551
+58991
+375
+0.0149
+0.0007
+0.0253
+0.0007
+0.2686
+0.0128
+J1136+1551
+58991
+425
+0.0167
+0.0009
+0.0250
+0.0006
+—
+—
+J1136+1551
+58991
+475
+0.0183
+0.0007
+0.0225
+0.0005
+–0.0402
+0.0123
+J1136+1551
+58991
+575
+0.0165
+0.0005
+0.0254
+0.0005
+0.0097
+0.0026
+J1136+1551
+58991
+625
+0.0178
+0.0008
+0.0271
+0.0005
+–0.0088
+0.0034
+J1136+1551
+58991
+675
+0.0173
+0.0007
+0.0261
+0.0004
+—
+—
+J1136+1551
+58991
+725
+0.0171
+0.0007
+0.0239
+0.0003
+0.00003
+0.4211
+J1136+1551
+59115
+325
+0.0084
+0.0002
+0.0099
+0.0002
+0.2219
+0.0095
+J1136+1551
+59115
+375
+0.0085
+0.0003
+0.0116
+0.0003
+0.1232
+0.0115
+J1136+1551
+59115
+425
+0.0095
+0.0005
+0.0139
+0.0004
+—
+—
+J1136+1551
+59115
+475
+0.0092
+0.0005
+0.0129
+0.0004
+0.0869
+0.0079
+J1136+1551
+59115
+575
+0.0095
+0.0003
+0.0111
+0.0002
+0.0057
+0.0014
+J1136+1551
+59115
+625
+0.0135
+0.0007
+0.0141
+0.0006
+—
+—
+J1136+1551
+59115
+675
+0.0143
+0.0004
+0.0154
+0.0003
+0.0159
+0.0057
+J1136+1551
+59115
+725
+0.0130
+0.0005
+0.0160
+0.0005
+0.0109
+0.0036
+J1136+1551
+59497
+325
+0.0065
+0.0002
+0.0070
+0.0003
+–0.1251
+0.0104
+J1136+1551
+59497
+375
+0.0078
+0.0004
+0.0077
+0.0005
+–0.0554
+0.0069
+J1136+1551
+59497
+425
+0.0079
+0.0002
+0.0072
+0.0002
+–0.0119
+0.0019
+J1136+1551
+59497
+475
+0.0070
+0.0003
+0.0068
+0.0004
+–0.0293
+0.0044
+J1136+1551
+59497
+325
+0.0070
+0.0001
+0.0067
+0.0002
+0.0076
+0.0030
+J1136+1551
+59497
+375
+0.0077
+0.0002
+0.0068
+0.0002
+0.2637
+0.0498
+J1136+1551
+59497
+425
+0.0072
+0.0002
+0.0076
+0.0002
+–0.0544
+0.0066
+Table 1 continued
+
+4
+J. E. TURNER ET AL.
+Table 1 (continued)
+Pulsar
+MJD
+Frequency
+ηL
+σηL
+ηR
+σηR
+dν/dt
+σdν/dt
+(MHz)
+(s3)
+(s3)
+(s3)
+(s3)
+(MHz/min)
+(MHz/min)
+J1136+1551
+59497
+475
+0.0069
+0.0001
+0.0067
+0.0002
+–0.0302
+0.0045
+J1136+1551
+59497
+325
+0.0075
+0.0002
+0.0075
+0.0002
+–0.0651
+0.0058
+J1136+1551
+59497
+375
+0.0076
+0.0002
+0.0076
+0.0003
+–0.0877
+0.0175
+J1136+1551
+59497
+425
+0.0082
+0.0002
+0.0075
+0.0002
+–0.0656
+0.0064
+J1136+1551
+59497
+475
+0.0067
+0.0002
+0.0074
+0.0003
+–0.0417
+0.0053
+J1136+1551
+59497
+575
+0.0063
+0.0004
+0.0079
+0.0013
+–0.0977
+0.0507
+J1136+1551
+59497
+675
+0.0059
+0.0004
+0.0075
+0.0007
+—
+—
+J1136+1551
+59497
+725
+0.0057
+0.0005
+0.0068
+0.0008
+—
+—
+J1509+5531
+58987
+575
+0.3155
+0.0191
+0.1798
+0.0063
+—
+—
+J1509+5531
+58987
+625
+0.3642
+0.0184
+0.2021
+0.0060
+–0.0284
+0.0224
+J1509+5531
+58987
+675
+0.3410
+0.0207
+0.1978
+0.0078
+–1.5130
+0.0117
+J1509+5531
+59064
+575
+0.2611
+0.0147
+0.2495
+0.0116
+0.0015
+0.0030
+J1509+5531
+59064
+625
+0.2890
+0.0170
+0.2767
+0.0118
+—
+—
+J1509+5531
+59064
+675
+0.2736
+0.0181
+0.2714
+0.0128
+0.0039
+0.0016
+J1509+5531
+59064
+725
+0.2921
+0.0186
+0.2930
+0.0142
+0.1192
+0.0728
+J1509+5531
+59115
+575
+0.0852
+0.0029
+0.0928
+0.0037
+—
+—
+J1509+5531
+59115
+625
+0.0868
+0.0029
+0.0964
+0.0044
+0.0173
+0.0066
+J1509+5531
+59115
+675
+0.0958
+0.0031
+0.0908
+0.0044
+–0.0970
+0.0628
+J1509+5531
+59115
+725
+0.1074
+0.0026
+0.0872
+0.0030
+—
+—
+J1509+5531
+59497
+575
+0.0992
+0.0034
+0.1139
+0.0045
+–0.9922
+0.0226
+J1509+5531
+59497
+625
+0.1057
+0.0032
+0.1257
+0.0043
+0.0186
+0.0140
+J1509+5531
+59497
+675
+0.1088
+0.0033
+0.1160
+0.0035
+—
+—
+J1509+5531
+59497
+725
+0.0973
+0.0026
+0.1337
+0.0036
+0.0517
+0.0395
+J1509+5531
+59497
+575
+0.0706
+0.0028
+0.1054
+0.0047
+—
+—
+J1509+5531
+59497
+625
+0.0741
+0.0026
+0.1101
+0.0046
+0.0059
+0.0047
+J1509+5531
+59497
+675
+0.0782
+0.0022
+0.1200
+0.0047
+0.0109
+0.0086
+J1509+5531
+59497
+725
+0.0808
+0.0021
+0.1265
+0.0040
+—
+—
+J1509+5531
+59497
+675
+0.0606
+0.0020
+0.1194
+0.0042
+—
+—
+J1645–0317
+59074
+575
+0.0832
+0.0051
+—
+—
+0.2015
+0.0091
+J1645–0317
+59074
+625
+0.0885
+0.0059
+—
+—
+–0.1688
+0.0098
+J1645–0317
+59074
+675
+0.0806
+0.0063
+—
+—
+–0.1812
+0.0091
+J1645–0317
+59074
+725
+0.0832
+0.0061
+—
+—
+–0.2235
+0.0105
+J1932+1059
+58997
+325
+0.0382
+0.0030
+0.0331
+0.0018
+–0.1092
+0.0227
+J1932+1059
+58997
+375
+0.0358
+0.0016
+0.0364
+0.0020
+—
+—
+J1932+1059
+58997
+425
+0.0346
+0.0006
+0.0335
+0.0007
+–0.0224
+0.0086
+J1932+1059
+58997
+475
+0.0365
+0.0007
+0.0335
+0.0005
+0.1987
+0.0427
+J1932+1059
+58997
+575
+0.0351
+0.0005
+0.0360
+0.0004
+0.0181
+0.0093
+J1932+1059
+58997
+625
+0.0366
+0.0005
+0.0335
+0.0004
+0.0614
+0.0352
+J1932+1059
+58997
+675
+0.0359
+0.0010
+0.0358
+0.0010
+–0.0062
+0.0016
+J1932+1059
+58997
+725
+0.0373
+0.0007
+0.0348
+0.0012
+–0.0074
+0.0022
+Table 1 continued
+
+SIMULTANEOUS DUAL-FREQUENCY SCINTILLATION ARC SURVEY
+5
+Table 1 (continued)
+Pulsar
+MJD
+Frequency
+ηL
+σηL
+ηR
+σηR
+dν/dt
+σdν/dt
+(MHz)
+(s3)
+(s3)
+(s3)
+(s3)
+(MHz/min)
+(MHz/min)
+J1932+1059
+59062
+325
+0.0330
+0.0006
+0.0271
+0.0004
+0.0669
+0.0111
+J1932+1059
+59062
+375
+0.0352
+0.0007
+0.0285
+0.0008
+0.1331
+0.0219
+J1932+1059
+59062
+425
+0.0347
+0.0005
+0.0293
+0.0004
+–0.0326
+0.0256
+J1932+1059
+59062
+475
+0.0345
+0.0007
+0.0278
+0.0006
+–0.1200
+0.0297
+J1932+1059
+59062
+575
+0.0337
+0.0007
+0.0303
+0.0004
+—
+—
+J1932+1059
+59062
+625
+0.0332
+0.0004
+0.0323
+0.0004
+–0.0221
+0.0095
+J1932+1059
+59062
+675
+0.0350
+0.0006
+0.0356
+0.0006
+–0.0221
+0.0095
+J1932+1059
+59062
+725
+0.0297
+0.0005
+0.0304
+0.0006
+–0.0192
+0.0152
+J1932+1059
+59497
+325
+0.0365
+0.0008
+0.0279
+0.0009
+–0.8215
+0.0090
+J1932+1059
+59497
+375
+0.0351
+0.0007
+0.0294
+0.0004
+0.1062
+0.0302
+J1932+1059
+59497
+425
+0.0399
+0.0004
+0.0320
+0.0005
+0.1479
+0.0287
+J1932+1059
+59497
+475
+0.0389
+0.0007
+0.0341
+0.0008
+0.2495
+0.3407
+J1932+1059
+59497
+575
+0.0405
+0.0016
+0.0363
+0.0012
+—
+—
+J1932+1059
+59497
+625
+0.0393
+0.0014
+0.0358
+0.0013
+–0.0815
+0.0392
+J1932+1059
+59497
+675
+0.0433
+0.0017
+0.0385
+0.0012
+–0.0222
+0.0103
+J1932+1059
+59497
+725
+0.0461
+0.0022
+0.0416
+0.0013
+—
+—
+J2048–1616
+59062
+325
+0.0221
+0.0006
+0.0108
+0.0009
+–0.4158
+0.0081
+J2048–1616
+59062
+375
+0.0198
+0.0007
+0.0113
+0.0023
+–0.0211
+0.0077
+J2048–1616
+59062
+425
+0.0247
+0.0020
+0.0108
+0.0012
+–0.1529
+0.0139
+J2048–1616
+59062
+625
+0.0152
+0.0004
+0.0132
+0.0005
+–0.0058
+0.0012
+J2048–1616
+59062
+675
+0.0152
+0.0004
+0.0138
+0.0005
+0.0208
+0.0090
+J2048–1616
+59062
+725
+0.0148
+0.0004
+0.0135
+0.0005
+0.00005
+0.0001
+NOTE—Scintillation arc measurements and drift rates in the left and right primary arms of each epoch at all
+frequencies where measurable. ηL and ηR are the the arc curvature measurements for the left and right arms,
+respectively, and dν/dt is the measured scintillation drift rate, with the matching σ’s representing the corre-
+sponding uncertainties. All curvatures and their errors have been scaled to 1 GHz and errors on curvature here
+are fit uncertainties. Some pulsars on MJD 59497 have multiple curvature measurements at a given frequency,
+due to this epoch being 155 minutes instead of the 40 minutes of the other observations, and so a new η was
+measured after every 40 minutes.
+4.1. Scintillation Arc Curvature Scaling Behavior
+As mentioned earlier, Hill et al. (2003) demonstrated
+through both theoretical and observational means that the
+arc curvature η should follow a ν−2 dependence, implying
+that scattering is dominated by one or several thin screens
+along the LOS. While over 2 GHz of bandwidth was used in
+those observations (10-12.5 MHz of bandwidth centered at
+430 MHz and either 50 or 100 MHz of bandwidth centered at
+1175 MHz, 1400 MHz, and 2250 MHz), the frequency cov-
+erage was discontinuous and all η measurements used in their
+corresponding fits were from different epochs. Generally the
+latter point should not be an issue as long as the observations
+were taken within a period shorter than the pulsar’s refrac-
+tive timescale. Indeed, for the data used in their fits, their
+measured arc curvatures at a given frequency did not vary
+significantly on day or week timescales, making them suit-
+able for this type of analysis. However, the ideal situation
+would be to obtain many measurements at many frequencies
+during the same observation, preferably at the same time for
+optimal consistency. With our high resolution and sufficient
+observing time, we have the ability to make up to eight con-
+current arc measurements over 450 MHz of bandwidth at low
+frequency and can consequentially provide a more definitive
+examination of the theory.
+Following the methodology of Hill et al. (2003), for a scal-
+ing index α, we performed a weighted linear least-squares fit
+of the form
+log10 η = α log10 ν + β
+(3)
+on the unscaled curvatures for each pulsar at each MJD. Ex-
+ample fits can be seen in Figure 1, with all measured in-
+
+6
+J. E. TURNER ET AL.
+dices listed in Table 2. We find that, overall, our scaling
+indices are consistent with a theoretical index of −2, which
+assumes thin screen scattering (Stinebring et al. 2001), with
+PSRs J1136+1551 and J1932+1059 being especially consis-
+tent. This effect can also be seen in Table 1, where arc cur-
+vatures at all frequencies from a given pulsar at the same
+MJD are generally in strong agreement after being scaled to
+1 GHz. Interestingly, a weighted average of all curvature
+fits shows that our left arm fits are overall more consistent
+with an index of −2 than our right arms, with a weighted av-
+erage of −1.99±0.03 across all left arm fits compared with
+−1.69±0.02 across all right arm fits, indicating that refrac-
+tion may play a role in how closely arc curvature scales as
+expected with frequency.
+4.2. Scattering Delay Scaling Behavior
+Our wide frequency coverage also allowed us to examine
+the scaling index of scattering delays. Under the assumption
+that ISM fluctuations follow behaviors consistent with a Kol-
+mogorov medium and that scattering can be modeled as the
+result of interactions of pulsar emission with an infinite, thin,
+scattering screen during its propagation, we should expect
+that scattering delays scale with frequency as τd ∝ ν−4.4
+(Romani et al. 1986; Cordes & Rickett 1998).
+Previous
+studies examining the scattering indices of various pulsars
+have done so using a number of methods, including simul-
+taneous multi-frequency measurements (Bhat et al. 2004;
+Bansal et al. 2019), splitting up measurements from a single
+frequency band into multiple subbands (Levin et al. 2016;
+Turner et al. 2021), and using measurements from many
+epochs taken at two observing bands non-simultaneously
+(Turner et al. 2021). Since more measurements and more fre-
+quency coverage in a single epoch is ideal, the method used
+in Bhat et al. (2004) and Bansal et al. (2019) is the most pre-
+ferred of the three. The method in this paper utilizes a com-
+bination of this approach and the subband approach to max-
+imize the number of delay measurements per epoch, which
+can be done thanks to our high frequency resolution and sen-
+sitivity in both observing bands.
+Similar to Equation 3 used to determine the arc curva-
+ture scaling index, our scattering delay scaling indices ξ at
+each epoch were determined by performing a weighted lin-
+ear least-squares fit of the form
+log10 τd = ξ log10 ν + b.
+(4)
+Example fits can be seen in Figure 2, with all measured in-
+dices listed in Table 3. We find that half of our measured
+indices are consistent with a Kolmogorov medium, while the
+other half are consistent with a shallower medium. This be-
+havior agrees well with previous studies, as both Bhat et al.
+(2004) and Bansal et al. (2019) found indices either consis-
+tent with a Kolmogorov medium or shallower than a Kol-
+mogorov medium, while Levin et al. (2016) and Turner et al.
+(2021) only found indices that were shallower than a Kol-
+mogorov medium.
+Many explanations have been given for why shallower-
+than-Kolmogorov medium behavior has been observed so
+frequently. Physical arguments have called into question the
+validity of the simple infinite, thin screen model, demonstrat-
+ing that shallower scaling indices are more consistent with
+finite, thin screens (Rickett et al. 2009). This is expected to
+be much more common among low DM pulsars (Cordes &
+Lazio 2001), which agrees with our results, as all of the pul-
+sars have dispersion measures below 40 pc cm−3. Shallower
+indices have also been attributed to the existence of multi-
+ple finite screens along the LOS (Lewandowski et al. 2013).
