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+arXiv:2301.00769v1  [math.AP]  2 Jan 2023
+SHARP NORM ESTIMATES FOR THE CLASSICAL HEAT EQUATION
+ERIK TALVILA
+Abstract. Sharp estimates of solutions of the classical heat equation are proved in Lp
+norms on the real line.
+1. Introduction
+In this paper we give sharp estimates of solutions of the classical heat equation on the
+real line with initial value data that is in an Lp space (1 ≤ p ≤ ∞).
+For u:R × (0, ∞) → R write ut(x) = u(x, t).
+The classical problem of the heat equation on the real line is, given a function f ∈ Lp
+for some 1 ≤ p ≤ ∞, find a function u : R × (0, ∞) → R such that ut ∈ C2(R) for each
+t > 0, u(x, ·) ∈ C1((0, ∞)) for each x ∈ R and
+∂2u(x, t)
+∂x2
+− ∂u(x, t)
+∂t
+= 0 for each (x, t) ∈ R × (0, ∞)
+(1.1)
+lim
+t→0+∥ut − f∥p = 0.
+(1.2)
+If p = ∞ then f is also assumed to be continuous.
+A solution is given by the convolution ut(x) = F ∗Θt(x) =
+� ∞
+−∞ F(x−y)Θt(y) dy where
+the Gauss–Weierstrass heat kernel is Θt(x) = exp(−x2/(4t))/(2
+√
+πt). For example, see
+[4]. Under suitable growth conditions on u the solution is unique. See [5] and [9]. Refer-
+ences [3] and [9] contain many results on the classical heat equation, including extensive
+bibliographies.
+The heat kernel has the following properties.
+Let t > 0 and let s ̸= 0 such that
+1/s + 1/t > 0. Then
+Θt ∗ Θs = Θt+s
+(1.3)
+∥Θt∥q =
+αq
+t(1−1/q)/2 where αq =
+
+
+
+1,
+q = 1
+1
+(2√π)1−1/q q1/(2q) ,
+1 < q < ∞
+1
+2√π,
+q = ∞.
+(1.4)
+The last of these follows from the probability integral
+� ∞
+−∞ e−x2 dx = √π.
+Theorem 1.1. Let 1 ≤ p ≤ ∞ and f ∈ Lp.
+(a) If p ≤ s ≤ ∞ then f ∗ Θt ∈ Ls.
+(b) Let q, r ∈ [1, ∞] such that 1/p + 1/q = 1 + 1/r. There is a constant Kp,q such that
+∥f ∗ Θt∥r ≤ Kp,q∥f∥p t−(1−1/q)/2 for all t > 0. The estimate is sharp in the sense that if
+ψ : (0, ∞) → (0, ∞) such that ψ(t) = o(t−(1−1/q)/2) as t → 0+ or t → ∞ then there is
+Date: Preprint January 2, 2023.
+2020 Mathematics Subject Classification. Primary 35K05, 46E30; Secondary 26A42.
+Key words and phrases. Heat equation, Lebesgue space.
+1
+
+2
+ERIK TALVILA
+G ∈ Lp such that ∥G ∗ Θt∥r/ψ(t) is not bounded as t → 0+ or t → ∞. The constant
+Kp,q = (cpcq/cr)1/2αq, where cp = p1/p/(p′)1/p′ with p, p′ being conjugate exponents. It
+cannot be replaced with any smaller number.
+(c) If 1 ≤ s < p then f ∗ Θt need not be in Ls.
+When r = p and q = 1 the inequality in part (b) reads ∥f ∗ Θt∥p ≤ ∥f∥p. When r = ∞
+then p and q are conjugates and the inequality in part (b) reads ∥f ∗Θt∥∞ ≤ ∥f∥pt−1/(2p).
+The condition for sharpness in Young’s inequality is that both functions be Gaussians.
+This fact is exploited in the proof of part (b). See [7, p. 99], [2] and [8]. Our proof also
+uses ideas from [5, Theorem 9.2, p. 195] and [1, pp. 115-120].
+The estimates are known, for example [6, Proposition 3.1], but we have not been able
+to find a proof in the literature that they are sharp.
+Proof. (a), (b) Young’s inequality gives
+(1.5)
+∥f ∗ Θt∥r ≤ Cp,q∥f∥p∥Θt∥q = Cp,q∥f∥pαq
+t(1−1/q)/2 ,
+where αq is given in (1.4). The sharp constant, given in [7, p. 99], is Cp,q = (cpcq/cr)1/2
+where cp = p1/p/(p′)1/p′ with p, p′ being conjugate exponents. Note that c1 = c∞ = 1.
+Also, 0 < Cp,q ≤ 1. We then take Kp,q = Cp,qαq.
+To show the estimate ∥ut∥r = O(t−(1−1/q)/2) is sharp as t → 0+ and t → ∞, let ψ be as
+in the statement of the theorem. Fix p ≤ r ≤ ∞. Define the family of linear operators St:
+Lp → Lr by St[f](x) = f ∗ Θt(x)/ψ(t). The estimate ∥St[f]∥r ≤ Kp,q∥f∥pt−(1−1/q)/2/ψ(t)
+shows that, for each t > 0, St is a bounded linear operator. Let ft = Θt. Then, from (1.3)
+and (1.4),
+∥St[ft]∥r
+∥ft∥p
+= ∥Θt ∗ Θt∥r
+ψ(t)∥Θt∥p
+=
+∥Θ2t∥r
+ψ(t)∥Θt∥p
+=
+αr
+αp2(1−1/r)/2ψ(t)t(1−1/q)/2 .