+This hypothesis agrees well with our measured indices for
+PSR B1133+16, as its indices are consistently shallower than
+that of a Kolmogorov medium and it is also known to have at
+least six distinct scattering screens (McKee et al. 2022).
+Quality-of-data arguments have also been proposed.
+Turner et al. (2021) suggested their shallower indices may
+be at least partially attributable to an imbalance of lower fre-
+quency data to higher frequency data for their multiple epoch
+approach as well as a lack of sufficient frequency resolution
+in their lower frequency band in some epochs. However, nei-
+ther of these issues should affect our results, as our observa-
+tions have a consistently even balance of low and high fre-
+quency measurements at all epochs and all of our measure-
+ments are well-resolved in frequency.
+4.3. 155 Minute Observation
+The inclusion of a 155 minute observation in our survey
+on MJD 59497 allowed for an analysis of short-term arc cur-
+vature variation in some pulsars, as observations had to be
+paused every 40 minutes for a five minute phase calibration,
+resulting in multiple 40 minute sub-observations. For pulsars
+with at least two measurements in a given scintillation arc at a
+given frequency, we examined overall variation in that arc at
+that frequency by looking at the percent difference between a
+given curvature measurement and the weighted average cur-
+vature for that arm and frequency over the entire epoch.
+For PSR J1136+1551, all observing frequencies centered
+at or below 475 MHz had three measurements in each pri-
+mary arm (the brightest arm, overwhelmingly often the arm
+with the lowest curvature) at each frequency, with the accu-
+mulation of all percent differences yielding a bimodal distri-
+bution with peaks around percent differences of 2% and 7%.
+The largest percent difference away from a weighted mean
+was 7.9±0.2% and the smallest was 0.14±1.91%, although
+the majority of all percent differences was below 3%. All of
+this strongly indicates the ISM underwent very little change
+along the LOS to this pulsar over the course of a given ob-
+servation. This result is supported by this pulsar’s incredibly
+low dispersion measure, meaning it does not sample a size-
+
+SIMULTANEOUS DUAL-FREQUENCY SCINTILLATION ARC SURVEY
+7
+(a) α fit for PSR J1932+1059 on MJD 58997
+(b) α fit for PSR J1136+1551 on MJD 58997. The inclusion of multiple points at
+certain frequencies is the result of this epoch containing a 155 minute
+observation instead of the 40 minutes of the other observations, and so a new η
+was measured after every 40 minutes.
+Figure 1. Example fits for the arc curvature scaling index
+Table 2. Fitted Pulsar Scintillation Arc Curvature Scaling Indices
+Pulsar
+MJD
+Scaling Index Left Arc
+Scaling Index Error Left Arc
+Nη
+Scaling Index Right Arc
+Scaling Index Error Right Arc
+Nη
+J0630–2834
+58987
+—
+—
+—
+–2.48
+0.31
+8
+J1136+1551
+58987
+–1.79
+0.11
+7
+–1.89
+0.12
+7
+J1136+1551
+58991
+–1.65
+0.12
+8
+–1.94
+0.08
+8
+J1136+1551
+59115
+–1.36
+0.13
+8
+–1.52
+0.15
+8
+J1136+1551
+59497
+–2.12
+0.12
+15
+–1.99
+0.10
+15
+J1509+5531
+58987
+–1.49
+0.83
+3
+–1.34
+0.55
+3
+J1509+5531
+59064
+–1.62
+0.26
+4
+–1.41
+0.22
+4
+J1509+5531
+59115
+–0.93
+0.21
+4
+–2.31
+0.18
+4
+J1509+5531
+59497
+–2.01
+0.87
+9
+–1.34
+0.21
+9
+J1645–0317
+59074
+–2.09
+0.26
+4
+—
+—
+—
+J1932+1059
+58997
+–1.93
+0.05
+8
+–1.93
+0.09
+8
+J1932+1059
+59062
+–2.08
+0.07
+8
+–1.75
+0.07
+8
+J1932+1059
+59497
+–1.77
+0.09
+8
+–1.53
+0.04
+8
+J2048–1616
+59062
+–2.52
+0.07
+6
+–1.68
+0.05
+6
+NOTE—Fitted arc curvature scaling indices for both left and right primary arcs. Nη indicates the number of arc curvature measurements used
+in each fit. Measurements on MJD 59497 may have Nη > 8 due to this epoch being 155 minutes rather than the 40 minutes of the other
+observations, and so a new η was measured after every 40 minutes. Arc curvature measurements used in these fits were left unscaled.
+able portion of the ISM along its LOS relative to many pul-
+sars that are observed (Bilous et al. 2016; Manchester et al.
+2005; Pilkington et al. 1968).
+For PSR J1509+5531, all observing frequencies centered
+at or above 575 MHz had at least two measurements in each
+arm at each frequency, with the accumulation of all per-
+cent differences resulting in a one-sided distribution peaked
+
+J1932+1059MJD 5899T
+4 × 10-1
+nL; Scaling Index= -1.93 ± 0.05
+nR; Scaling Index = -1.93 ± 0.09
+3 × 10-1.
+2 × 10-1
+n
+10-1
+6 × 10-2
+400
+500
+600
+700
+Frequency [MHz]J1136+1551MJD 5949T
+nL; Scaling Index= -2.12 ± 0.12
+6 × 10-2
+nr; Scaling Index = -1.99 ± 0.10
+4 × 10-2
+3 × 10-2
+n
+2 × 10-2
+10-1
+400
+500
+600
+700
+Frequency [MHz]8
+J. E. TURNER ET AL.
+(a) Scattering delay scaling index fit for PSR J1932+1059 on MJD 59497
+(b) Scattering delay scaling index fit for PSR J1136+1551 on MJD 58991.
+Figure 2. Example fits for the scattering delay scaling index
+Table 3. Fitted Pulsar Scattering Delay Scaling Indices
+Pulsar
+MJD
+Scaling Index
+Index Error
+Nτd
+J0630–2834
+58987
+–4.19
+1.43
+8
+J1136+1551
+58987
+–1.44
+0.71
+6
+J1136+1551
+58991
+–3.78
+0.62
+7
+J1136+1551
+59115
+–2.72
+1.05
+6
+J1136+1551
+59497
+–1.71
+0.57
+13
+J1645–0317
+59074
+–4.60
+0.75
+4
+J1932+1059
+58997
+–1.83
+0.31
+7
+J1932+1059
+59062
+–1.74
+0.31
+6
+J1932+1059
+59497
+–4.14
+0.39
+6
+J2048–1616
+59062
+–3.77
+1.39
+6
+NOTE—Fitted scattering delay scaling indices, with a minimum
+of four delay measurements (Nτd) required in a given epoch to
+obtain a scaling index. Errors are the parameter uncertainties
+from parameter fits. Half of our measured indices were con-
+sistent with a Kolmogorov medium, while the other half were
+consistent with a shallower medium. Measurements on MJD
+59497 may have Nτd > 8 due to this epoch being 155 minutes
+rather than the 40 minutes of the other observations, and so a
+new �� was measured after every 40 minutes.
+around 6%.
+The smallest percent difference away from
+a weighted mean was 1.2±2.1%, while the largest was
+36±0.1%, although the next largest after that was only
+22±0.1%, meaning this maximum was an extreme outlier.
+The majority of all percent differences was below 7%. As
+with the previous pulsar, this also strongly indicates the ISM
+underwent very little change along the LOS to this pul-
+sar over the course of a given observation, a result again
+supported by this pulsar’s fairly low dispersion measure
+(Huguenin et al. 1968; Manchester et al. 2005). The fact that
+this pulsar shows higher variation of this observation com-
+pared to PSR J1136+1551 is likely due to PSR J1509+5531
+having a dispersion measure four times higher and a trans-
+verse velocity 45% larger (Bilous et al. 2016; Huguenin et al.
+1968; Manchester et al. 2005; Pilkington et al. 1968; Stovall
+et al. 2015), so a significantly larger fraction of the ISM was
+sampled during its observation, increasing the likelihood of
+larger scintillation-based variations.
+The next few subsections will be dedicated to highlighting
+the features of a few pulsars in the survey.
+4.4. J0630-2834
+In the one epoch for which we were able to resolve a scin-
+tillation arc, only the right arm was resolvable across all fre-
+quencies, with its relative brightness relative to the left side
+of the fringe frequency axis consistently decreasing as fre-
+quency increased. An example of this asymmetry can be seen
+in Figure 3.This strong asymmetry is known to be the result
+of refraction leading to scintillation drifting in the dynamic
+spectra (Cordes et al. 2006). Interestingly, despite our asym-
+metry appearing to decrease with frequency, the magnitude
+of our measured scintillation drift rates seem to mildly favor
+an increase with frequency, whereas one would expect an in-
+crease in scintillation drift to coincide with an increase in the
+asymmetry.
+4.5. J1136+1551
+
+J1932+1059 MJD 59497
+Scaling Index= -4.14 ± 0.39
+102
+(ns)
+101
+400
+500
+600
+700
+Frequency [MHz]J1136+1551MJD 58991
+Scaling Index= -3.78 ± 0.62
+102
+(ns)
+101
+400
+500
+600
+700
+Frequency [MHz]SIMULTANEOUS DUAL-FREQUENCY SCINTILLATION ARC SURVEY
+9
+Figure 3. An example dynamic (top) and secondary (bottom) spec-
+trum from PSR J0630-2834 on MJD 58987 centered at 425 MHz.
+There is a clear asymmetry in the secondary spectrum, with the right
+arm being the dominant feature. This is likely the result of refraction
+along the line of sight. The green line represents the arc curvature
+fit and the arc curvature measurement quoted is unscaled. Scaled
+uncertainties of the arc curvature can be found in Table 1.
+This pulsar is well known for having the uncommon fea-
+ture of multiple scintillation arcs, implying multiple scatter-
+ing screens along its LOS (Hill et al. 2003; Stinebring et al.
+2019). In the literature six distinct sets of arcs have been
+found over a ∼34 year span of observations (McKee et al.
+2022). In three of the four epochs in we which observed
+this pulsar, we observed multiple arcs, an example of which
+can be seen in Figure 4. After scaling our measurements
+to 1400 MHz and using the convention from McKee et al.
+(2022), we can conclude that we detected arcs E, C, and B
+on MJDs 58987 and 58991 and arcs D and C on MJD 59115,
+with arc C being the only detectable arc on MJD 59497. All
+multiple-arc detections were made only in the observations
+using uGMRT’s band 4, which was centered at 650 MHz.
+The fact that the two epochs closest to each other in our sur-
+vey (58987 and 58991) both detected the same sets of arcs
+may hint at a timescale over which certain screens have a
+larger influence over the pulsar signal propagation.
+An examination of the power in each of the arms show
+notable levels of asymmetry along the delay axis, and conse-
+quently a notable amount of refraction, in all detectable arms
+and across all frequencies in the first two epochs, with the
+right arm having more power and extending further out on the
+delay axis. This asymmetry clearly decreases over the course
+of our observations across all frequencies until our final ob-
+servation, where the arcs have approximately even levels of
+power or the left arc starts to dominate in the asymmetry.
+This trend is generally supported by the measured scintilla-
+tion drift rates as well, especially for data taken at band 4
+(650 MHz), i.e., the same band where the multiple arcs were
+visible, as measured drifts are generally positive during the
+first three epochs and then considerably negative during the
+final epoch.
+Perhaps the most interesting finding from our observations
+of this pulsar is the discovery of a strong correlation between
+the measured arc curvatures and the arc asymmetry index,
+which is a metric that describes the relative power between
+the left and right arcs and is found by comparing the average
+power along each arc via
+A = PR(fν) − PL(fν)
+PR(fν) + PL(fν)
+,
+(5)
+with a larger index magnitude indicating greater asymmetry.
+We believe this phenomenon has never before been reported
+and is therefore worth further examination in future observa-
+tions. As briefly mentioned earlier, asymmetry in arcs has
+long been attributed to either the refraction of pulsar emis-
+sion at the scattering screen or as the result of an asymmetric
+distribution of the material within the screen (Cordes et al.
+2006), while arc curvature is known to indicate the distance
+between a given scattering screen and the observer (Stine-
+bring et al. 2001). The correlation between the two suggests
+further study of this effect may result in a better understand-
+ing of how screen asymmetry and/or refraction affects pulsar
+emission depending on the scattering screen’s proximity to
+the pulsar.
+An example dynamic and secondary spectrum pair is
+shown in Figure 5, with its corresponding normalized sec-
+ondary spectrum power profile, which is used to determine
+the asymmetry index, shown in Figure 6, while the scatter
+plot showing the relation between measured arc curvature
+and arc asymmetry index across all measurements taken in
+the 650 MHz band is shown in Figure 7. Of particular note in
+Figure 7 are the three distinct clumps, which we believe are
+the result of our observations being dominated by a differ-
+ent scattering screen at each epoch (two of our observations
+were taken four days apart, and so are dominated by the same
+screen). It is likely that this pulsar’s at least six known scat-
+tering screens are the main reason why we were able to see
+this correlation in our data in the first place, as individual
+scattering screens likely do not vary enough in distance over
+time for this trend to become apparent. Indeed, the limited
+number of pulsars with multiple known screens is probably
+the main reason why this trend has not been reported in ear-
+lier studies.
+4.6. J1509+5531
+In the observations of this pulsar in the 650 MHz band,
+all secondary spectra featured patchy rather than continuous
+
+PSR J0630-2834 MJD 58987
+450
+Flux Density (Arbitrary Units)
+0.15
+[MHz]
+440
+0.10
+430
+Frequency [
+0.05
+420
+0.00
+410
+-0.05
+400
+0
+10
+20
+30
+40
+Time [Min]
+10
+nL =1.534 s3
+40
+nR =1.623 s3
+Log Power (dB)
+5
+Delay [μs]
+20
+0
+0
+-5
+20
+-40
+-10
+-40
+-20
+0
+20
+40
+Fringe Frequency [10-3 Hz]10
+J. E. TURNER ET AL.
+(a) Scintillation arcs without overlaid fits
+(b) Scintillation arcs with overlaid fits
+Figure 4. Secondary spectrum of PSR J1136+1551 at 650 MHz on MJD 58987 showing the detection of three distinct scintillation arcs.
+Figure 5. Dynamic (top) and secondary (bottom) spectra of PSR
+J1136+1551 centered at 650 MHz on MJD 58987. The top half of
+the secondary spectrum shows the overlaid arc fits in green. Scaled
+uncertainties of the arc curvature can be found in Table 1.
+arcs, particularly in the left arm. This patchiness indicates
+a detection of this pulsar’s arclets, which result from sub-
+structures in the ISM thought to arise from scattering inter-
+ference between an inhomogeniously scattered distribution
+of material and some distinct offset region (Walker & Stine-
+bring 2005; Cordes et al. 2006). In the particular case of this
+Figure 6. Normalized secondary spectrum power profile of PSR
+J1136+1551 centered at 650 MHz on MJD 58987. The vertical
+dashed lines indicate where the arcs fall on the normalized delay
+axis.
+pulsar these substructures are roughly AU in scale. Unique
+to these arclets is their distinctly flat nature, which has been
+attributed to its exceptionally high transverse velocity of over
+960 km s−1 (Manchester et al. 2005). Interestingly, the arc
+curvatures measured in the last two epochs (MJDs 59115 and
+59497) are a factor of two to three times smaller than the first
+
+PSR J1136+1551MJD 5898T
+14000
+70
+12000
+60
+10000
+50
+8000
+({_w) f
+40
+6000
+30
+4000
+20
+2000
+10
+0
+-20
+0
+20
+-40
+40
+ft (mHz)PSR J1136+1551MJD5898T
+14000
+70
+12000
+60
+10000
+50
+8000
+({_w) f
+40
+6000
+30
+4000
+20
+2000
+10
+-20
+0
+20
+-40
+40
+ft (mHz)PSR J1136+1551MJD 5898T
+750
+( )s
+0.35
+0.30
+Frequency [MHz]
+700
+0.25
+650
+0.15
+0.10
+600
+0.05
+550
+0.00
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Time [Min]
+10
+nL =0.056 s3
+60
+nR =0.060 s3
+5
+ Power (dB)
+40
+Delay [μs]
+20
+0
+0
+Log
+20
+-10
+-40
+-20
+0
+20
+40
+Fringe Frequency [10-3 Hz]18
+16
+12
+10
+8
+-2
+0
+-1
+2
+Normalized ftSIMULTANEOUS DUAL-FREQUENCY SCINTILLATION ARC SURVEY
+11
+Figure 7. Scatter plot showing measured arc curvatures and the
+corresponding asymmetry indices for all measurements of PSR
+J1136+1551 taken with Band 4. All arc curvatures have been scaled
+to their corresponding 1 GHz equivalent. The three distinct clumps
+are the result of the observations being dominated by three different
+scattering screens.
+two epochs (MJDs 59064 and 58987), possibly indicating a
+detection of multiple scattering screens along the LOS to this
+pulsar. This result augments the results of Sprenger et al.
+(2022), who also found significant variability along the LOS
+to this pulsar during the same period of time. An example
+observation from the earlier two epochs is shown in Figure
+8, while an example from the later two epochs is shown in
+Figure 9.
+4.7. J1932+1059
+Due to having the lowest DM in our survey, this pulsar
+showed the least variation in arc curvature from epoch to
+epoch across all frequencies. Its close proximity to Earth also
+resulted in wide scintles in frequency, leading to high scintle
+resolution and, consequently, very bright, narrow, and well
+defined arcs. The sharpness of these arcs may also indicate
+scattering that is highly anisotropic along the LOS (Walker
+et al. 2004; Cordes et al. 2006), as well as originating from
+a discrete, localized source (Stinebring et al. 2001). Overall
+this was our most consistent pulsar in all aspects of scintilla-
+tion.
+This consistency lines up with its other astrophysical pa-
+rameters, as its dispersion measure of 3.18 pc cm−3 (Large
+et al. 1968; Manchester et al. 2005) was the lowest in our
+survey and its transverse velocity of 152 km s−1 (Bilous
+et al. 2016; Manchester et al. 2005) was the second low-
+est. While its transverse velocity is a bit larger than PSR
+Figure 8. Dynamic (top) and secondary (bottom) spectra of PSR
+J1509+5531 centered at 625 MHz on MJD 58987. The top half of
+the secondary spectrum shows the overlaid arc fits in green. Scaled
+uncertainties of the arc curvature can be found in Table 1. During
+this period of observations, visible arcs were considerably narrower
+than later observations.
+Figure 9. Dynamic (top) and secondary (bottom) spectra of PSR
+J1509+5531 centered at 650 MHz on MJD 59115. The top half of
+the secondary spectrum shows the overlaid arc fits in green. Scaled
+uncertainties of the arc curvature can be found in Table 1. During
+this period of observations, visible arcs were considerably wider
+than later observations.
+J0630−2834 and their distances are almost equivalent, PSR
+J0630−2834 has a dispersion measure 10 times higher than
+PSR J1932+1059 (Large et al. 1968, 1969; Manchester et al.
+
+J1136+1551 High Frequencies PL =0.92; PR =0.88
+Referenced at 1 GHz
+nL
+0.12
+NR
+0.10
+0.08
+0.06
+0.04
+0.02
+0.00
+-0.02
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+n (s3)PSR J1509+5531MJD 58987
+650
+Flux Density (Arbitrary Units)
+0.125
+[MHz]
+640
+0.100
+0.075
+630
+Frequency 
+0.050
+620
+0.025
+610
+0.000
+-0.025
+600
+0
+10
+20
+30
+40
+Time [Min]
+10
+nL =0.932 s3
+40
+NR =0.518 s3
+5
+Log Power (dB)
+20
+Delay [μs]
+0
+0
+20
+-5
+一
+-40
+-10
+-40
+-20
+0
+20
+40
+Fringe Frequency [10-3 Hz]PSR J1509+5531 MJD 59115
+750
+(n ) s
+0.35
+0.30
+Frequency [MHz]
+700
+0.25
+0.20
+650
+0.15
+0.10
+600
+0.05
+0.00
+550
+0
+5
+10
+15
+20
+25
+30
+35
+Time [Min]
+10
+60
+nL =0.224 s3
+NR =0.187 s3
+40
+5
+ Power (dB)
+Delay [μs]
+20
+0
+0
+Logl
+-5
+-20
+-10
+-40
+-20
+0
+20
+40
+Fringe Frequency [10-3 Hz]12
+J. E. TURNER ET AL.
+2005). This means that a much denser ISM was sampled
+in PSR J0630−2834 than in PSR J1932+1059, meaning that
+PSR J1932+1059 had decisively the least amount of ISM
+sampled over our survey, making it the least likely to ex-
+perience large scintillation-related variations. An example
+observation is shown in Figure 10.
+Figure 10. Dynamic (top) and secondary (bottom) spectra of PSR
+J1932+1059 centered at 725 MHz on MJD 58987. The top half of
+the secondary spectrum shows the overlaid arc fits in green. Scaled
+uncertainties of the arc curvature can be found in Table 1.
+5. CONCLUSIONS & FUTURE WORK
+We performed simultaneous dual-frequency observations
+of six bright canonical pulsars using the uGMRT. We ex-
+tracted scintillation arc, bandwidth, and drift rate measure-
+ments for each of these pulsars to examine a variety of sci-
+ence. We examined how arc curvature scaled with frequency
+and found our observations to be consistent with the index
+predicted by theory, while at the same time using a more as-
+tronomically ideal setup to perform these measurements. We
+also measured scattering delay scaling indices for five of our
+six pulsars and found indices consistent with or shallower
+than what is expected from a Kolmogorov medium, agreeing
+with previous literature. Finally, we find an interesting and
+strong correlation between arc curvature and arc asymmetry
+in PSR J1136+1551, demonstrating a potential connection
+between screen asymmetry and/or refraction and scattering
+screen location along the LOS, and the which we intend to
+follow up with additional observations.
+This study demonstrates the value of array-based tele-
+scopes such as uGMRT to the pulsar astronomy community,
+as well as the strengths of simultaneous multiband studies of
+pulsars and the wide variety of science that can be done with
+such an approach. This also shows strong promise for the
+future observations using ultrawideband (UWB) receivers,
+which are coming online at instruments such as the Green
+Bank Telescope.
+We thank the staff at the uGMRT who have made these
+observations possible. The uGMRT is run by the National
+Centre for Radio Astrophysics of the Tata Institute of Funda-
+mental Research. We gratefully acknowledge support of this
+effort from the NSF Physics Frontiers Center grants 1430284
+and 2020265 to NANOGrav. Some of the data processing
+in this work utilized the resources of the Bowser computing
+cluster at West Virginia University.
+Software:
+SCINTOOLS Reardon et al. (2020), PYPULSE
+Lam (2017), SCIPY Virtanen et al. (2020), NUMPY van der
+Walt et al. (2011), and MATPLOTLIB Hunter (2007).
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+(   
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+Frequency [MHz]
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+ Power (dB)
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+

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+Asynchronous training of quantum reinforcement learning
+Samuel Yen-Chi Chen1
+1Wells Fargo
+(Dated: January 13, 2023)
+Abstract
+The development of quantum machine learning (QML) has received a lot of interest recently
+thanks to developments in both quantum computing (QC) and machine learning (ML). One of
+the ML paradigms that can be utilized to address challenging sequential decision-making issues is
+reinforcement learning (RL). It has been demonstrated that classical RL can successfully complete
+many difficult tasks. A leading method of building quantum RL agents relies on the variational
+quantum circuits (VQC). However, training QRL algorithms with VQCs requires significant amount
+of computational resources. This issue hurdles the exploration of various QRL applications. In
+this paper, we approach this challenge through asynchronous training QRL agents. Specifically,
+we choose the asynchronous training of advantage actor-critic variational quantum policies. We
+demonstrate the results via numerical simulations that within the tasks considered, the asyn-
+chronous training of QRL agents can reach performance comparable to or superior than classical
+agents with similar model sizes and architectures.
+1
+arXiv:2301.05096v1  [quant-ph]  12 Jan 2023
+
+I.
+INTRODUCTION
+Quantum computing (QC) has been posited as a means of achieving computational supe-
+riority for certain tasks that classical computers struggle to solve [1]. Despite this potential,
+the lack of error-correction in current quantum computers has made it challenging to ef-
+fectively implement complex quantum circuits on these ”noisy intermediate-scale quantum”
+(NISQ) devices [2]. To harness the quantum advantages offered by NISQ devices, the devel-
+opment of specialized quantum circuit architectures is necessary.
+Recent advances in the hybrid quantum-classical computing framework [3] that utilizes
+both classical and quantum computing. Under this approach, certain computational tasks
+that are expected to benefit from quantum processing are executed on a quantum computer,
+while others, such as gradient calculations, are performed on classical computers. This hybrid
+approach aims to take advantage of the strengths of both types of computing to address a
+wide range of tasks. Hybrid algorithms that utilize variational quantum circuits (VQC)
+have proven to be effective in a variety of machine learning tasks. VQCs are a subclass of
+quantum circuits that possess tunable parameters, and their incorporation into QML models
+has demonstrated success in a wide range of tasks [3, 4].
+Reinforcement learning (RL) is a branch of machine learning that deals with sequential
+decision making tasks. Deep neural network-based RL has achieved remarkable results in
+complicated tasks with human-level [5] or super-human performance [6]. However, quantum
+RL is a developing field with many unresolved issues and challenges. The majority of existing
+quantum RL models are based on VQC [7–11]. Although these models have been shown to
+perform well in a variety of benchmark tasks, training them requires a significant amount
+of computational resources. The long training time limits the exploration of quantum RL’s
+broad application possibilities. We propose an asynchronous training framework for quantum
+RL agents in this paper. We focus on the asynchronous training of advantage actor-critic
+quantum policies using multiple instances of agents running in parallel.
+We show, using numerical simulations, that quantum models may outperform or be sim-
+ilar to classical models in the various benchmark tasks considered. Furthermore, the sug-
+gested training approach has the practical advantage of requiring significantly less time for
+training, allowing for more quantum RL applications.
+The structure of this paper is as follows: In SectionII, we provide an overview of relevant
+2
+
+prior work and compare our proposal to these approaches. In SectionIII, we provide a brief
+overview of the necessary background in reinforcement learning. In SectionIV, we introduce
+the concept of variational quantum circuits (VQCs), which serve as the building blocks of
+our quantum reinforcement learning agents. In SectionV, we present our proposed quantum
+A3C framework. In Section VI, we describe our experimental setup and present our results.
+Finally, in Section VII, we offer some concluding remarks.
+II.
+RELEVANT WORKS
+The work that gave rise to quantum reinforcement learning (QRL) [12] may be traced
+back to [13]. However, the framework demands a quantum environment, which may not
+be met in most real-world situations. Further studies utilizing Grover-like methods include
+[14, 15]. Quantum linear system solvers are also used to implement quantum policy iteration
+[16]. We will concentrate on recent advancements in VQC-based QRL dealing with classical
+environments.
+The first VQC-based QRL [7], which is the quantum version of deep Q-
+learning (DQN), considers discrete observation and action spaces in the testing environments
+such as Frozen-Lake and Cognitive-Radio. Later, more sophisticated efforts in the area of
+quantum DQN take into account continuous observation spaces like Cart-Pole [8, 9]. A
+further development along this direction includes the using of quantum recurrent neural
+networks such as QLSTM as the value function approximator [17] to tackle challenges such
+as partial observability or environments requiring longer memory of previous steps. Various
+methods such as hybrid quantum-classical linear solver are developed to find value functions
+[18]. A further improvement of DQN which can improve the agent convergence such as
+Double DQN (DDQN) are also implemented within VQC framework in the work [19], in
+which the authors apply QRL to solve robot navigation task. Recent advances in QRL
+have led to the development of frameworks that aim to learn policy functions, denoted as
+π, directly. These frameworks are able to learn the optimal policy for a given problem,
+in addition to learning value functions such as the Q-function.
+For example, the paper
+[10] describes the quantum policy gradient RL through the use of REINFORCE algorithm.
+Then, the work [11] consider an improved policy gradient algorithm called PPO with VQCs
+and show that even with a small number of parameters, quantum models can outperform
+their classical counterparts. Provable quantum advantages of policy gradient are shown in
+3
+
+the work [20]. Additional research, such as the work in [21], has explored the impact of
+various post-processing methods for VQC on the performance of quantum policy gradients.
+Several improved quantum policy gradient algorithms have been proposed in recent years,
+including actor-critic [22] and soft actor-critic (SAC) [23, 24]. These modifications seek to
+further improve the efficiency and effectiveness of QRL methods. QRL has also been applied
+to the field of quantum control [25] and has been extended to the multi-agent setting [26–
+28]. The work [29] were the first to explore the use of evolutionary optimization for QRL.
+In their work, multiple agents were initialized and run in parallel, with the top performing
+agents being selected as parents to generate the next generation of agents. In the work [30],
+the authors studied the use of advanced quantum policy gradient methods, such as the deep
+deterministic policy gradient (DDPG) algorithm, for QRL in continuous action spaces.
+In this work, we extend upon previous research on quantum policy gradient [10, 11, 22] by
+introducing an asynchronous training method for quantum policy learning. While previous
+approaches have employed single-threaded training, our method utilizes an asynchronous
+approach, which may offer practical benefits such as reduced training time through the use
+of multi-core CPU computing resources and the potential for utilizing multiple quantum
+processing units (QPUs) in the future.
+Our approach shares some similarities with the
+evolutionary QRL method presented in [29], which also utilizes parallel computing resources.
+However, our approach differs in that individual agents can share their gradients directly
+with the shared global gradient asynchronously, rather than waiting for all agents to finish
+before calculating fitness and creating the next generation of agents. This characteristic
+may further improve the efficiency of the training process. These contributions represent a
+novel advancement in the field of quantum reinforcement learning.
+III.
+REINFORCEMENT LEARNING
+Reinforcement learning (RL) is a machine learning framework in which an agent learns to
+accomplish a given goal by interacting with an environment E in discrete time steps [31]. The
+agent observes a state st at each time step t and then chooses an action at from the action
+space A based on its current policy π. The policy is a mapping from a specific state st to the
+probabilities of choosing one of the actions in A. After performing the action at, the agent
+gets a scalar reward rt and the state of the following time step st+1 from the environment.
+4
+
+For episodic tasks, the procedure is repeated across a number of time steps until the agent
+reaches the terminal state or the maximum number of steps permitted. Seeing the state
+st along the training process, the agent aims to maximize the expected return, which can
+be expressed as the value function at state s under policy π, V π(s) = E [Rt|st = s], where
+Rt = �T
+t′=t γt′−trt′ is the return, the total discounted reward from time step t. The value
+function can be further expressed as V π(s) = �
+a∈A Qπ(s, a)π(a|s), where the action-value
+function or Q-value function Qπ(s, a) = E[Rt|st = s, a] is the expected return of choosing
+an action a ∈ A in state s according to the policy π. The Q-learning is RL algorithm to
+optimize the Qπ(s, a) via the following formula
+Q (st, at) ← Q (st, at)
++ α
+�
+rt + γ max
+a
+Q (st+1, a) − Q (st, at)
+�
+.
+(1)
+In contrast to value-based reinforcement learning techniques, such as Q-learning, which
+rely on learning a value function and using it to guide decision-making at each time step,
+policy gradient methods focus on directly optimizing a policy function, denoted as π(a|s; θ),
+parametrized by θ. The parameters θ are updated through a gradient ascent procedure
+on the expected total return, E[Rt]. A notable example of a policy gradient algorithm is
+the REINFORCE algorithm, introduced in [32]. In the standard REINFORCE algorithm,
+the parameters θ are updated along the direction ∇θ log π (at|st; θ) Rt, which is an unbiased
+estimate of ∇θE [Rt]. However, this policy gradient estimate often suffers from high variance,
+making training difficult.
+To reduce the variance of this estimate while maintaining its
+unbiasedness, a term known as the baseline can be subtracted from the return. This baseline,
+denoted as bt(st), is a learned function of the state st.
+The resulting update becomes
+∇θ log π (at|st; θ) (Rt − bt (st)). A common choice for the baseline bt(st) in RL is an estimate
+of the value function V π(st).
+Using this choice for the baseline often results in a lower
+variance estimate of the policy gradient [31]. The quantity Rt − bt = Q(st, at) − V (st) can
+be interpreted as the advantage A(st, at) of action at at state st. Intuitively, the advantage
+can be thought of as the ”goodness or badness” of action at relative to the average value
+at state st. This approach is known as the advantage actor-critic (A2C) method, where the
+policy π is the actor and the baseline, which is the value function V , is the critic [31].
+The asynchronous advantage actor-critic (A3C) algorithm [33] is a variant of the A2C
+method that employs multiple concurrent actors to learn the policy through parallelization.
+5
+
+Asynchronous training of RL agents involves executing multiple agents on multiple instances
+of the environment, allowing the agents to encounter diverse states at any given time step.