+This is not bounded in the limit t → 0+. Hence, St is not uniformly bounded. By the
+Uniform Bounded Principle it is not pointwise bounded. Therefore, there is a function
+f ∈ Lp such that ∥f ∗ Θt∥r ̸= O(ψ(t)) as t → 0+. And, the growth estimate ∥f ∗ Θt∥r =
+O(t−(1−1/q)/2)) as t → 0+ is sharp. Similarly for sharpness as t → ∞.
+Now show the constant Kp,q cannot be reduced. A calculation shows we have equality
+in (1.5) when f = Θβ
+t and β is given by the equation
+(1.6)
+β1−1/q
+(β + 1)1−1/r = cpcq
+cr
+�αpαq
+αr
+�2
+=
+�
+1 − 1
+p
+�1−1/p �
+1 − 1
+q
+�1−1/q �
+1 − 1
+r
+�−(1−1/r)
+.
+First consider the case p ̸= 1 and q ̸= 1. Notice that 1 − 1/r = (1 − 1/q) + (1 − 1/p) >
+1 − 1/q. Let g(x) = xA(x + 1)−B with B > A > 0. Then g is strictly increasing on
+(0, A/(B − A)) and strictly decreasing for x > A/(B − A) so there is a unique maximum
+for g at A/(B − A). Put A = 1 − 1/q and B = 1 − 1/r. Then
+g
+�
+A
+B − A
+�
+=
+β1−1/q
+(β + 1)1−1/r =
+�
+1 − 1
+p
+�1−1/p �
+1 − 1
+q
+�1−1/q �
+1 − 1
+r
+�−(1−1/r)
+.
+Hence, (1.6) has a unique positive solution for β given by β = (1 − 1/q)/(1 − 1/p).
+
+HEAT EQUATION
+3
+If p = 1 then q = r. In this case, (1.6) reduces to (1 + 1/β)1−1/q = 1 and the solution
+is given in the limit β → ∞. Sharpness of (1.5) is then given in this limit. It can also be
+seen that taking f to be the Dirac distribution gives equality.
+If q = 1 then p = r. Now, (1.6) reduces to (β +1)1−1/p = 1 and β = 0. There is equality
+in (1.5) when f = 1. This must be done in the limit β → 0+.
+If p = q = r = 1 then there is equality in (1.5) for each β > 0.
+Hence, the constant in (1.5) is sharp.
+(c) Suppose f ≥ 0 and f is decreasing on [c, ∞) for some c ∈ R. Let x > c. Then
+f ∗ Θt(x)
+≥
+� x
+c
+f(y)Θt(x − y) dy ≥ f(x)
+� x
+c
+Θt(x − y) dy
+=
+f(x)
+√π
+� (x−c)/(2
+√
+t)
+0
+e−y2 dy ∼ f(x)/2
+as x → ∞.
+Now put f(x) = 1/[x1/p log2(x)] for x ≥ e and f(x) = 0, otherwise. For p = ∞ replace
+x1/p by 1.
+□
+References
+[1] S. Axler, P. Bourdon and W. Ramey, Harmonic function theory, New York, Springer-Verlag, 2001.
+[2] W. Beckner, Inequalities in Fourier analysis, Ann. of Math. (2) 102(1975), 159–182.
+[3] J.R. Cannon, The one-dimensional heat equation, Menlo Park, Addison–Wesley, 1984.
+[4] G.B. Folland, Introduction to partial differential equations, Princeton, Princeton University Press,
+1995.
+[5] I.I. Hirschman and D.V. Widder, The convolution transform, Princeton, Princeton University Press,
+1955.
+[6] T. Iwabuchi, T. Matsuyama and K. Taniguchi, Boundedness of spectral multipliers for Schr¨odinger
+operators on open sets, Rev. Mat. Iberoam. 34(2018), 1277–1322.
+[7] E.H. Lieb and M. Loss, Analysis, Providence, American Mathematical Society, 2001.
+[8] G. Toscani, Heat equation and the sharp Young’s inequality, arXiv:1204.2086 (2012).
+[9] D.V. Widder, The heat equation, New York, Academic Press, 1975.