+This diminished correlation between states or observations enhances the numerical stability
+of on-policy RL algorithms such as actor-critic [33]. Furthermore, asynchronous training does
+not require the maintenance of a large replay memory, thus reducing memory requirements
+[33]. By harnessing the advantages and gradients computed by a pool of actors, A3C exhibits
+impressive sample efficiency and robust learning performance, making it a prevalent choice
+in the realm of reinforcement learning.
+IV.
+VARIATIONAL QUANTUM CIRCUIT
+Variational quantum circuits (VQCs), also referred to as parameterized quantum circuits
+(PQCs), are a class of quantum circuits that contain tunable parameters. These parame-
+ters can be optimized using various techniques from classical machine learning, including
+gradient-based and non-gradient-based methods. A generic illustration of a VQC is in the
+central part of Figure 1.
+The three primary components of a VQC are the encoding circuit, the variational circuit,
+and the quantum measurement layer. The encoding circuit, denoted as U(x), transforms
+classical values into a quantum state, while the variational circuit, denoted as V (θ), serves
+as the learnable part of the VQC. The quantum measurement layer, on the other hand,
+is utilized to extract information from the circuit. It is a common practice to repeatedly
+execute the circuit, also known as ”shots,” in order to obtain the expectation values of
+each qubit. A common choice is to use the Pauli-Z expectation values. Instead of being
+binary integers, the values are received as floats. Additionally, other components, such as
+additional VQCs or classical components such as DNN, can process the values obtained from
+the circuit.
+The VQC can operate with other classical components such as tensor networks (TN) [29,
+34, 35] and deep neural networks (NN) to perform data pre-processing such as dimensional
+reduction or post-processing such as scaling. We call such VQCs as dressed VQC, as shown
+in Figure 1. The whole model can be trained in an end-to-end manner via gradient-based
+[34, 35] or gradient-free methods [29]. For the gradient-based methods, the whole model
+can be represented as a directed acyclic graph (DAG) and then back-propagation can be
+6
+
+applied. The success of such end-to-end optimization relies on the capabilities of calculating
+the quantum gradients such as parameter-shift rule [36]. VQC-based QML models have
+shown success in areas such as classification [34–38], natural language processing [39–41]
+and sequence modeling [42, 43].
+Hybrid VQC
+U(x)
+V(θ)
+|0⟩
+|0⟩
+|0⟩
+|0⟩
+NN
+NN
+FIG. 1. Hybrid variational quantum circuit (VQC) architecture. The hybrid VQC archi-
+tecture includes a VQC and classical neural networks (NN) before and after the VQC. NN can be
+used to reduce the dimensionality of the input data and refine the outputs from the VQC.
+V.
+QUANTUM A3C
+The proposed quantum asynchronous advantage actor-critic (QA3C) framework consists
+of two main components: a global shared memory and process-specific memories for each
+agent. The global shared memory maintains the dressed VQC policy and value parameters,
+which are modified when an individual process uploads its own gradients for parameter up-
+dates. Each agent has its own process-specific memory that maintains local dressed VQC
+policy and value parameters. These local models are used to generate actions during an
+episode within individual processes. When certain criteria are met, the gradients of the
+local model parameters are uploaded to the global shared memory, and the global model
+parameters are modified accordingly. The updated global model parameters are then im-
+mediately downloaded to the local agent that just uploaded its own gradients. The overall
+concept of QA3C is depicted in Figure 2.
+We construct the quantum policy π (at | st; θ) and value V (st; θv) function with the
+dressed VQC as shown in Figure 1, in which the VQC follows the architecture shown in
+7
+
+⋯
+Worker 1
+Worker 2
+Worker 3
+Worker n
+Environment 1
+Environment 2
+Environment 3
+Environment n
+Global Parameter
+st
+π(at|st)
+V(st)
+FIG. 2.
+Quantum asynchronous advantage actor-critic (A3C) learner.
+The proposed
+quantum A3C includes a global shared parameters and multiple parallel workers.
+The action
+generation process within each local agent is performed using the dressed VQC policy and value
+functions stored in the process-specific memories. Upon meeting certain criteria, the gradients of
+the local model parameters are uploaded to the global shared memory, where the global model
+parameters are updated. The updated global model parameters are then immediately downloaded
+to the local agent that just uploaded its own gradients.
+Figure 3. This VQC architecture has been studied in the work such as quantum recurrent
+neural networks (QRNN) [42], quantum recurrent RL [17], quantum convolutional neural
+networks [44], federated quantum classification [38] and has demonstrated superior perfor-
+mance over their classical counterparts under certain conditions. In addition, we employ
+the classical DNN before and after the VQC to dimensionally reduce the data and fine-tune
+the outputs from the VQC, respectively. The neural network components in this hybrid
+architecture consist of single-layer networks for dimensionality conversion. Specifically, the
+network preceding the VQC is a linear layer with an input dimension equal to the size of the
+observation vector and an output dimension equal to the number of qubits in the VQC. The
+networks following the VQC are linear layers with input dimensions equal to the number
+of qubits in the VQC and output dimensions equal to the number of actions (for the actor
+function π (at | st; θ)) or 1 (for the critic function V (st; θv)). These layers serve to convert
+8
+
+the output of the VQC for use in the actor-critic algorithm. The policy and value function
+are updated after every S steps or when the agent reaches the terminal state. The details
+of the algorithm such as the gradient update formulas are presented in Algorithm 1.
+|0⟩
+H
+Ry(arctan(x1))
+Rz(arctan(x2
+1))
+•
+•
+R(α1, β1, γ1)
+|0⟩
+H
+Ry(arctan(x2))
+Rz(arctan(x2
+2))
+•
+•
+R(α2, β2, γ2)
+|0⟩
+H
+Ry(arctan(x3))
+Rz(arctan(x2
+3))
+•
+•
+R(α3, β3, γ3)
+|0⟩
+H
+Ry(arctan(x4))
+Rz(arctan(x2
+4))
+•
+•
+R(α4, β4, γ4)
+FIG. 3. VQC architecture for quantum A3C. The VQC used here includes Ry and Rz for
+encoding classical values x, multiple CNOT gates to entangle qubits, general unitary rotations R
+and the final measurement. The output of the VQC consists of Pauli-Z expectation values, which
+are obtained through multiple runs (shots) of the circuit.
+These values are then processed by
+classical neural networks for further use. We use a 4-qubit system as an example here, however, it
+can be enlarge or shrink based on the problem of interest. In this work, the number of qubit is 8.
+VI.
+EXPERIMENTS AND RESULTS
+A.
+Testing Environments
+1.
+Acrobot
+The Acrobot environment from OpenAI Gym [45] consists of a system with two linearly
+connected links, with one end fixed. The joint connecting the two links can be actuated by
+applying torques. The goal is to swing the free end of the chain over a predetermined height,
+starting from a downward hanging position, using as few steps as possible. The observation
+in this environment is a six-dimensional vector comprising the sine and cosine values of the
+two rotational joint angles, as well as their angular velocities. The agents are able to take
+one of three actions: applying −1, 0, or +1 torque to the actuated joint. An action resulting
+in the free end reaching the target height receives a reward of 0 and terminates the episode.
+Any action that does not lead to the desired height receives a reward of −1. The reward
+threshold is −100.
+9
+
+FIG. 4. The Acrobat environment from OpenAI Gym.
+2.
+Cart-Pole
+Cart-Pole is a commonly used evaluation environment for simple RL models that has
+been utilized as a standard example with in OpenAI Gym [45] (see Figure 5).
+A fixed
+junction connects a pole to a cart traveling horizontally over a frictionless track in this
+environment. The pendulum initially stands upright, and the aim is to keep it as near to its
+starting position as possible by moving the cart left and right. Each time step, the RL agent
+learns to produce the right action according on the observation it gets. The observation in
+this environment is a four dimensional vector st containing values of the cart position, cart
+velocity, pole angle, and pole velocity at the tip. Every time step where the pole is near to
+being upright results in a +1 award. An episode ends if the pole is inclined more than 15
+degrees from vertical or the cart moves more than 2.4 units away from the center.
+FIG. 5. The Cart-Pole environment from OpenAI Gym.
+10
+
+3.
+MiniGrid-SimpleCrossing
+The MiniGrid-SimpleCrossing environment [46] is more sophisticated, with a lot bigger
+observation input for the RL agent. In this scenario, the RL agent receives a 7 × 7 × 3 =
+147 dimensional vector through observation and must choose an action from the action
+space A, which offers six options. It is important to note that the 147-dimensional vector
+is a compact and efficient representation of the environment rather than the real pixels.
+There are six actions 0,· · · ,5 in the action space A for the agent to choose.
+They are
+turn left, turn right, move forward, pick up an object, drop the object being carried and
+toggle. Only the first three of them are having actual effects in this case. The agent is
+expected to learn this fact. In this environment, the agent receives a reward of 1 upon
+reaching the goal.
+A penalty is subtracted from this reward based on the formula 1 −
+0.9 × (number of steps/max steps allowed), where the maximum number of steps allowed is
+defined as 4 × n × n, and n is the grid size [46]. In this work, n is set to 9. This reward
+scheme presents a challenge because it is sparse, meaning that the agent does not receive
+rewards until it reaches the goal. As shown in Figure 6, the agent (shown in red triangle)
+is expected to find the shortest path from the starting point to the goal (shown in green).
+We consider three cases in this environment: MiniGrid-SimpleCrossingS9N1-v0, MiniGrid-
+SimpleCrossingS9N2-v0 and MiniGrid-SimpleCrossingS9N3-v0. Here the N represents the
+number of valid crossings across walls from the starting position to the goal.
+B.
+Hyperparameters and Model Size
+In the proposed QA3C, we use the Adam optimizer with learning rate 1×10−4, β1 = 0.92
+and β2 = 0.999. The local agents will update the parameters with the global shared memory
+every S = 5 steps. The discount factor γ is set to be 0.9. For the VQC, we set the number
+of qubits to be 8 and two variational layers are used. Therefore, for each VQC, there are
+8 × 3 × 2 = 48 quantum parameters. Actor and critic both have their own VQC, thus
+the total number of quantum parameters is 96. The VQC architecture are the same across
+various testing environments considered in this work. As we described in the Section V,
+single layer networks are used before and after the VQC to convert the dimensions of data.
+The networks preceding the VQC have input dimensions based on the environments that
+11
+
+(a)
+(b)
+(c)
+FIG. 6. The SimpleCrossing environment from MiniGrid. The three environments from
+MiniGrid-SimpleCrossing we consider in this work.
+In each environment, there are also walls
+which span 1 unit on each side (not shown in the figure).
+(a), (b) and (c) represent exam-
+ples from the MiniGrid-SimpleCrossingS9N1-v0, MiniGrid-SimpleCrossingS9N2-v0 and MiniGrid-
+SimpleCrossingS9N3-v0 environments, respectively.
+the agent is to solve. For the classical benchmarks, we consider the model which are very
+similar to the dressed VQC model. Specifically, we keep the architecture of classical model
+similar to the one presented in Figure 1 while we replace the 8-qubit VQC with a single
+layer with input and output dimensions equal to 8. This makes the architecture very similar
+to the quantum model and the number of parameters are also very close. We summarize
+the number of parameters in Table I. We utilize the open-source PennyLane package [47]
+QA3C
+Classical
+Classical Quantum Total
+Total
+Acrobot
+148
+96
+244
+292
+Cart-Pole
+107
+96
+203
+251
+SimpleCrossing
+2431
+96
+2527
+2575
+TABLE I. Number of parameters. We provide details on the number of parameters in the
+proposed QA3C model, which includes both quantum and classical components.
+The classical
+benchmarks were designed with architectures similar to the quantum model, resulting in similar
+model sizes.
+to construct the quantum circuit models and the PyTorch as a overall machine learning
+12
+
+framework. The number of CPU cores and hence the number of parallel agents is 80 in
+this work. We present simulation results in which the scores from the past 100 episodes are
+averaged.
+C.
+Results
+1.
+Acrobot
+We begin by evaluating the performance of our models on the Acrobot environment. The
+simulation results of this experiment are presented in Figure7. The total number of episodes
+was 100,000. As shown in the figure, the quantum model exhibits a gradual improvement
+during the early training episodes, while the classical model struggles to improve its policy.
+In terms of average score, the quantum model demonstrates superior performance compared
+to the classical model. Furthermore, the quantum model exhibits a more stable convergence
+pattern, without significant fluctuations or collapses after reaching optimal scores. These
+results suggest that the quantum model may be more robust and reliable in this environment.
+0
+20000
+40000
+60000
+80000
+100000
+Episode #
+500
+400
+300
+200
+100
+0
+Average Score
+Quantum
+Classical
+FIG. 7. Results: Quantum A3C in the Acrobot environment.
+13
+
+2.
+Cart-Pole
+The next experiment was conducted in the Cart-Pole environment. The total number of
+episodes was 100,000. As illustrated in Figure 8, the quantum model achieved significantly
+higher scores than the classical model. While the classical model demonstrated faster learn-
+ing in the early training episodes, the quantum model eventually surpassed it and reached
+superior scores. These results suggest that the quantum model may be more effective in this
+environment.
+0
+20000
+40000
+60000
+80000
+100000
+Episode #
+0
+100
+200
+300
+400
+500
+Average Score
+Quantum
+Classical
+FIG. 8. Results: Quantum A3C in the CartPole environment.
+3.
+MiniGrid-SimpleCrossing
+The final experiment was conducted in the MiniGrid-SimpleCrossing environment, com-
+prising a total of 100,000 episodes.
+As depicted in Figure 9, among the three scenar-
+ios, MiniGrid-SimpleCrossingS9N1-v0, MiniGrid-SimpleCrossingS9N2-v0, and MiniGrid-
+SimpleCrossingS9N3-v0, the quantum model outperformed the classical model in two of
+the three scenarios, MiniGrid-SimpleCrossingS9N2-v0 and MiniGrid-SimpleCrossingS9N3-
+v0, demonstrating faster convergence and higher scores. Even in the remaining scenario,
+MiniGrid-SimpleCrossingS9N1-v0, the difference in performance between the two models
+14
+
+was minor.
+0
+20000
+40000
+60000
+80000
+100000
+Episode #
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Average Score
+S9N1-Quantum
+S9N1-Classical
+0
+20000
+40000
+60000
+80000
+100000
+Episode #
+Average Score
+S9N2-Quantum
+S9N2-Classical
+0
+20000
+40000
+60000
+80000
+100000
+Episode #
+Average Score
+S9N3-Quantum
+S9N3-Classical
+FIG. 9. Results: Quantum A3C in the MiniGrid-SimpleCrossing environment.
+VII.
+CONCLUSION
+In this study, we demonstrate the effectiveness of an asynchronous training framework for
+quantum RL agents. Through numerical simulations, we show that in the benchmark tasks
+considered, advantage actor-critic quantum policies trained asynchronously can outperform
+or match the performance of classical models with similar architecture and sizes.
+This
+technique affords a strategy for expediting the training of quantum RL agents through
+parallelization, and may have potential applications in various real-world scenarios.
+ACKNOWLEDGMENTS
+The views expressed in this article are those of the authors and do not represent the views
+of Wells Fargo. This article is for informational purposes only. Nothing contained in this
+article should be construed as investment advice. Wells Fargo makes no express or implied
+warranties and expressly disclaims all legal, tax, and accounting implications related to this
+article.
+15
+
+Appendix A: Algorithms
+1.