+Department of Mathematics & Statistics, University of the Fraser Valley, Abbots-
+ford, BC Canada V2S 7M8
+Email address: [email protected]
+

-NAyT4oBgHgl3EQf3flQ/content/tmp_files/load_file.txt

@@ -0,0 +1,146 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf,len=145
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='00769v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='AP]  2 Jan 2023  SHARP NORM ESTIMATES FOR THE CLASSICAL HEAT EQUATION  ERIK TALVILA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Sharp estimates of solutions of the classical heat equation are proved in Lp  norms on the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Introduction  In this paper we give sharp estimates of solutions of the classical heat equation on the  real line with initial value data that is in an Lp space (1 ≤ p ≤ ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  For u:R × (0, ∞) → R write ut(x) = u(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  The classical problem of the heat equation on the real line is, given a function f ∈ Lp  for some 1 ≤ p ≤ ∞, find a function u : R × (0, ∞) → R such that ut ∈ C2(R) for each  t > 0, u(x, ·) ∈ C1((0, ∞)) for each x ∈ R and  ∂2u(x, t)  ∂x2  − ∂u(x, t)  ∂t  = 0 for each (x, t) ∈ R × (0, ∞)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='1)  lim  t→0+∥ut − f∥p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='2)  If p = ∞ then f is also assumed to be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  A solution is given by the convolution ut(x) = F ∗Θt(x) =  � ∞  −∞ F(x−y)Θt(y) dy where  the Gauss–Weierstrass heat kernel is Θt(x) = exp(−x2/(4t))/(2  √  πt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' For example, see  [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Under suitable growth conditions on u the solution is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' See [5] and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Refer-  ences [3] and [9] contain many results on the classical heat equation, including extensive  bibliographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  The heat kernel has the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  Let t > 0 and let s ̸= 0 such that  1/s + 1/t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Then  Θt ∗ Θs = Θt+s  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='3)  ∥Θt∥q =  αq  t(1−1/q)/2 where αq =  \uf8f1  \uf8f2  \uf8f3  1,  q = 1  1  (2√π)1−1/q q1/(2q) ,  1 < q < ∞  1  2√π,  q = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='4)  The last of these follows from the probability integral  � ∞  −∞ e−x2 dx = √π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Let 1 ≤ p ≤ ∞ and f ∈ Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  (a) If p ≤ s ≤ ∞ then f ∗ Θt ∈ Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  (b) Let q, r ∈ [1, ∞] such that 1/p + 1/q = 1 + 1/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' There is a constant Kp,q such that  ∥f ∗ Θt∥r ≤ Kp,q∥f∥p t−(1−1/q)/2 for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' The estimate is sharp in the sense that if  ψ : (0, ∞) → (0, ∞) such that ψ(t) = o(t−(1−1/q)/2) as t → 0+ or t → ∞ then there is  Date: Preprint January 2, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Primary 35K05, 46E30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Secondary 26A42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Heat equation, Lebesgue space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  1  2  ERIK TALVILA  G ∈ Lp such that ∥G ∗ Θt∥r/ψ(t) is not bounded as t → 0+ or t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' The constant  Kp,q = (cpcq/cr)1/2αq, where cp = p1/p/(p′)1/p′ with p, p′ being conjugate exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' It  cannot be replaced with any smaller number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  (c) If 1 ≤ s < p then f ∗ Θt need not be in Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  When r = p and q = 1 the inequality in part (b) reads ∥f ∗ Θt∥p ≤ ∥f∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' When r = ∞  then p and q are conjugates and the inequality in part (b) reads ∥f ∗Θt∥∞ ≤ ∥f∥pt−1/(2p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  The condition for sharpness in Young’s inequality is that both functions be Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  This fact is exploited in the proof of part (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' See [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' 99], [2] and [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Our proof also  uses ideas from [5, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' 195] and [1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' 115-120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  The estimates are known, for example [6, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='1], but we have not been able  to find a proof in the literature that they are sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' (a), (b) Young’s inequality gives  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='5)  ∥f ∗ Θt∥r ≤ Cp,q∥f∥p∥Θt∥q = Cp,q∥f∥pαq  t(1−1/q)/2 ,  where αq is given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' The sharp constant, given in [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' 99], is Cp,q = (cpcq/cr)1/2  where cp = p1/p/(p′)1/p′ with p, p′ being conjugate exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Note that c1 = c∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  Also, 0 < Cp,q ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' We then take Kp,q = Cp,qαq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  To show the estimate ∥ut∥r = O(t−(1−1/q)/2) is sharp as t → 0+ and t → ∞, let ψ be as  in the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Fix p ≤ r ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Define the family of linear operators St:  Lp → Lr by St[f](x) = f ∗ Θt(x)/ψ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' The estimate ∥St[f]∥r ≤ Kp,q∥f∥pt−(1−1/q)/2/ψ(t)  shows that, for each t > 0, St is a bounded linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Let ft = Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Then, from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='3)  and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='4),  ∥St[ft]∥r  ∥ft∥p  = ∥Θt ∗ Θt∥r  ψ(t)∥Θt∥p  =  ∥Θ2t∥r  ψ(t)∥Θt∥p  =  αr  αp2(1−1/r)/2ψ(t)t(1−1/q)/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  This is not bounded in the limit t → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Hence, St is not uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' By the  Uniform Bounded Principle it is not pointwise bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Therefore, there is a function  f ∈ Lp such that ∥f ∗ Θt∥r ̸= O(ψ(t)) as t → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' And, the growth estimate ∥f ∗ Θt∥r =  O(t−(1−1/q)/2)) as t → 0+ is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Similarly for sharpness as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  Now show the constant Kp,q cannot be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' A calculation shows we have equality  in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='5) when f = Θβ  t and β is given by the equation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='6)  β1−1/q  (β + 1)1−1/r = cpcq  cr  �αpαq  αr  �2  =  �  1 − 1  p  �1−1/p �  1 − 1  q  �1−1/q �  1 − 1  r  �−(1−1/r)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  First consider the case p ̸= 1 and q ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Notice that 1 − 1/r = (1 − 1/q) + (1 − 1/p) >  1 − 1/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Let g(x) = xA(x + 1)−B with B > A > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Then g is strictly increasing on  (0, A/(B − A)) and strictly decreasing for x > A/(B − A) so there is a unique maximum  for g at A/(B − A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Put A = 1 − 1/q and B = 1 − 1/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Then  g  �  A  B − A  �  =  β1−1/q  (β + 1)1−1/r =  �  1 − 1  p  �1−1/p �  1 − 1  q  �1−1/q �  1 − 1  r  �−(1−1/r)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  Hence, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='6) has a unique positive solution for β given by β = (1 − 1/q)/(1 − 1/p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  HEAT EQUATION  3  If p = 1 then q = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' In this case, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='6) reduces to (1 + 1/β)1−1/q = 1 and the solution  is given in the limit β → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Sharpness of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='5) is then given in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' It can also be  seen that taking f to be the Dirac distribution gives equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  If q = 1 then p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Now, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='6) reduces to (β +1)1−1/p = 1 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' There is equality  in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='5) when f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' This must be done in the limit β → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  If p = q = r = 1 then there is equality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='5) for each β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  Hence, the constant in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='5) is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  (c) Suppose f ≥ 0 and f is decreasing on [c, ∞) for some c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Let x > c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Then  f ∗ Θt(x)  ≥  � x  c  f(y)Θt(x − y) dy ≥ f(x)  � x  c  Θt(x − y) dy  =  f(x)  √π  � (x−c)/(2  √  t)  0  e−y2 dy ∼ f(x)/2  as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  Now put f(x) = 1/[x1/p log2(x)] for x ≥ e and f(x) = 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' For p = ∞ replace  x1/p by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  □  References  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Axler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Bourdon and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Ramey, Harmonic function theory, New York, Springer-Verlag, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Beckner, Inequalities in Fourier analysis, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' (2) 102(1975), 159–182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Cannon, The one-dimensional heat equation, Menlo Park, Addison–Wesley, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Folland, Introduction to partial differential equations, Princeton, Princeton University Press,  1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  [5] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Hirschman and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Widder, The convolution transform, Princeton, Princeton University Press,  1955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  [6] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Iwabuchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Matsuyama and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Taniguchi, Boundedness of spectral multipliers for Schr¨odinger  operators on open sets, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Iberoam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' 34(2018), 1277–1322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Lieb and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Loss, Analysis, Providence, American Mathematical Society, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Toscani, Heat equation and the sharp Young’s inequality, arXiv:1204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='2086 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content=' Widder, The heat equation, New York, Academic Press, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='  Department of Mathematics & Statistics, University of the Fraser Valley, Abbots-  ford, BC Canada V2S 7M8  Email address: Erik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='Talvila@ufv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}
+page_content='ca' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'}

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+Growth of α-Ga2O3 on α-Al2O3 by conventional molecular-beam epitaxy
+and metal-oxide-catalyzed epitaxy
+J. P. McCandless∗ and V. Protasenko
+School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA
+D. Rowe and N. Pieczulewski
+Department of Material Science and Engineering, Cornell University, Ithaca, New York 14853, USA
+M. Alonso-Orts, M. S. Williams, and M. Eickhoff
+Institute of Solid-State Physics, University Bremen, Otto-Hahn-Allee 1, 28359 Bremen, Germany
+H. G. Xing
+School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA
+Department of Material Science and Engineering, Cornell University, Ithaca, New York 14853, USA and
+Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, New York 14853, USA
+D. A. Muller
+Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, New York 14853, USA and
+School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA
+D. Jena
+Department of Material Science and Engineering, Cornell University, Ithaca, New York 14853, USA
+School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA and
+Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, New York 14853, USA
+P. Vogt
+Department of Material Science and Engineering, Cornell University, Ithaca, New York 14853, USA and
+Institute of Solid-State Physics, University Bremen, Otto-Hahn-Allee 1, 28359 Bremen, Germany
+We report the growth of α-Ga2O3 on 𝑚-plane α-Al2O3 by conventional plasma-assisted molecular-beam epitaxy
+(MBE) and In-mediated metal-oxide-catalyzed epitaxy (MOCATAXY). We report a growth-rate-diagram for
+α-Ga2O3(10¯10), and observe (i) a growth rate increase, (ii) an expanded growth window, and (iii) reduced out-of-
+lane mosaic spread when MOCATAXY is employed for the growth of α-Ga2O3. Through the use of In-mediated
+catalysis, growth rates over 0.2 μm hr−1 and rocking curves with full width at half maxima of ∆𝜔 ≈ 0.45◦ are
+achieved. Faceting is observed along the α-Ga2O3film surface and is explored through scanning transmission
+electron microscopy.
+INTRODUCTION
+Over the past decade, Ga2O3 has gained much attention as
+a wide-band gap semiconductor. Monoclinic β-Ga2O3 pos-
+sesses an ultra-wide bang gap of ∼ 4.7 eV [1], and it has
+been the most studied phase owing to its thermal stability
+and the availability of large-area, native, semi-insulating and
+conductive substrates [2, 3].
+To further increase its band
+gap β-Ga2O3 can be alloyed with Al to form β-(Al,Ga)2O3,
+but achieving high Al content has remained challenging due
+to the tendency to have phase segregation [4]. In contrast,
+α-(Al,Ga)2O3 becomes more stable as the Al is increased be-
+cause the crystal is isostructural with the α-Al2O3 substrate,
+and the lattice mismatch is reduced as the Al concentration
+is increased [5]. This has enabled the entire compositional
+range of α-(Al,Ga)2O3 to be readily achieved [5, 6], and it has
+enabled band gaps exceeding those of AlN, BN, or diamond
+[7, 8].
+With the recent advances enabling α-Ga2O3 to remain stable
+during high-temperature anneals [9], the next challenge is
+to achieve electrical conductivity. To date, conductivity in
+α-Ga2O3 has been achieved by chemical vapor deposition
+(CVD) [10, 11], but has remained elusive for films grown by
+molecular-beam epitaxy (MBE). Additionally, conductive β-
+Ga2O3 films grown by MBE on α-Al2O3 have yet to be
+achieved [7]. While the exact reasons these films remain
+insulating are unknown, the thermodynamics during MBE
+growth and the low formation energy of defects may cause
+these Ga2O3 films on Al2O3 to be insulating.
+Multiple compensating point defects (e.g., cation vacancies,
+oxygen vacancies, donor impurities [12, 13]) and extended
+crystallographic defects (e.g., rotational domains and thread-
+ing dislocations[14]) occur within the Ga2O3 films grown
+on sapphire.
+Reasons for the emergence of these defects
+include the lattice mismatch between the film and the sub-
+strate [14], and the growth regime in which the films are
+grown [12, 13, 15]. For example, Ga vacancies (𝑉Ga) may
+act as compensating acceptors for introduced 𝑛-type dopants
+arXiv:2301.13053v1  [cond-mat.mtrl-sci]  30 Jan 2023
+
+2
+in grown Ga2O3 thin films [15].
+O-rich growth environ-
+ments are likely to generate a significant amount of 𝑉Ga (due
+to their low formation energy) whereas Ga-rich growth en-
+vironments are found to significantly suppress the formation
+of 𝑉Ga (due to their high formation energy) [13]. Thus, the
+growth of Ga2O3 in the highly Ga-rich regime—accessed by
+new epitaxial growth concepts [16]—may improve the trans-
+port properties of Ga2O3 thin films since the Ga-rich growth
+regimes lead to higher 𝑉Ga formation energies, resulting in
+lower 𝑉Ga densities within the Ga2O3 layers.
+One approach to address these issues is through the use of
+metal-oxide-catalyzed epitaxy (MOCATAXY) [17]. This is
+a growth process where a catalytic element (e.g. In) is in-
+troduced to the growth system and results in metal-exchange
+catalysis [18]. This growth mode has been observed for β-
+(Al,Ga)2O3 on different substrates and surface orientations,
+as well as for different epitaxial growth techniques [19–23].
+Many benefits arise from using MOCATAXY during the
+growth of Ga2O3. For example: (i) It can improve the surface
+morphologies of β-Ga2O3-based films [20]. (ii) The synthe-
+sis of Ga2O3 can occur in previously inaccessible kinetic and
+thermodynamic growth regimes (e.g. in highly metal-rich
+regimes) which can be beneficial for the suppression of un-
+desired point (such as𝑉Ga) defects in Ga2O3 [12, 13, 18]. (iii)
+The formation of thermodynamically unstable Ga2O3 phases
+becomes energetically favorable [16, 18, 23], e.g., the for-
+mation of the ϵ/κ-phase of Ga2O3, which has enabled novel
+ϵ/κ-Ga2O3-based heterostructures [22]. (iv) The growth rate
+(𝛤), possible growth temperatures (𝑇G), and crystalline qual-
+ity of β-(Al,Ga)2O3-based thin films can be vastly enhanced
+[17].
+In this work, we introduce the growth of α-Ga2O3 by MO-
+CATAXY, resulting in an expansion of the α-Ga2O3 growth
+window combined with an increased 𝛤 and an improvement
+in its out-of-plane mosaic spread. It is the first demonstration
+of a catalytic growth process during the growth of α-Ga2O3.
+EXPERIMENTAL
+Samples were grown in a Veeco GEN930 plasma MBE sys-
+tem with standard Ga and In effusion cells. For all samples,
+the substrates were cleaned in acetone and isopropanol for 10
+minutes and the α-Ga2O3 samples were grown for 60 minutes.
+The growth temperature (𝑇G) was measured by a thermocou-
+ple located within the substrate heater. The Ga flux (𝜙Ga)
+and In flux (𝜙In) were monitored by beam equivalent pres-
+sure (BEP) chamber readings. For conventional MBE and
+MOCATAXY, the O2 flux (𝜙O) was measured in standard
+cubic centimeters per minute (SCCM) and a radio-frequency
+plasma power of 250 W was employed during all growths. To
+convert the measured values of 𝜙Ga (BEP), 𝜙In (BEP), and 𝜙O
+(SCCM) into units of nm−2 s−1 conversion factors are taken
+from Ref. [24]. Note, when using In-mediated catalysis, the
+available 𝜙O for Ga to Ga2O3 oxidation is about 2.8 times
+larger than for Ga oxidation in the absence of In [16, 18].
+TABLE I. Collected growth parameters used in this work, val-
+ues of 𝜙Ga, 𝜙In, 𝜙O, and 𝑇G, for samples grown by conven-
+tional MBE and MOCATAX are listed.