+Quantum-A3C
+Algorithm 1 Quantum asynchronous advantage actor-critic learning (algorithm for each
+actor-learner process)
+Define the global update parameter S
+Assume global shared hybrid VQC policy parameter θ
+Assume global shared hybrid VQC value parameter θv
+Assume global shared episode counter T = 0
+Assume process-specific hybrid VQC policy parameter θ′
+Assume process-specific hybrid VQC value parameter θ′
+v
+Initialize process-specific counter t = 1
+while T < Tmax do
+Reset gradients dθ ← 0 and dθv ← 0
+Set tstart = t
+Reset the environment and get state st
+while st non-terminal or t − tstart < tmax do
+Perform at according to policy π(at|st; θ′)
+Receive reward rt and the new state st+1
+Update process-specific counter t ← t + 1
+if t mod S = 0 or reach terminal state then
+Set R =
+�
+�
+�
+0
+for terminal st
+V (st, θ′
+v)
+for non-terminal st
+for i ∈ {t − 1, . . . , tstart } do
+R ← ri + γR
+Accumulate gradients wrt θ′: dθ ← dθ + ∇θ′ log π (ai | si; θ′) (R − V (si; θ′
+v))
+Accumulate gradients wrt θ′
+v: dθv ← dθv + ∂ (R − V (si; θ′
+v))2 /∂θ′
+v
+end for
+Perform asynchronous update of θ using dθ and of θv using dθv
+Update process-specific parameters from global parameters: θ′ ← θ and θ′
+v ← θv
+end if
+end while
+end while
+16
+
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+20
+

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+arXiv:2301.03562v1  [math.FA]  9 Jan 2023
+(Non-)amenability of B(E) and Banach space geometry
+Matthew Daws and Matthias Neufang
+Abstract
+Let E be a Banach space, and B(E) the algebra of all bounded linear operators on E. The question
+of amenability of B(E) goes back to Johnson’s seminal memoir [39] from 1972. We present the first
+general criteria applying to very wide classes of Banach spaces, given in terms of the Banach space
+geometry of E, which imply that B(E) is non-amenable. We cover all spaces for which this is known so
+far (with the exception of one particular example), with much shorter proofs, such as ℓp for p ∈ [1, ∞]
+and c0, but also many new spaces: the numerous classes of spaces covered range from all Lp-spaces
+for p ∈ (1, ∞) to Lorentz sequence spaces and reflexive Orlicz sequence spaces, to the Schatten classes
+Sp for p ∈ [1, ∞], and to the James space J, the Schlumprecht space S, and the Tsirelson space T ,
+among others. Our approach also highlights the geometric difference to the only space for which B(E)
+is known to be amenable, the Argyros–Haydon space, which solved the famous scalar-plus-compact
+problem.
+1
+Introduction
+Amenability of Banach algebras is a central notion in functional analysis; cf., e.g., [56, 59]. We mention
+Johnson’s fundamental result [39] that a locally compact group G is amenable if and only if its group
+algebra L1(G) is amenable. Amenability is also a key concept in operator algebra theory. We note the
+deep classical result, due to Connes [21], Bunce–Paschke [14] and Haagerup [38], that for C∗-algebras,
+amenability is equivalent to nuclearity (the forward implication is due to Connes and Bunce–Paschke).
+Together with work of Wassermann [66], it follows that the algebra B(H) of all bounded linear operators
+on a Hilbert space H is non-amenable. However, it was a long-standing open problem whether this holds
+for B(ℓp) for all p ∈ [1, ∞]. Indeed, Johnson already asked in 1972 in [39], where he introduced the very
+notion of amenability, whether B(E) is ever amenable for an infinite-dimensional Banach space E. Since
+then, determining if B(E) is amenable or not for a given infinite-dimensional Banach space E, has proven
+a very difficult open problem; cf. the survey article [58].
+Let us briefly recall the history of this problem. For many years, ℓ2 remained the only space for
+which the problem was solved. Then Read showed in [54] that B(ℓ1) is non-amenable. His proof was
+simplified by Pisier in [51], using expanders, and further simplified by Ozawa in [47], where also a proof
+of non-amenability of B(ℓ2) is given avoiding the use of the above results of Connes, Bunce–Paschke and
+Wassermann. Moreover, it is shown by Ozawa in [47] that B(ℓ∞) and (implicitly) B(c0) are not amenable.
+However, the case of B(ℓp) for p ∈ (1, ∞)\{2} remained open, as noted by Ozawa in [47], by Pisier in [51],
+and by Read in [54]. In [57], Runde finally established the celebrated result that B(ℓp) is not amenable
+for any p ∈ (1, ∞), through a technically involved proof, building in particular on his earlier work with
+Daws ([26], [27]) and on Ozawa’s work. More generally, it is shown in [57] that B(ℓp(E)) is non-amenable
+for all Lp-spaces E, where p ∈ (1, ∞). Moreover, Choi shows in [19] that B(L1[0, 1]) is non-amenable;
+this is also proven by Aldabbas in [2]. Further, Choi gives a criterion, applying to E = T (ℓ2), the trace
+class operators on ℓ2, showing that B(E) is not amenable; see Remark 4.19 below. However, as noted by
+Dales (cf. [57, Corollary 2.5]), the Argyros–Haydon space, which solved the famous scalar-plus-compact
+problem, provides an example of an infinite-dimensional space E such that B(E) is amenable. Note that
+this space is a hereditarily indecomposable L∞-space.
+1
+
+In this paper, we develop entirely different methods from the ones used so far and establish, employing
+our new techniques, the first general principles which encode non-amenability of B(E), for very wide classes
+of spaces E, in terms of the Banach space geometry of E. For instance, we show that if E is reflexive
+with the approximation property and isomorphic to its square, then B(E) is non-amenable. We also show
+that if E is infinite-dimensional and reflexive with a subsymmetric basis, then B(E) is non-amenable.
+Our results cover all spaces known so far (with the exception of one particular example, cf. Remark 4.19
+below), with much shorter proofs, and many new (classes of) spaces. The following is a list of spaces E
+– obviously always assumed infinite-dimensional – for which we show that B(E) is non-amenable, as a
+result of our general approach (we write “∼=” to denote a Banach space isomorphism):
+1. all Lp(Ω, B, µ)-spaces and, more generally, all Lp spaces for p ∈ (1, ∞) – this generalises the main
+result of [57];
+2. the Lorentz sequence spaces d(w, p), the Garling sequence spaces g(w, p), and the Baernstein spaces
+Bp, for all p ∈ (1, ∞);
+3. all reflexive Orlicz sequence spaces whose Orlicz function satisfies the ∆2 condition at 0;
+4. all separable reflexive rearrangement invariant (r.i.) function spaces;
+5. the Schatten classes Sp for all p ∈ [1, ∞];
+6. the non-commutative Lp-spaces Lp(M) for all p ∈ [2, ∞), whenever M is an infinite-dimensional
+von Neumann algebra such that Lp(M) has the approximation property;
+7. c0, ℓ1, ℓ∞;
+8. the non-commutative counterparts of the above, i.e., the spaces K(H), T (H) and B(H) of compact,
+trace class and all bounded linear operators on a separable Hilbert space H, and, more generally,
+K(ℓp), N(ℓp) and B(ℓp) for all p ∈ (1, ∞);
+9. C(K) for any infinite compact metric space K (that is, all separable C(K) spaces), and all separable
+L1(Ω, B, µ)-spaces (note that L∞[0, 1] ∼= ℓ∞, so this space is covered as well);
+10. the Hardy spaces Hp for all p ∈ [1, ∞) (note that Hp ∼= Lp[0, 1] for p ∈ (1, ∞));
+11. the vector-valued spaces Lp(µ, X) for all p ∈ (1, ∞), for σ-finite µ with Lp(µ) infinite-dimensional,
+whenever X∗ has the bounded approximation property and the Radon-Nikodym Property (for ex-
+ample, X is reflexive with the approximation property);
+12. the vector-valued spaces C(K, X) for any infinite compact metric space K, whenever X∗ has the
+bounded approximation property and the Radon-Nikodym Property;
+13. particular spaces such as the James space J, the Schlumprecht space S, and the Tsirelson space T.
+The paper is organized as follows. In Section 2, we first establish a result on a new hereditary property
+of amenability which generalises a well-known theorem of Gourdeau and Ghahramani–Loy–Willis. From
+this we shall derive, in Section 3, the general criteria for non-amenability of B(E) in the case of reflexive
+spaces E. We treat the non-reflexive situation in Section 4. In the last Section, we present an alternative,
+short proof of the non-amenability of B(ℓp) for all p ∈ (1, ∞] which uses operator algebra techniques
+and harmonic analysis, instead of Banach space geometry. We also present an “elementary” proof of the
+non-amenability of B(ℓ2), which has long been sought after: it is even shorter and of course avoids the
+use of nuclearity for C∗-algebras.
+2
+
+2
+A new hereditary property of amenability
+The central idea of this paper is the following.
+For a Banach space E, write A(E) for the space of
+approximable operators, the norm closure of the finite-rank operators in B(E).
+Write K(E) for the
+compact operators on E. Often A(E) = K(E); this is true when E has the approximation property
+(AP), [60, Chapter 4]. When E is reflexive with the AP, we can identify the bidual K(E)∗∗ with B(E),
+where K(E)∗∗ is given the (first) Arens product.
+Indeed, K(E) is Arens regular, and so both Arens
+products agree. This result goes back to [48, Theorem 2], see also [24, Section 6], and in more generality,
+[25]. Consequently, to study B(E), one might study the bidual of A = K(E). A well-known result in
+this direction if that when A∗∗ is amenable, also A is amenable, a result shown by Gourdeau ([34], [35,
+Theorem 2.3]) and, independently, Ghahramani–Loy–Willis [33, Theorem 1.8]. This result is not directly
+useful to us, as K(E) is often amenable (cf. [11, 36]). We will thus substantially generalise this result
+below, and this generalisation will be central to our new approach.
+With this motivation outlined, we start with some generality, and consider a Banach algebra A and
+its bidual A∗∗ equipped with the first Arens product. Let us recall the Arens products. We turn A∗, A∗∗
+into A-bimodules in the usual way. We then define bilinear maps A∗∗ × A∗ → A∗, A∗ × A∗∗ → A∗ by
+⟨F · µ, a⟩ = ⟨F, µ · a⟩,
+⟨µ · F, a⟩ = ⟨F, a · µ⟩
+(a ∈ A, µ ∈ A∗, F ∈ A∗∗).
+We then define bilinear maps ✷, ✸ : A∗∗ × A∗∗ → A∗∗ by
+⟨F✷G, µ⟩ = ⟨F, G · µ⟩,
+⟨F✸G, µ⟩ = ⟨G, µ · F⟩
+(F, G ∈ A∗∗, µ ∈ A∗).
+Direct calculations show that these are Banach algebra products, the first and second Arens products,
+respectively. The canonical map κA : A → A∗∗ is a homomorphism, and κA(a)✷F = a·F, F✷κA(a) = F·a,
+and similarly for ✸. Henceforth, we always equip A∗∗ with ✷ unless otherwise stated.
+Our first step, following [33], is to link A∗∗ �⊗A∗∗ with (A�⊗A)∗∗, where �⊗ denotes the completed
+projective Banach space tensor product. We wish to use slightly more concrete identifications than [33],
+and we shall hence identify (A�⊗A)∗ with B(A, A∗), say B(A, A∗) ∋ T ↔ ϕT ∈ (A�⊗A)∗ by
+⟨ϕT , a ⊗ b⟩ = ⟨T(a), b⟩
+(a, b ∈ A).
+(1)
+Compare with [60, Section 2.2], for example. Define Ψ : A∗∗ �⊗A∗∗ → (A�⊗A)∗∗ by
+⟨Ψ(F ⊗ G), ϕT ⟩ = ⟨F, T ∗(G)⟩
+(F, G ∈ A∗∗).
+(2)
+Clearly Ψ extends by bi-linearity and continuity to a contraction. Let us just check that this agrees with
+[33, Lemma 1.7], so choose bounded nets (ai), (bj) in A converging weak∗ to F, G ∈ A∗∗, respectively.
+Then
+⟨Ψ(F ⊗ G), ϕT ⟩ = ⟨F, T ∗(G)⟩ = lim
+i ⟨T ∗(G), ai⟩ = lim
+i lim
+j ⟨T(ai), bj⟩ = lim
+i lim
+j ⟨ϕT , ai ⊗ bj⟩,
+which is the same extension given by [33, Lemma 1.6] (cf. [5, §3]).
+We now record some useful facts about this map.
+Proposition 2.1 ([33, Lemma 1.7]). We have the following commutative diagram
+A∗∗ �⊗A∗∗
+Ψ
+� (A�⊗A)∗∗
+A�⊗A
+��
+κA⊗κA
+�
+��
+κA �
+⊗A
+�q
+q
+q
+q
+q
+q
+q
+q
+q
+q
+With πA : A�⊗A → A the product map, and similarly for πA∗∗, we have that (πA)∗∗ ◦ Ψ = πA∗∗. Further-
+more, Ψ is an A-bimodule map.
+3
+
+An obvious, but key, property is that A(E) is always an ideal in B(E). We abstract this idea as follows.
+Firstly identify A with κA(A) ⊆ A∗∗. Suppose that B ⊆ A∗∗ is some closed subalgebra containing A as
+an ideal; we write A ✂ B ⊆ A∗∗. As A is an ideal in B, naturally A is a B-bimodule, and hence also A�⊗A
+is a B-bimodule. In the standard way, hence also A∗ and A∗∗, so also A∗∗ �⊗A∗∗, and (A�⊗A)∗∗, become
+B-bimodules. However, as B ⊆ A∗∗, also A∗∗ and A∗ have B-actions for the restriction of the A∗∗ actions.
+Lemma 2.2. The two B actions on A∗ agree, while the left action of B on A∗∗ agrees with ✸, and the
+right action of B on A∗∗ agrees with ✷.
+Proof. In this proof, to avoid confusion, let us write b ⊲ a and a ⊳ b for a ∈ A, b ∈ B, to denote the
+B-bimodule actions arising from viewing A as an ideal in B, and similarly for the actions of B on A∗ and
+A∗∗. Given b ∈ B, a ∈ A, µ ∈ A∗, we find that
+⟨b ⊲ µ, a⟩ = ⟨µ, a ⊳ b⟩ = ⟨a · b, µ⟩ = ⟨b, µ · a⟩ = ⟨b · µ, a⟩,
+and so b ⊲ µ = b · µ. To be precise, when we write “a · b” we are considering b ∈ A∗∗ and the natural A
+action on A∗∗; similarly b · µ is the A∗∗ action on A∗ where we view B as a subalgebra of A∗∗.
+Similarly
+⟨µ ⊳ b, a⟩ = ⟨µ, b ⊲ a⟩ = ⟨b · a, µ⟩ = ⟨b, a · µ⟩ = ⟨µ · b, a⟩,
+so that µ ⊳ b = µ · b.
+Then, given b ∈ B, F ∈ A∗∗, µ ∈ A∗, we have
+⟨b ⊲ F, µ⟩ = ⟨F, µ ⊳ b⟩ = ⟨F, µ · b⟩ = ⟨b✸F, µ⟩,
+⟨F ⊳ b, µ⟩ = ⟨F, b ⊲ µ⟩ = ⟨F, b · µ⟩ = ⟨F✷b, µ⟩,
+as claimed.
+As both A∗∗ �⊗A∗∗ and (A�⊗A)∗∗ are B-bimodules, we might ask if Ψ is a B-bimodule map. Again, the
+situation is slightly complicated as both Arens products arise.
+Lemma 2.3. Given b ∈ B and F, G ∈ A∗∗ we have
+b · Ψ(F ⊗ G) = Ψ(b✸F ⊗ G),
+Ψ(F ⊗ G) · b = Ψ(F ⊗ G✷b).
+Proof. Again, we identify T ∈ B(A, A∗) with ϕT ∈ (A�⊗A)∗ as in (1), and in the proof we continue to
+write ⊲, ⊳ for the B-bimodule actions. For b ∈ B, a1, a2 ∈ A,
+⟨b ⊲ ϕT , a1 ⊗ a2⟩ = ⟨ϕT , a1 ⊗ a2 ⊳ b⟩ = ⟨T(a1), a2 ⊳ b⟩ = ⟨b ⊲ T(a1), a2⟩,
+⟨ϕT ⊳ b, a1 ⊗ a2⟩ = ⟨ϕT , b ⊲ a1 ⊗ a2⟩ = ⟨T(b ⊲ a1), a2⟩.
+Set b ⊲ ϕT = ϕT1 and ϕT ⊳ b = ϕT2, so that T1(a1) = b ⊲ T(a1) and T2(a1) = T(b ⊲ a1). Then, for G ∈ A∗∗,
+⟨T ∗
+1 (G), a1⟩ = ⟨G, b ⊲ T(a1)⟩ = ⟨T ∗(G ⊳ b), a1⟩,
+⟨T ∗
+2 (G), a1⟩ = ⟨G, T(b ⊲ a1)⟩ = ⟨T ∗(G), b ⊲ a1⟩ = ⟨T ∗(G) ⊳ b, a1⟩.
+Then, for F, G ∈ A∗∗, from (2), and using Lemma 2.2,
+⟨b · Ψ(F ⊗ G), ϕT ⟩ = ⟨Ψ(F ⊗ G), ϕT2⟩ = ⟨F, T ∗
+2 (G)⟩ = ⟨F, T ∗(G) ⊳ b⟩ = ⟨Ψ(b✸F ⊗ G), ϕT ⟩,
+⟨Ψ(F ⊗ G) · b, ϕT ⟩ = ⟨Ψ(F ⊗ G), ϕT1⟩ = ⟨F, T ∗
+1 (G)⟩ = ⟨F, T ∗(G ⊳ b)⟩ = ⟨Ψ(F ⊗ G✷b), ϕT ⟩,
+as claimed.