+The conversion for
+𝜙Ga and 𝜙In from BEP to nm min−1 to nm−2 s−1 are 𝜙Ga =
+2.5 × 10−8 Torr ˆ= 1.1 nm min−1 ˆ= 1 nm−2 s−1 and 𝜙In = 1.1 ×
+10−7 Torr ˆ= 2.6 nm−2 s−1, respectively.
+Growth parameters
+Conventional MBE
+MOCATAXY
+𝜙Ga (nm−2 s−1)
+0.8 – 2.0
+1.1 – 5.5
+𝜙In (nm−2 s−1)
+0
+2.6 – 2.8
+𝜙O (SCCM)
+1.4
+0.7 – 1.0
+𝜙O (nm−2 s−1)
+2.2
+3.2 – 4.6
+𝑇G (◦C)
+640 – 800
+680
+For samples grown by conventional MBE and MOCATAXY,
+the impact of 𝜙Ga and 𝑇G is studied. In the case of MO-
+CATAXY growth, the impact of 𝜙In is also investigated. All
+growth parameters used in this work are collected in Table I.
+For scanning transmission electron microscopy (STEM),
+samples were prepared using Thermo Fisher Helios G4 UX
+Focused Ion Beam with a final milling step of 5 keV to
+minimize damage. Carbon and Au-Pd layers were sputtered
+to reduce charging during sample preparation. Carbon and
+platinum protective layers were also deposited to minimize
+ion-beam damage. STEM measurements were taken with
+an aberration-corrected Thermo Fisher Spectra 300 CFEG
+operated at 300 keV.
+RESULTS AND DISCUSSION
+Figure 1 shows the growth-rate-diagram of α-Ga2O3(10¯10)
+grown on α-Al2O3(10¯10) by conventional MBE (the gray
+shaded area) and MOCATAXY (the purple shaded area). For
+conventionally grown samples two distinct growth regimes
+are observed: (i) the O-rich regime where O adsorbates are in
+excess over Ga adsorbates (i.e., the Ga flux limited regime),
+and (ii) the 𝛤-plateau regime (i.e., the Ga2O desorption lim-
+ited regime). The O-rich regime is characterized by an in-
+creasing 𝛤 with increasing 𝜙Ga, whereas the plateau regime
+is characterized by a constant 𝛤, being independent of 𝜙Ga.
+Within this regime, however, 𝛤 may decrease with increasing
+𝑇G (see inset in Fig. 1) as the desorption of the volatile sub-
+oxide Ga2O becomes thermally more active [27]. The data in
+the inset of Fig. 1 plot 𝛤 as a function of 𝑇G: (i) for α-Ga2O3
+grown the O-rich regime and (ii) for α-Ga2O3 grown in the
+𝛤-plateau regime.
+To expand the accessible growth window of α-Ga2O3 to
+higher 𝜙Ga and higher 𝑇G, combined with increased 𝛤 and
+improved crystalline quality, In-mediated catalysis was em-
+ployed to the formation of α-Ga2O3 [18]. The red stars in
+Fig. 1 show the resulting 𝛤 as a function of 𝜙Ga at con-
+stant 𝑇G.
+The gray shaded and purple shaded areas in
+Fig. 1 depict model-based descriptions of 𝛤 for α-Ga2O3
+grown by conventional MBE and MOCATAXY, respec-
+
+3
+1.6
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+Growth Rate,  Γ (nm
+-2s
+-1)
+3.5
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+Growth Rate,  Γ (nm/min)
+10
+8
+6
+4
+2
+0
+Ga Flux, φGa (nm
+-2s
+-1)
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+Ga Flux, φGa (10
+7× torr)
+1.6
+1.2
+0.8
+0.4
+0.0
+Growth Rate (nm/min)
+800
+750
+700
+650
+Growth Temp(ºC)
+MOCATAXY
+T = 680°C
+Conventional
+T = 680°C
+1.2
+0.8
+0.4
+0.0
+Growth Rate (nm/min)
+800
+750
+700
+650
+Growth Temp(ºC)
+1.6
+1.2
+0.8
+0.4
+0.0
+Growth Rate (nm/min)
+800
+750
+700
+650
+Growth Temp(ºC)
+Fig 1: Shows the growth rate (Γ) of α-Ga O (10-10) grown on α-Al O (10-10) as a function of the
+1.6
+1.2
+0.8
+0.4
+0.0
+Growth Rate (nm/min)
+800
+750
+700
+650
+Growth Temp(ºC)
+φGa = 0. 9
+φGa = 1.6
+3.5
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+Growth Rate (nm/min)
+10
+8
+6
+4
+2
+0
+Ga Flux (atoms/nm
+2s)
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+Growth Rate (atoms/nm
+2s)
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+Ga BEP (torr)×10
+-7
+MOCATAXY
+T = 680°C
+Conventional
+T = 680°C
+(i) φGa = 0. 9
+(ii) φGa = 1.6
+(i)
+(ii)
+3.5
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+Growth Rate (nm/min)
+10
+8
+6
+4
+2
+0
+Ga Flux (atoms/nm
+2s)
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+Growth Rate (atoms/nm
+2s)
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+Ga BEP (torr)×10
+-7
+1.6
+1.2
+0.8
+0.4
+0.0
+Γ (nm/min)
+800
+750
+700
+650
+TG (ºC)
+MOCATAXY
+Conventional
+Ga Flux, φGa (nm-2 s-1)
+FIG. 1.