+4
+
+We write Zt(A∗∗) for the (first) topological centre (denoted by Z(1)
+t
+(A∗∗) in [24, 25]), that is,
+Zt(A∗∗) = {F ∈ A∗∗ : F✷G = F✸G (G ∈ A∗∗)}.
+The following is now immediate from Lemmas 2.2 and 2.3.
+Corollary 2.4. Let A ✂ B ⊆ A∗∗ and suppose that B ⊆ Zt(A∗∗). Then the B-bimodule actions on A∗∗
+agree with the product ✷, and Ψ is a B-bimodule map.
+We now state and prove the main result of this Section, which shows that amenability of A∗∗ (or more
+generally a subalgebra) passes to B when A ✂ B. This generalises [35, Theorem 2.3] and [33, Theorem
+1.8], where the statement is shown for the special case B = A and C = A∗∗.
+Theorem 2.5. Let A✂B ⊆ A∗∗ and suppose that B ⊆ Zt(A∗∗). Let C ⊆ A∗∗ be a closed subalgebra which
+is amenable, with B ⊆ C. Then B is amenable.
+Proof. The canonical maps
+A�⊗A → B�⊗B → C �⊗C → A∗∗ �⊗A∗∗
+are all contractions, but the overall composition is an isometry, [60, Corollary 2.14], and so each individual
+map must also be an isometry, and we identify each tensor product as a closed subspace of A∗∗ �⊗A∗∗. As
+B ⊆ Zt(A∗∗), Lemma 2.2 shows that each inclusion is also a B-bimodule map.
+As C is amenable, [56, Theorem 2.2.4] or [59, Theorem 2.2.5], it has a bounded approximate diagonal
+(di) ⊆ C �⊗C ⊆ A∗∗ �⊗A∗∗, that is,
+∥c · di − di · c∥ → 0,
+∥πC(di)✷c − c∥ → 0
+(c ∈ C).
+(3)
+For each i let ni = Ψ(di) ∈ (A�⊗A)∗∗. As B ⊆ C, (3) holds for each member of B, and so it follows from
+Corollary 2.4 that for each b ∈ B,
+∥b · ni − ni · b∥ = ∥Ψ(b · di) − Ψ(di · b)∥ ≤ ∥b · di − di · b∥ → 0,
+(4)
+∥(πA)∗∗(ni)✷b − b∥ = ∥(πA)∗∗(Ψ(di))✷b − b∥ = ∥πA∗∗(di)✷b − b∥ = ∥πC(di)✷b − b∥ → 0,
+(5)
+the second claim using Proposition 2.1.
+The bi-adjoint of the inclusion A�⊗A → B�⊗B gives an (isometric) inclusion (A�⊗A)∗∗ → (B�⊗B)∗∗.
+For each i let mi be the image of ni in (B�⊗B)∗∗, and let m ∈ (B�⊗B)∗∗ be a weak∗-cluster point of the
+bounded net (mi). As the B-bimodule actions on (B�⊗B)∗∗ are weak∗-continuous, it follows from (4) that
+b · m = m · b for each b ∈ B. As A is a subalgebra of B, it follows that (πB)∗∗(mi) = (πA)∗∗(ni) for each
+i, and so (5) shows that (πB)∗∗(m)✷b = b for each b ∈ B. That is, m is a virtual diagonal for B, showing
+that B is amenable, [56, Theorem 2.2.4] or [59, Theorem 2.2.5].
+We immediately obtain the following
+Corollary 2.6. Let A ✂ B ⊆ A∗∗ and suppose that A is Arens regular. If A∗∗ is amenable, then so is B.
+3
+The general criteria in the reflexive case, and applications
+In this Section, we apply Theorem 2.5 in the classical situation when E is reflexive with the AP. As
+explained above, it follows that if we set A = A(E) then A = K(E) and A∗∗ is isomorphic to B(E). As
+A is Arens regular in this case, the condition on Zt(A∗∗) is vacuous; see Corollary 2.6. Our aim is to use
+contradiction to show that B(E) cannot be amenable, by applying Corollary 2.6 to a suitable algebra B
+which is “obviously” not amenable. Our technique for finding such B is the following idea which, combined
+with Theorem 2.5 or Corollary 2.6, is key to our simplified approach.
+5
+
+Proposition 3.1. Let E be a Banach space, and suppose there exists T ∈ B(E) which is not compact,
+but with T 2 ∈ K(E). Let B be the Banach algebra generated by K(E) and T. Then B is the linear span
+of K(E) and T, and B is not amenable.
+Proof. As T 2 ∈ K(E) it is easy to see that B is the linear span of K(E) and T. Consider the quotient
+algebra C = B/K(E), and let x be the image of T in this quotient. As T is non-compact, x ̸= 0, but x2 = 0
+as T 2 is compact. Thus C is the one-dimensional algebra spanned by x. As C is obviously not unital, it
+cannot be amenable. Thus B also cannot be amenable, as amenability passes to quotient algebras.
+Corollary 3.2. Let E be a reflexive Banach space with the AP such that there exists T ∈ B(E) which is
+not compact, but with T 2 ∈ K(E). Then B(E) is not amenable.
+Proof. This follows from Corollary 2.6 applied with A = A(E), so that A∗∗ = B(E), and with B as in
+Proposition 3.1.
+Theorem 3.3. Let E be a reflexive Banach space with the AP such that E ∼= E0 ⊕ E1 with E0, E1
+isomorphic as Banach spaces. When E is infinite-dimensional, B(E) is not amenable.
+Proof. Let T ∈ B(E) be the composition of the projection from E onto E0, the isomorphism from E0
+to E1, and the inclusion E1 → E. Then T 2 = 0. As E is infinite dimensional, so are E0, E1, and hence
+T is not compact. Indeed, by the Riesz Lemma, we can find a sequence of unit vectors (xn) in E0 with
+∥xn − xm∥ ≥ 1/2 for n ̸= m. Treating E0 ⊆ E, we have that (T(xn)) is a bounded sequence of vectors,
+with ∥T(xn) − T(xm)∥ ≥ c/2 for each n ̸= m, where c > 0 is a constant depending on the isomorphisms
+E ∼= E0 ⊕ E1 and E0 ∼= E1. Thus T cannot be compact. The result then follows from Corollary 3.2.
+Corollary 3.4. Let E be a reflexive Banach space with the AP. If E ∼= E ⊕E then B(E) is not amenable.
+For the definition of tree translation equivalent Banach spaces used below, we refer the reader to [10,
+Section 4]. This technical definition is useful to us precisely because it allows us to prove the following.
+Corollary 3.5. Let E be a reflexive, tree translation equivalent Banach space. Then B(E) is not amenable.
+Proof. By assumption, E has a tree translation equivalent basis, so in particular has a basis and hence
+has the AP, and by [10, Theorem 4.6], satisfies E ∼= E ⊕ E. Corollary 3.4 now yields the claim.
+Generalising the above idea, we obtain the following.
+Theorem 3.6. Let E be a reflexive Banach space with the AP, such that there exist closed, infinite-
+dimensional, isomorphic subspaces E0 and E1, and a projection P from E onto E0 with P(E1) finite-
+dimensional. Then B(E) is not amenable.
+Proof. Let T0 : E0 → E1 be an isomorphism, let P : E → E0 be a projection, and set T = T0P : E →
+E1 ⊆ E. As PT0P(x) ∈ P(E1) for any x ∈ E, and as P(E1) is finite-dimensional, PT0P is compact,
+and so T 2 is compact. As T(x) = T0(x) for each x ∈ E0, and as E0 is infinite-dimensional, and T0 an
+isomorphism, it follows that T is not compact. The result now follows from Corollary 3.2.
+For the next result, recall the notion of a subsymmetric basis, [46, Definition 3.a.2], which is an
+unconditional basis (en) such that (eni) is equivalent to (en) for all increasing sequences (ni).
+Corollary 3.7. Let E be a reflexive Banach space with the AP. If E has an infinite-dimensional com-
+plemented subspace with a subsymmetric basis, then B(E) is not amenable.
+In particular, if E is an
+infinite-dimensional reflexive Banach space with a subsymmetric basis, then B(E) is not amenable.
+6
+
+Proof. Let (ei) be the subsymmetric basis of the infinite-dimensional complemented subspace X. Put
+E0 := lin{e2i | i ∈ N} and E1 := lin{e2i−1 | i ∈ N}. As (ei) is subsymmetric, by definition the map
+ei �→ e2i extends linearly and continuously to an isomorphism between E and E0; similarly E and E1 are
+isomorphic, whence also E0 ∼= E1. As X is complemented, we have a projection P from E onto X. As (ei)
+is unconditional, we have a projection Q from X onto E0. Obviously, QP(E) = E0 and QP(E1) = {0}.
+By Theorem 3.6, we obtain that B(E) is non-amenable.
+We now apply the above results to various classes of Banach spaces. We start with Lp-spaces; for
+details on this very important class of spaces, we refer the reader to [3, Section 5] or [44, 45]. We use [28,
+Section 23] below, which takes a different definition, but [28, Section 23.3] shows that the latter covers
+the Lp-spaces.
+Corollary 3.8. Let E be any infinite-dimensional Lp-space, where p ∈ (1, ∞); for instance, E is any
+infinite-dimensional Lp(Ω, B, µ) space. Then B(E) is not amenable.
+Proof. Let E be an Lp-space, for p ∈ (1, ∞). By [44, Theorem 7.1], E is isomorphic to a complemented
+subspace of an Lp space, so certainly reflexive. Further, by [28, Section 21.6, Corollary 1], taking account
+of the aforementioned [28, Section 23.3], E has the bounded approximation property. Finally, by [44,
+Proposition 7.3], E contains a complemented subspace isomorphic to ℓp. As ℓp has a symmetric basis,
+Corollary 3.7 yields the claim.
+Remark 3.9. The above corollary generalises [57, Theorem 4.4], the main result of [57], and provides
+a much shorter proof. More precisely, it is shown in [57, Theorem 4.4] that B(ℓp(E)) is non-amenable
+for any Lp-space E. We note that ℓp(E) is again an Lp- or an L2-space. Indeed, E is isomorphic to
+a complemented subspace of some Lp-space, by [45, Corollary 1]. Hence, ℓp(E) is also isomorphic to a
+complemented subspace of some Lp-space. Thus, ℓp(E) is an Lp- or an L2-space, by [45, Corollary 1].
+(See also the introduction to [3, Section 5].)
+The Baernstein space B2 was introduced by Baernstein in [8], and the p generalisations Bp by Seifert
+in his dissertation [63]. They are now viewed as being strongly related to Tsirelson’s space, see [16].
+Corollary 3.10. The Baernstein spaces Bp satisfy that B(Bp) is non-amenable for all p ∈ (1, ∞).
+Proof. Each Bp is reflexive and has a basis, hence the AP, and contains a complemented subspace iso-
+morphic to ℓp; cf. [16, Theorem 0.15]. As above, the claim now follows from Corollary 3.7.
+We now consider the most fundamental examples of non-commutative Lp spaces, namely the Schatten
+classes Sp. Recall that Sp is the collection of operators u in B(ℓ2) with Tr(|u|p) < ∞, and norm ∥u∥p =
+Tr(|u|p)1/p. For p ∈ (1, ∞), Sp is reflexive (having canonically dual Sq for 1
+p + 1
+q = 1). Letting Pn ∈ B(ℓ2)
+be the projection onto the first n coordinates, we have that PnuPn ∈ Sp for each u ∈ Sp, with ∥PnuPn∥p ≤
+∥u∥p. Further, PnuPn → u is norm, hence showing that Sp has the (metric) approximation property.
+Corollary 3.11. For p ∈ (1, ∞) the Schatten class Sp satisfies that B(Sp) is not amenable.
+Proof. The operation of projecting an operator in B(ℓ2) onto its diagonal restricts to Sp and gives a
+projection of Sp onto a subspace isomorphic to ℓp, see the discussion in [4, pages 84–85] for example. The
+claim now follows from Corollary 3.7.
+More generally, given a von Neumann algebra M, one can consider the non-commutative Lp-spaces
+over M, denoted by Lp(M). Again, for p ∈ (1, ∞), Lp(M) is reflexive (having canonically dual Lq(M) for
+1
+p + 1
+q = 1). To our knowledge, it is not in general known when Lp(M) has the (Banach space) AP, but
+there has been some study of when Lp(M) possesses various Operator Space approximation properties,
+7
+
+all of which imply the AP; see for example [41] which shows in particular that for a discrete group Γ with
+the (group) approximation property, and with M = V N(Γ) the group von Neumann algebra, Lp(M) has
+the Operator Space Approximation Property, [41, Theorem 1.1].
+Corollary 3.12. Let p ∈ [2, ∞), and let M be an infinite-dimensional von Neumann algebra such that
+Lp(M) has the AP. Then B(Lp(M)) is not amenable.
+Proof. By [53, Theorem 0.2], Lp(M) contains a complemented subspace isomorphic to ℓ2 or ℓp. Under
+our hypothesis, Corollary 3.7 now applies to give the result.
+In the following, we consider various classes of “classical” Banach spaces; in the statement of the result
+we give references regarding the properties of the spaces needed for Corollary 3.7 to apply.
+Corollary 3.13. For the following infinite-dimensional Banach spaces E we have that B(E) is non-
+amenable:
+(i) the Lorentz sequence spaces d(w, p) for all p ∈ (1, ∞), see [46, Section 3.a, Section 4.e];
+(ii) the reflexive Orlicz sequence spaces whose Orlicz function satisfies the ∆2 condition at 0, see [46,
+Section 4.a, Proposition 4.a.4], and for when such a space is reflexive, [46, Proposition 4.b.2];
+(iii) the Garling sequence spaces g(w, p) for all p ∈ (1, ∞), see [1, Proposition 2.4, Theorem 3.1];
+(iv) the Schlumprecht space S, [62], which is reflexive, [18, Theorem 2.1] or [6, Corollary 8.3].
+Proof. Each Banach space above is infinite-dimensional and reflexive with a subsymmetric basis, which
+the given references show. Corollary 3.7 hence implies the claims.
+For the Tsirelson space T, we follow the modern convention and view T as the dual of the original
+construction of Tsirelson, so T is as defined in [31, Section 2].
+Corollary 3.14. For the following Banach spaces E we have that B(E) is non-amenable:
+(i) all infinite-dimensional separable reflexive rearrangement invariant (r.i.) function spaces;
+(ii) the Tsirelson space T.
+Proof. Each Banach space above is infinite-dimensional, reflexive (cf. [31, Section 2] for T) and, by
+[10, Examples 4.3 and 4.4], tree translation equivalent (note that the Haar system is a tree translation
+equivalent basis in case (i)). Hence Corollary 3.5 yields the claims.
+Remark 3.15. The isomorphism T ∼= T ⊕T, used by Corollary 3.5, also appears in [9]. There, the author
+defines a generalised way of constructing “Tsirelson-like” spaces, which are denoted by S. The space T
+follows as a special case, see [9, page 209]. The remarks after [9, Corollary 4.7] show that S ∼= S ⊕ T and
+so in particular, T ∼= T ⊕ T.
+In this direction, we should also remark that A(T) is known to be non-amenable, [11, Corollary 5.8],
+and so [33, 35] shows also that B(T) cannot be amenable.
+Finally, we note that the uniformly convex Banach space E (of Tsirelson type) with symmetric basis
+which contains no isomorphic copy of any ℓp for p ∈ (1, ∞), constructed by Figiel–Johnson in [31, Section
+4], also satisfies that B(E) is non-amenable, by Corollary 3.7.
+We finish this Section by comparing these constructions with the one known example of an infinite-
+dimensional space E with B(E) amenable.
+8
+
+Remark 3.16. Note that the Argyros–Haydon space X, [7], for which B(X) is amenable, [57, Corol-
+lary 2.5], is a hereditarily indecomposable L∞-space. Its dual X∗ is isomorphic to ℓ1 and hence has the
+AP, so X has the AP, [60, Corollary 4.7]. Yet, besides being non-reflexive, the key geometric property
+of X of being hereditarily indecomposable is in stark contrast to the Banach space properties which we
+employ above, all of which say that the space is “very decomposable”.
+4
+The case of non-reflexive Banach spaces
+In this Section, we treat the case when E is not reflexive. Then A(E) is not Arens regular, and so we
+need to consider the (first) topological centre when applying Theorem 2.5.
+We follow [25], which is a dense article, so we recall some of the definitions. Given a Banach space E
+we write N(E) for the nuclear operators on E, the image of E∗ �⊗E in B(E) equipped with the quotient
+norm. Write I(E) for the integral operators; there is always a norm-decreasing inclusion N(E) ⊆ I(E).
+Given x∗ ∈ E∗ and x ∈ E we write θx,x∗ for the rank-one operator y �→ ⟨x∗, y⟩x. Then A(E) is by
+definition the closed linear span of such operators in B(E). Trace duality gives a natural pairing between
+A(E) and I(E∗) which extends the pairing
+⟨J, θx,x∗⟩ = Tr(Jθ∗
+x,x∗) = ⟨J(x∗), x⟩
+(J ∈ I(E∗), θx,x∗ ∈ A(E)).
+With respect to this pairing, we have that A(E)∗ = I(E∗).
+For T ∈ B(E∗∗) define
+η(T) = κ∗
+E ◦ T ∗ ◦ κE∗ ∈ B(E∗),
+and let Q(T) = η(T)∗. Direct calculation shows that η(T ∗) = T for T ∈ B(E∗), as κ∗
+ET ∗∗κE∗ = κ∗
+EκE∗T =
+T. Similarly, given S ∈ B(E∗∗) we see that η(T ∗S) = κ∗
+ES∗T ∗∗κE∗ = η(S)T and so Q(T ∗S) = T ∗Q(S).
+Define a bilinear map ⋆ on B(E∗∗) by T ⋆ S = Q(T) ◦ S. Then ⋆ is a Banach algebra product, see [25,
+Proposition 2.5].
+We write W(E) for the ideal of weakly compact operators in B(E).
+Proposition 4.1. Let E be a Banach space. Suppose that A(E) admits a bounded approximate identity
+(eα). This holds if and only if E∗ has the bounded approximation property (BAP). Let Φ0 ∈ A(E)∗∗ be a
+weak∗-cluster point of the net (eα). There are bounded maps ψ1, ψ2 : B(E∗∗) → A(E)∗∗ = I(E∗)∗ given
+by
+⟨ψ1(T), J⟩ = ⟨Φ0, η(TJ∗)⟩,
+⟨ψ2(T), J⟩ = ⟨Φ0, η(T)J⟩
+(T ∈ B(E∗∗), J ∈ I(E∗)).
+Then ψ1 is an isomorphism onto its range, and a homomorphism B(E∗∗) → (A(E)∗∗, ✷), and ψ2 is
+a homomorphism (B(E∗∗), ⋆) → (A(E)∗∗, ✸).
+The map ψ2, restricted to {T ∗ : T ∈ B(E∗)}, is an
+isomorphism onto its range. For a ∈ A(E) we have that ψ1(a∗∗) = ψ2(a∗∗) = κA(E)(a). For T ∈ W(E)
+we have that ψ1(T ∗∗) = ψ2(T ∗∗).
+We have the identification
+Zt(A(E)∗∗) =
+�
+ψ2(T ∗) : T ∈ B(E∗), TI(E∗) ⊆ N(E∗)
+�
+.
+Proof. The equivalence of A(E) having a bai and E∗ having the BAP is shown in [37], building on the
+work of many authors. [25, Theorem 5.17] shows the claims about ψ1, ψ2, while that ψ1(T ∗∗) = ψ2(T ∗∗)
+for T ∈ W(E) is observed at the end of the proof of [25, Corollary 5.22]. Finally [25, Corollary 5.22]
+shows the claim about Zt(A(E)∗∗), which is denoted by Z(1)
+t
+(A(E)∗∗) in [25].
+To obtain a class of operators T ∈ B(E∗) with TI(E∗) ⊆ N(E∗), we use [25, Theorem 3.31] or [60,
+Theorem 5.47], which shows that when T ∈ W(E∗), we have TI(E∗) ⊆ N(E∗).
+9
+
+Theorem 4.2. Let E be a Banach space such that E∗ has the BAP. Suppose there is S ∈ W(E) \ K(E)
+with S2 ∈ K(E). Then B(E) is not amenable.
+Proof. That S is weakly compact means that S∗ is weakly compact by Gantmacher’s Theorem, [22,
+Theorem 5.5] for example, and so ψ2(S∗∗) is in Zt(A(E)∗∗). Further, ψ1(S∗∗) = ψ2(S∗∗).
+Set A = A(E) and let C = {ψ1(T ∗∗) : T ∈ B(E)} ⊆ A∗∗, an algebra isomorphic to B(E), as
+B(E) → B(E∗∗), T �→ T ∗∗ is a homomorphism. Let B0 be the algebra generated by A(E) = K(E) and S,
+so as S2 ∈ K(E), B0 is the linear span of A(E) and S. Let B be the image of B0 in C. As ψ1(S∗∗) = ψ2(S∗∗)
+and ψ1, ψ2 agree on A, we see that B ⊆ Zt(A∗∗).
+If B(E) is amenable, so is C, so by Theorem 2.5, B is amenable, hence B0 is amenable. Thus B0/A
+is amenable, which as in the proof of Proposition 3.1 leads to the required contradiction. So B(E) is not
+amenable.
+Corollary 4.3. Let K be an uncountable compact metric space. Then B(C(K)) is not amenable.
+Proof. By Milutin’s Theorem, [55, Theorem 2.1], we have C(K) ∼= C[0, 1], so it is enough to consider
+E = C[0, 1]. We apply Theorem 4.2, so we seek S ∈ W(E) \ K(E) with S2 ∈ K(E). Firstly, suppose we
+can find any T ∈ W(E) \ K(E). Define linear maps
+T0 : C[0, 1] → C[0, 1],
+T0(f)(s) = f(2s)
+(f ∈ C[0, 1], s ∈ [0, 1]),
+and also T1 : C[0, 1] → C[0, 1] in the following way.
+Pick a small δ > 0 and a linear bijection ϕ :
+[1/2+δ, 1−δ] → [0, 1]. For f ∈ C[0, 1] define T1(f) as follows. For 1/2+δ ≤ t ≤ 1−δ let T1(f)(t) = f(ϕ(t)),
+and for t < 1/2 let T1(f)(t) = 0.
+Set T1(f)(1/2) = T1(f)(1) = 0, and linearly interpolate on the
+intervals [1/2, 1/2 + δ] and [1 − δ, 1].
+Then T0 is a metric surjection, T1 is an isometry, and hence
+S = T1TT0 ∈ W(E)\K(E). As T0T1 = 0 also S2 = 0. In fact, the square of any weakly compact operator
+on any C(K) is always compact by a result of Grothendieck, [55, Corollary 4.2], and so already T works
+in Theorem 4.2, but we prefer to give this explicit construction.
+It hence remains to find a suitable T. We use [52, Example 6] which gives an example of an absolutely
+summing but non-compact map R : C[0, 1] → c0; for instance, with rn : [0, 1] → {±1} the Rademacher
+functions, we may define
+R(f) =
+� � 1
+0
+rn(t)f(t) dt
+�∞
+n=1.
+By [60, Corollary 6.20], R is weakly compact as it is absolutely summing. It remains to find an isometry
+c0 → C[0, 1] which when composed with R will give us our required T. This is well-known (cf., e.g.,
+[55, Lemma 2.5 (d)]) but we give a construction.
+Let f1 ∈ C[0, 1] be the piece-wise linear function
+with f1(0) = f1(1/2) = f1(1) = 0, f1(3/4) = 1, let f2 ∈ C[0, 1] be the piece-wise linear function with
+f2(0) = f2(1/4) = f2(1/2) = f2(1) = 0, f2(3/8) = 1, and so forth. Then (fn) is a copy of the standard
+unit vector basis of c0, and the map c0 → C[0, 1], (an) �→ �
+n anfn is our isometry. Here the sum is to be
+interpreted pointwise, but as (an) ∈ c0, it actually converges absolutely.
+Corollary 4.4. B(L1[0, 1]) is not amenable.
+Proof. We apply Theorem 4.2 to E = L1[0, 1]. As E∗ = L∞[0, 1] has the BAP, we need only find a
+suitable S ∈ W(E). As E ∼= E ⊕ E, we can obtain S2 = 0 so long as we can find any R ∈ W(E) \ K(E).
+To find such an R, we can follow [19, Example 3.4], for instance.
+Corollary 4.5. Let E be a Banach space containing a complemented subspace isomorphic to ℓp for some
+p ∈ (1, ∞). If E∗ has the BAP then B(E) is not amenable.
+10
+
+Proof. We can easily find R ∈ B(ℓp) with R non-compact but R2 = 0. Indeed, if (en) is the standard
+unit vector basis of ℓp, define R by R(e2n) = e2n+1 and R(e2n−1) = 0 for each n ∈ N. Let P from E
+onto ℓp be a projection, so as ℓp is reflexive, RP is weakly compact. As R2 = 0 also (RP)2 = 0. As P
+is a projection onto ℓp and by the construction of R, we see that RP is not compact. Now Theorem 4.2
+implies the result.
+Remark 4.6. Note that the above yields another proof of the non-amenability of B(E) for any Lp-space
+E, where p ∈ (1, ∞).
+Corollary 4.7. The Hardy spaces Hp for all p ∈ [1, ∞) satisfy that B(Hp) is not amenable.
+Proof. For p ∈ (1, ∞) we have Hp ∼= Lp[0, 1] by the classical result [12], so the claim follows from
+Corollary 3.8. Now consider H1. By [42, Section 3], H1 contains a complemented subspace isomorphic
+to ℓ2 (this is due to Paley, cf. the references in [42]). Also, by [40, Corollary 1], the dual H∗
+1 ∼= BMO
+has the uniform approximation property, hence the BAP. Now Corollary 4.5 entails that B(H1) is not
+amenable.
+Next we derive a result concerning vector-valued Lp spaces over σ-finite measure spaces. So let µ be a
+σ-finite measure, E a Banach space, and consider Lp(µ, E). When E∗ has the Radon-Nikodym Property
+(RNP), then Lp(µ, E)∗ = Lq(µ, E∗), where 1
+p + 1
+q = 1, cf. [30, Section IV, Theorem 1]. For the RNP, see,
+e.g., [30, Chapter III] or [25, Definition 3.16]. All reflexive spaces, and all separable dual spaces have the
+RNP. More generally, E∗ has the RNP if and only if every separable subspace of E has separable dual,
+[30, Section VII.2, Corollary 8]. We note that [30] works only with finite measures, but the σ-finite case
+is a routine generalisation from this.
+To apply our result, we wish to know when Lq(µ, E∗) has the BAP. The following result is surely
+known, but as we have not found a suitable reference, we give a proof.
+Lemma 4.8. Let E have the BAP, and let p ∈ [1, ∞). Let µ be a σ-finite measure. Then Lp(µ, E) has
+the BAP.
+Proof. We shall use the ∆p norm from [28, Chapter 7] which is not quite a “tensor norm” on Lp(µ) ⊗ E;
+in particular, it fails the usual mapping property. Nevertheless, Lp(µ) ⊗ E is dense in Lp(µ, E) for the
+norm ∆p.
+It is easy to see that we can witness that Lp(µ) has the metric approximation property by finite-rank,
+positive operators (Ti), cf. [30, Section VIII.3, Example 11] for instance. For any positive operator T on
+Lp(µ), we do have that T ⊗ idE is bounded for ∆p, with bound ∥T∥, and so extends to Lp(µ, E), see [28,
+Theorem 7.3].
+Any S ∈ B(E) extends to idLp(µ) ⊗ S on Lp(µ, E) with norm ∥S∥. Thus if (Sj) is a bounded net of
+finite-rank operators on E witnessing that E has the BAP, then (Ti ⊗ Sj) is a bounded net of finite-rank
+operators on Lp(µ, E) which tends in the point-norm topology to the identity on Lp(µ) ⊗ E, and thus by
+boundedness and density, on all of Lp(µ, E). Hence Lp(µ, E) has the BAP.
+Corollary 4.9. Let E be a Banach space such that E∗ has the BAP and the RNP (for example, E is
+reflexive with the AP), and let p ∈ (1, ∞). Let µ be a σ-finite measure with Lp(µ) infinite-dimensional.
+Then B(Lp(µ, E)) is not amenable.
+Proof. The discussion above shows that under these hypotheses, Lp(µ, E) has dual Lq(µ, E∗). By Lemma
+4.8, also Lq(µ, E∗) has the BAP, as E∗ does. Furthermore, Lp(µ, E) contains a complemented subspace
+isomorphic to ℓp, by [17, Proposition 1.4.1]. Now Corollary 4.5 implies the result.
+For the definition of the James space J, we refer the reader to [46, Example 1.d.2].
+11
+
+Corollary 4.10. The James space J satisfies that B(J) is not amenable.
+Proof. By combining [15, Theorem 5] with the remark after [15, Corollary 4] and [15, Corollary 3], we
+find that J admits a complemented subspace isomorphic to ℓ2. Also, as J has a shrinking basis, J∗ has
+a basis by [46, Proposition 1.b.1], and so the BAP. Again, Corollary 4.5 yields the claim.
+4.1
+When the nuclear and integral operators on E∗ agree
+If we assume that N(E∗) = I(E∗) then we can say more.
+Recall the discussion of the RNP after
+Corollary 4.7 above. A useful result is that when E∗ has the RNP, then N(E∗) = I(E∗) with equal
+norms, [25, Theorem 3.18] and [60, Section 5.3].
+We continue with the notation from Proposition 4.1.
+Lemma 4.11. Let E be a Banach space such that E∗ has the BAP, and N(E∗) = I(E∗). Then ψ1(T ∗) =
+ψ2(T ∗) for each T ∈ B(E∗).
+Proof. Firstly, for general E, given T ∈ B(E∗) and J ∈ I(E∗), we always have
+⟨ψ1(T ∗), J⟩ = ⟨Φ0, η(T ∗J∗)⟩ = ⟨Φ0, JT ⟩,
+⟨ψ2(T ∗), J⟩ = ⟨Φ0, η(T ∗)J⟩ = ⟨Φ0, TJ⟩.
+(6)
+As in Proposition 4.1, here Φ0 is a weak∗-cluster point of a bai (eα) for A(E). Now let J ∈ N(E∗), say
+J = θx∗,x∗∗. Then
+⟨Φ0, JT ⟩ = ⟨Φ0, θx∗,T ∗(x∗∗)⟩ = lim
+α ⟨θx∗,T ∗(x∗∗), eα⟩ = lim
+α ⟨T ∗(x∗∗), e∗
+α(x∗)⟩ = ⟨T ∗(x∗∗), x∗⟩,
+here using that e∗
+α(x∗) → x∗ for each x∗ ∈ E∗. Indeed, as θx,x∗eα = θx,e∗α(x∗) and ∥θx,x∗ − θx,x∗eα∥ → 0,
+it follows that ∥x∥∥x∗ − e∗
+α(x∗)∥ → 0. We similarly see that
+⟨Φ0, TJ⟩ = ⟨Φ0, θT(x∗),x∗∗⟩ = lim
+α ⟨θT(x∗),x∗∗, eα⟩ = lim
+α ⟨x∗∗, e∗
+α(T(x∗))⟩ = ⟨x∗∗, T(x∗)⟩.
+It follows that ⟨Φ0, TJ⟩ = ⟨Φ0, JT⟩, and by linearity and continuity, this holds for all J ∈ N(E∗). The
+result now follows from (6).
+Theorem 4.12. Let E be a Banach space such that E∗ has the BAP, and N(E∗) = I(E∗). Let B be
+a closed subalgebra with K(E) ✂ B ⊆ B(E). If any of B(E), B(E∗) or B(E∗∗) is amenable, then B is
+amenable.
+Proof. Set A = A(E), and let C = {ψ1(T ∗∗) : T ∈ B(E)} ⊆ A∗∗, an algebra isomorphic to B(E).
+As I(E∗) = N(E∗), we know from Proposition 4.1 that Zt(A∗∗) = {ψ2(T ∗) : T ∈ B(E∗)}, and by
+Lemma 4.11, this equals {ψ1(T ∗) : T ∈ B(E∗)}. Thus C ⊆ Zt(A∗∗), and so when B(E) is amenable, also
+C is amenable, and hence Theorem 2.5 yields the result.
+When B(E∗) is amenable, we instead set C = {ψ1(T ∗) : T ∈ B(E∗)} ⊆ A∗∗, an algebra anti-isomorphic
+to B(E∗), so that C is amenable. Now C = Zt(A∗∗). We identify B with {ψ1(T ∗∗) : T ∈ B}, so that B ⊆ C,
+and A ✂ B. Again, Theorem 2.5 yields the claim.