+Growth-rate-diagram of α-Ga2O3(10¯10) grown on α-
+Al2O3(10¯10). The growth rate 𝛤 as a function of 𝜙Ga at 𝑇G =
+680 ◦C is plotted for the growth by conventional MBE (blue trian-
+gles) and MOCATAXY (red stars). The 𝛤-data is fit by a 𝛤-model
+taken from Ref. [25]. The gray shaded region shows the parameter
+space under which the formation of α-Ga2O3by conventional MBE
+may occur. The purple shaded area depicts the growth regime of
+α-Ga2O3assisted by MOCATAXY. Both fitted data sets were ob-
+tained at constant 𝑇G and 𝜙O (values given in Table I). Inset: 𝛤 as
+a function of 𝑇G at two different fluxes of (i) 𝜙Ga = 0.9 nm−2 s−1
+(the O-rich regime, solid squares) and (ii) 𝜙Ga = 1.6 nm−2 s−1 (the
+𝛤-plateau regime, solid discs). A growth-rate-diagram of α-Ga2O3
+as a function of 𝜙O is given in Ref. [26].
+tively. The maximum 𝛤 obtained for each growth technique
+is 𝛤 ≈ 1.5 nm min−1 and 𝛤 ≈ 3.3 nm min−1, respectively.
+Using MOCATAXY, a more than 2-times increase in 𝛤 for
+α-Ga2O3 at given growth conditions, as well as a shift far
+into the adsorption-controlled regime (i.e, far into the Ga rich
+flux regime) is observed. This direct comparison between
+the two growth types clearly shows the expanded growth
+window made possible with MOCATAXY, for example, en-
+abling 𝛤 ≈ 1.8 nm min−1 for α-Ga2O3 at 𝜙Ga = 5.5 nm−2 s−1.
+In contrast, at these growth conditions, no growth of α-Ga2O3
+is obtained by conventional MBE. The catalytic effect on 𝛤
+of α-Ga2O3 is modeled as a function of 𝜙O within the sup-
+plemental section [26]. We note that the depicted models
+use arbitrary kinetic parameters, based on kinetic parameters
+extracted for the growth of β-Ga2O3 [25].
+To describe the growth of α-Ga2O3 by MOCATAXY, 𝜙O
+is scaled by a factor of 2.8 compared with the growth of
+α-Ga2O3 by conventional MBE. This additional O comes
+from the catalytic nature of In forming a catalytic adlayer
+(𝐴) with O adsorbates, e.g., 𝐴 = In–O, which provides more
+active O for the Ga to α-Ga2O3 oxidation. In other words,
+𝐴 increases the reaction probability of Ga with O on the
+respective growth surface, facilitating the formation of the
+final Ga2O3 compound at much higher 𝜙Ga and 𝑇G, which
+-1
+0
+1
+ω(º)
+10
+-1
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+7
+10
+8
+Intensity (a.u.)
+68
+67
+66
+65
+64
+2θ-ω (°)
+[email protected]
+3
+ɑ-Ga2O3 with MOCATAXY 
+Conventional
+MOCATAXY
+Fig. 2 Longitudinal XRD scans recorded for optimized Ga2O3 films grown on α-Al2O3(10
+by conventional MBE and MOCATAXY. The reflections of the films coincide with the
+Ga2O3(10-10) phase grown by conventional MBE (the black trace) and MOCATAXY (the blue
+trace). The Ga2O3 films were grown at !Ga … and TG = …, respectively, where an O flux
+… was provided. The growth rates and surface morphologies of both films are shown in Figs
+1(?) and 2(?) and depicted as … and …
+MOCATAXY AFM
+(a)
+(d)
+(e)
+Conventional
+Rq = 0.64 nm
+MOCATAXY
+Rq = 0.96nm
+Best individual AFM and Best individual XRD RC
+Conventional AFM
+X-ray Intensity (arb. unit)
+30$30
+ɑ-Ga2O3
+30$30
+ɑ-Al2O3
+1.92 in = 10um
+2μm
+2μm
+Δω = 0.55°
+(b)
+(c) Δω = 0.45°
+X-ray Intensity (arb. unit)
+FIG. 2. (a) Longitudinal XRD scans of optimized α-Ga2O3 films.
+The reflections of the films coincide with the α-Ga2O3(10¯10)
+phase grown by conventional MBE (the blue trace) and MO-
+CATAXY (the red trace). The used growth parameters were 𝜙Ga
+= 2.9 nm−2 s−1, 𝜙O = 1.4 SCCM ˆ= 2.2 nm−2 s−1, and 𝑇G = 750 ◦C
+(conventional MBE), and 𝜙Ga = 2.9 nm−2 s−1, 𝜙In = 2.8 nm−2 s−1,
+𝜙O = 0.7 SCCM ˆ= 3.2 nm−2 s−1, and 𝑇G = 680 ◦C (MOCATAXY).
+(b) and (c) Transverse XRD scans across the 30¯30 peak with
+their FWHM of ∆𝜔 = 0.55◦ (conventionally MBE-grown) and
+∆𝜔 = 0.45◦ (MOCATAXY-grown). These obtained ∆𝜔 are de-
+picted in Fig. 3 at given 𝜙Ga and 𝑇G. (d) and (e) Surface morpholo-
+gies obtained by 10 × 10 μm AFM scans for α-Ga2O3(10¯10) sur-
+faces grown by conventional MBE and MOCATAXY, respectively.