+Finally, suppose that B(E∗∗) is amenable. As A∗ = I(E∗) = N(E∗) by hypothesis, and as E∗ has the
+BAP so that N(E∗)∗ = B(E∗∗), it follows that A∗∗ = B(E∗∗) with ψ1 being an isomorphism. For this,
+see [25, Section 5.2] and [24, Section 6]. Now set C = A∗∗ which is thus amenable, and again identify B
+with {ψ1(T ∗∗) : T ∈ B}, so that B ⊆ Zt(A∗∗). Again Theorem 2.5 implies the result.
+Corollary 4.13. Let K be an infinite countable compact metric space. Then B(C(K)) is not amenable.
+12
+
+Proof. By [55, Lemma 2.5(d)], we know that C(K) contains an isometric copy of c0. Using Sobczyk’s
+theorem, see [65], as C(K) is separable, there is a projection P from C(K) onto c0. As K is countable,
+we have C(K)∗ = M(K) ∼= ℓ1(K) which is a separable dual space, and hence has the RNP; also it has
+the BAP. We then apply Theorem 4.12 with B = K(C(K)) ⊕ CS for a suitable operator S. Again, we use
+Proposition 3.1, so we seek S non-compact with S2 compact. It is easy to find T ∈ B(c0) non-compact
+with T 2 = 0, compare the proof of Corollary 4.16 below. Set S = TP, so that S is non-compact as P is
+a projection onto c0, while PT = T so S2 = TPTP = T 2P = 0. Theorem 4.12 now yields that B(C(K))
+is not amenable.
+Corollary 4.14. Let K be an infinite compact metric space, equivalently, let K be an infinite compact
+space with C(K) separable. Then B(C(K)) is not amenable.
+Proof. This follows immediately from Corollaries 4.3 and 4.13.
+We now consider the vector-valued spaces C(K, E) for a Banach space E. These can be realised as the
+injective tensor product C(K)ˇ⊗E, see [60, Section 3.2]. The following generalises the previous corollary,
+in that we can take E = C.
+Corollary 4.15. Let K be an infinite compact metric space, and let E be a Banach space such that E∗
+has the BAP and the RNP. Then B(C(K, E)) is not amenable.
+Proof. We may identify C(K, E)∗ with the space of regular vector measures of bounded variation, defined
+on the Borel subsets of K, with values in E∗, see for example [60, page 112]. As E∗ has the RNP, [60,
+Corollary 5.23] shows that this space coincides with M(K)�⊗E∗, paired against C(K)ˇ⊗E = C(K, E) in
+the canonical way. As both M(K) and E∗ have the BAP, the proof of Lemma 4.8 is readily adapted to
+show that M(K)�⊗E∗ = C(K, E)∗ has the BAP; cf. also [29, Corollary 1.18].
+As C(K, E) = C(K)ˇ⊗E, fixing x ∈ E and µ ∈ E∗ with ∥x∥ = ∥µ∥ = ⟨µ, x⟩ = 1, the maps
+C(K)ˇ⊗E → C(K), f ⊗ y �→ ⟨µ, y⟩f
+and
+C(K) → C(K)ˇ⊗E, f �→ f ⊗ x
+establish that C(K) is a complemented subspace of C(K, E). When K is uncountable, we may use the
+complemented copy of C(K) together with the proof of Corollary 4.3 to show that there is T ∈ B(C(K, E))
+which is weakly compact but not compact, and with T 2 compact. Thus Theorem 4.2 yields the result.
+Now suppose that K is countable. As E∗ has the RNP, we know that separable subspaces of E have
+separable duals. Let X ⊆ C(K, E) be separable, and let (fn) be a dense subset. Then {fn(k) : n ∈ N, k ∈
+K} is a countable subset of E and so its closed linear span is a separable subspace of E, say E0. Then
+X ⊆ C(K, E0), and as K is countable, it follows easily that C(K, E0) is separable. We know that E∗
+0 is
+separable, and so C(K, E0)∗ = ℓ1(K)�⊗E∗
+0 is separable, as ℓ1(K) is separable. We have hence shown that
+separable subspaces of C(K, E) have separable dual. So C(K, E)∗ has the RNP and the BAP in this case.
+As in the proof of Corollary 4.13, C(K) contains a complemented copy of c0, and hence so does C(K, E).
+We can now argue exactly as in the proof of Corollary 4.13 to see that B(C(K, E)) is not amenable.
+Corollary 4.16. B(c0), B(ℓ1) and B(ℓ∞) are not amenable.
+Proof. Set E = c0. Then E∗ = ℓ1 has the BAP, and as a separable dual space, has the RNP, so that
+N(E∗) = I(E∗). We can find S ∈ B(c0) \ K(c0) with S2 = 0. Indeed, if (en) is the standard unit vector
+basis of c0 then define S by S(e2n) = e2n+1 and S(e2n−1) = 0 for each n ∈ N. Choosing B = K(c0) ⊕ CS
+gives a non-amenable Banach algebra by Proposition 3.1, and then Theorem 4.12 shows that B(c0), B(ℓ1)
+and B(ℓ∞) are not amenable.
+Corollary 4.17. Let E be an infinite-dimensional separable L1 space. Then B(E) is not amenable.
+13
+
+Proof. There is a classification of such E, [67, p. 83]: indeed, either E ∼= L1[0, 1] or E ∼= ℓ1, so the result
+follows from Corollaries 4.4 and 4.16.
+We now establish the non-commutative version of Corollary 4.16.
+Corollary 4.18. Let H be an infinite-dimensional separable Hilbert space. Then B(K(H)), B(T (H)) and
+B(B(H)) are not amenable.
+Proof. Set E = K(H), so that E∗ = T (H), the trace class operators, and E∗∗ = B(H). Then E∗ has
+the BAP, indeed, even a (Schauder) basis, which follows from [60, Proposition 4.25] for example. Also
+T (H) is a separable dual space, and so has the RNP, hence N(E∗) = I(E∗). As in the proof of the
+next corollary, we can find an operator S ∈ B(E) which is non-compact with S2 = 0, thus showing that
+B = K(E) ⊕ CS is not amenable, by Proposition 3.1. Now Theorem 4.12 yields that B(K(H)), B(T (H))
+and B(B(H)) are not amenable.
+Remark 4.19. In [19, Section 4], Choi shows that when E is a Banach space with the BAP such that
+E∗ does not have the BAP, then B(E) is not amenable. This criterion is rather restrictive; the canonical
+example, thanks to Szankowski’s result [64], is E = T (H).
+Choi’s argument uses again [37] which shows that, under these hypotheses, A(E) has a one-sided but
+no two-sided bounded approximate identity. As A(E) is an ideal in B(E), this contradicts B(E) being
+amenable, cf. [19, Lemma 2.2]. Our proof above that B(T (H)) is non-amenable avoids the use of the very
+deep result of Szankowski.
+There is to our knowledge one way to construct spaces which Choi’s result covers, giving non-amenability
+of B(E), and where our methods do not apply. By [43, Corollary 3], see also [46, Theorem 1.e.7(b)], start-
+ing with any Banach space F which fails to have the AP, one can show that there exists a Banach space
+E with the BAP (indeed, a Schauder basis) such that E∗ does not have the AP.
+In fact, more generally, we obtain the following.
+Corollary 4.20. For all p ∈ (1, ∞) we have that B(K(ℓp)), B(N(ℓp)) and B(B(ℓp)) are not amenable.
+Proof. We proceed at first with some generality. Let F be a Banach space such that F ∗∗ is separable
+with the BAP. Also F ∗ is separable, and a dual space, and so has the RNP. Further, F ∗ has the BAP,
+see [25, Corollary 3.22] for example, and so there are bounded nets (ti), (sj) of finite-rank operators in
+B(F ∗), B(F ∗∗), respectively, converging in the point-norm topology to the identity. Set E = K(F) = A(F),
+so that E∗ = I(F ∗) = N(F ∗) = F ∗ �⊗F ∗∗ by the hypotheses on F. Then (ti ⊗ sj) is a bounded net of
+finite-rank operators converging in the point-norm topology to the identity on F ∗ �⊗F ∗∗, showing that E∗
+has the BAP. As F ∗∗ and F ∗ are separable, also E∗ is separable. Thus we can apply Theorem 4.12 to
+find that B(E), B(E∗) and B(E∗∗) are not amenable, provided a suitable B can be constructed.
+Now specialise to the case F = ℓp for p ∈ (1, ∞). Let (en) be the usual unit vector basis of ℓp, and
+set F0, F1 to be the closed span of (e2n), (e2n−1), respectively. For i = 0, 1 there is a natural projection
+Pi : F → Fi, and an inclusion ιi : Fi → F. Further, there is an isometry j : F0 → F1. For x ∈ A(ℓp)
+define S(x) = jP0xP0 ∈ A(ℓp). As P0j = 0, we see that S2(x) = jP0S(x)P0 = jP0jP0xP0 = 0 for each
+x. For y ∈ A(F0) we can set x = ι0yP0 ∈ A(ℓp), and then S(x) = jP0ι0yP0 = jyP0, so in particular,
+∥S(x)∥ = ∥yP0∥ = ∥y∥ = ∥x∥. This shows that S is non-compact, and so B = A(E) ⊕ CS is a suitable
+non-amenable algebra.
+Remark 4.21. Note that the Argyros–Haydon space X, for which B(X) is amenable, cf. Remark 3.16,
+satisfies that X∗ is isomorphic to ℓ1, so X∗ has the BAP and the RNP, whence N(X∗) = I(X∗). However,
+as B(X) = K(X) ⊕ CidX, there is obviously no operator S ∈ B(X) \ K(X) with S2 compact.
+14
+
+5
+Alternative proofs of the non-amenability of B(ℓp) for p ∈ (1, ∞]
+In this Section, we present alternative, quick proofs that B(ℓp) is non-amenable for all p ∈ (1, ∞] using
+operator algebra methods and harmonic analysis, rather than Banach space geometry. We first give a
+short proof of the non-amenability of B(ℓ2), avoiding the use of nuclearity for C∗-algebras (we remark
+that [13] quoted below was written when there was no relationship known between amenability and
+nuclearity for C∗-algebras, as noted therein). Given a discrete group G, we write C∗
+r (G) for its reduced
+group C∗-algebra. Recall that K(ℓ2(G)) is Arens regular being a C∗-algebra, and the Arens product on
+K(ℓ2(G))∗∗ = B(ℓ2(G)) is the usual composition of operators.
+Theorem 5.1. B(ℓ2) is not amenable.
+Proof. Realize ℓ2 as ℓ2(G) for some countable discrete non-amenable group G, such as F2.
+Suppose
+that B(ℓ2(G)) is amenable. Put A := K(ℓ2(G)), and consider the C∗-subalgebra B := A ⊕ C∗
+r (G) of
+B(ℓ2(G)) = A∗∗ (note that K(ℓ2(G)) ∩ C∗
+r (G) = {0} by [20, Proposition 3.2]). As in Section 3 we can
+apply Corollary 2.6 to see that amenability of B(ℓ2(G)) passes to B, and hence to the quotient C∗
+r (G).
+Thus G is amenable by [13, Proposition 2] – a contradiction.
+For the case of B(ℓp), p ∈ (1, ∞), we will argue similarly. We will consider the p-analogue of C∗
+r (G),
+i.e., the algebra PFp(G) of p-pseudofunctions on G, defined as the Banach algebra generated in B(ℓp(G))
+by λp(ℓ1(G)), where λp is the representation of ℓ1(G) on ℓp(G) given by left convolution.
+We are grateful to N.C. Phillips for pointing out the following
+Lemma 5.2. Let G be a countable discrete group, and p ∈ (1, ∞). Then the canonical quotient map
+q : B(ℓp(G)) → B(ℓp(G))/K(ℓp(G)) is isometric on PFp(G).
+Proof. This is shown for p = 2 in [49, Proposition 4.5], and inspection of the proof shows that the
+argument carries over, mutatis mutandis, to the case of general p.
+For the following, note that given a discrete group G, K(ℓp(G)) is Arens regular by [23, Theorem
+2.6.23], and the product on K(ℓp(G))∗∗ = B(ℓp(G)) is the usual composition of operators.
+Theorem 5.3. B(ℓp) is not amenable for any p ∈ (1, ∞).
+Proof. Let p ∈ (1, ∞). Realize ℓp as ℓp(G) for some countable discrete non-amenable group G, such as
+F2. Suppose that B(ℓp(G)) is amenable. Put A := K(ℓp(G)), and consider the space B := A ⊕ PFp(G)
+(note that K(ℓp(G)) ∩ PFp(G) = {0} as the proof of [20, Proposition 3.2] for p = 2 carries over to
+the case of general p).
+Let q : B(ℓp(G)) → B(ℓp(G))/K(ℓp(G)) be the canonical quotient map.
+By
+Lemma 5.2, q(PFp(G)) ⊆ B(ℓp(G))/K(ℓp(G)) is closed. Hence B = q−1(q(PFp(G))) is a closed subalgebra
+of B(ℓp(G)) = A∗∗. Again, by Corollary 2.6, B is amenable. So the quotient PFp(G) is amenable, whence
+G is amenable (see the proof of [32, Theorem 6.4], which uses work of Phillips [50]) – a contradiction.
+We shall now give an alternative proof of the non-amenability of B(ℓ∞). To this end, let G be a
+countable discrete group. We recall that K(c0(G))∗∗ = B(ℓ∞(G)), with the first Arens product being the
+usual composition of operators, and Zt(K(c0(G))∗∗) = Bσ(ℓ∞(G)), where the latter denotes the maps in
+B(ℓ∞(G)) which are weak∗-weak∗-continuous; see pages 59–61, in particular Example 6.2, in [24]. (This
+also follows from Proposition 4.1 in this special case when E = c0(G), as then E∗ = ℓ1(G) has the RNP
+and so the ideas of Section 4.1 apply.) We also recall that Φ : ℓ1(G) → B(c0(G)), where
+Φ(f)(g) = f ∗ g for all f ∈ ℓ1(G), g ∈ c0(G),
+is an isometric representation. We see B(c0(G)) as a subalgebra of B(ℓ∞(G)) (by taking second adjoints).
+So we have B(c0(G)) ⊆ Bσ(ℓ∞(G)).
+We have the following
+15
+
+Lemma 5.4. Let G be a countable discrete group. Then the canonical quotient map q : B(c0(G)) →
+B(c0(G))/K(c0(G)) is isometric on Φ(ℓ1(G)).
+Proof. Again, this follows, mutatis mutandis, as in the proof of [49, Proposition 4.5], replacing ℓ2(I) by
+c0(I). Note that all elements of Φ(ℓ1(G)) commute with right translations in B(c0(G)).
+We shall now prove
+Theorem 5.5. B(ℓ∞) is not amenable.
+Proof. Realize ℓ∞ as ℓ∞(G) for some countable discrete non-amenable group G, such as F2. Suppose that
+B(ℓ∞(G)) is amenable. Put A := K(c0(G)), and consider the space B := A ⊕ Φ(ℓ1(G)) ⊆ B(c0(G)) ⊆
+Bσ(ℓ∞(G)); note that K(c0(G))∩Φ(ℓ1(G)) = {0} follows from [61, proof of Theorem 1]. Let q : B(c0(G)) →
+B(c0(G))/K(c0(G)) be the canonical quotient map. By Lemma 5.4, q(Φ(ℓ1(G))) ⊆ B(c0(G))/K(c0(G)) is
+closed. Hence B = q−1(q(Φ(ℓ1(G)))) is a closed subalgebra of Bσ(ℓ∞(G)) = Zt(A∗∗). By Theorem 2.5, B
+is amenable. So the quotient Φ(ℓ1(G)) is amenable. Thus, ℓ1(G) is amenable, whence G is amenable by
+Johnson’s classical result, [59, Theorem 2.1.10] or [56, Theorem 2.1.8] – a contradiction.
+Acknowledgements
+The first named author is partially supported by EPSRC grant EP/T030992/1. For the purpose of open
+access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript
+version arising. No data were created or analysed in this study. The second named author is partially
+supported by NSERC; this support is gratefully acknowledged.
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+19
+
+Authors’ affiliations
+Matthew Daws
+Jeremiah Horrocks Institute, University of Central Lancashire, Preston, PR1 2HE, United Kingdom
+Email: [email protected]; [email protected]
+Matthias Neufang
+School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S
+5B6, Canada
+and
+Laboratoire de Math´ematiques Paul Painlev´e (UMR CNRS 8524), Universit´e de Lille, D´epartement de
+Math´ematiques, 59655 Villeneuve d’Ascq Cedex, France
+Email: [email protected]; [email protected]
+20
+

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