+Growth conditions for the samples plotted in (d) and (e) are the same
+as for the ones plotted in panels (a)–(c), except a slightly lower 𝑇G=
+730 ◦C used for the conventionally grown sample and a slightly
+higher supplied 𝜙O = 1.0 SCCM for the MOCATAXY grown film.
+This resulted in ∆𝜔 = 0.61◦ and 𝛤 ≈ 1.2 nm min−1 for the conven-
+tionally grown sample, and ∆𝜔 = 0.48◦ and 𝛤 > 3.0 nm min−1 for
+the MOCATAXY grown sample.
+enables excellent crystal quality [16, 18]. We further note
+that the same factor of 2.8 was needed for modeling the
+MOCATAXY growth of β-Ga2O3 on different substrates and
+different surface orientations [16, 18]. We note, however, that
+for a quantitative extraction of all kinetic growth parameters
+more 𝛤-studies of α-Ga2O3 are needed and are beyond the
+scope of this work. Nevertheless, the models help validate
+the 𝛤-data and provide insight into the growth regimes and
+growth mechanisms of α-Ga2O3.
+For example, once 𝜙Ga
+exceeds the active O flux, i.e., for 𝜙Ga > 𝜙O, the growth
+will enter the Ga-rich regime and 𝛤 will start to decrease, as
+shown by the gray shaded area in Fig. 1. Thus, this is the first
+direct indication that the growth of α-Ga2O3 is limited by the
+formation and subsequent desorption of Ga2O, like what is
+observed for β-Ga2O3 grown by conventional MBE [25].
+
+oμm
+2
+4
+6
+8
+10.0nm
+oum
+8.0
+N
+6.0
+4.0
+2.0
+0.04
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+Rocking Curve FWHM (º)
+760
+720
+680
+640
+Growth Temperature (ºC)
+6
+5
+4
+3
+2
+1
+0
+Ga Flux (atms/nm
+2s)
+6
+5
+4
+3
+2
+1
+0
+Ga Flux (atms/nm
+2s)
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+Rocking Curve FWHM (º)
+760
+720
+680
+640
+Growth Temperature (ºC)
+Fig. 3: (a) Full width at half maxima (FWHM) as a function of the growth temperature (TG), obtained by
+transverse XRD scans across the 30-30 peaks of Ga2O3 [e.g., see Fig. 3]. (b) The root means square (rms)
+roughnesses as a function of TG; measured by AFM [see Fig. ? or supplement ... ]. Three distinct growth regimes
+are indicated in panels (a) and (b): (i) the O-rich rich regime (depicted as squares), (ii) the adsorption-controlled
+regime (depicted as stars), and the MOCATAXY regime (depicted as triangles). Panels (c) and (d) show the
+impact of the Ga flux on the FWHM of the 30-30 peak and rms roughnesses, respectively, of the grown α-
+Ga2O3(10-10) thin films.
+(a)
+%-Plateau Regime
+MOCATAXY
+3500
+3000
+2500
+2000
+1500
+1000
+500
+0
+Rocking Curve FWHM (arcsec.)
+760
+720
+680
+640
+Growth Temperature (ºC)
+O-rich Regime
+Tsub = 680°C
+φGa = 0.95 (O-rich)
+φGa = 1.24 (%-Plateau)
+φGa = 5.5 (MOCATAXY)
+Ga Flux, φGa (nm-2 s-1)
+TG (°C)
+(b)
+FIG. 3. (a) and (b) FWHM (i.e., ∆𝜔 values) are plotted as a function
+of 𝑇G and 𝜙Ga are plotted, respectively. Values are obtained by
+transverse XRD scans of the 30¯30 peaks of α-Ga2O3 grown films
+(XRD data not shown). Three distinct growth regimes are studied
+in panels (a) and (b): (i) the O-rich rich regime (blue squares), (ii)
+the 𝛤-plateau regime (green circles), and (iii) the MOCATAXY
+regime (red stars). The lowest value of ∆𝜔 is indicated by a dashed
+line. Note that for the samples grown by MOCATAXY, 𝜙In = (2.6 –
+2.8) nm−2 s−1 was supplied and might explain the slight variations
+observed in ∆𝜔 for α-Ga2O3 grown at 𝜙Ga = 2.9 nm−2 s−1 in panel
+(b)].
+Figure 2 directly compares the impact of both MBE growth
+techniques on the structural quality of the epitaxially grown
+films. In Fig. 2 (a), 2𝜃-𝜔 XRD scans of two selected α-
+Ga2O3 films are shown, one grown by conventional MBE
+(depicted as the blue trace) and one grown by MOCATAXY
+(depicted as the red trace). The reflections of the films co-
+incide with the α-Ga2O3 30¯30 peak. This, along with the
+absence of other diffraction peaks, indicates phase-pure α-
+Ga2O3(10¯10) with In incorporation of < 1% in the grown
+α-Ga2O3 layers, similar to what is observed for β-(Al,Ga)2O3
+grown by MOCATAXY [17]. Fig. 2(b) and 2(c) plot trans-
+verse scans (rocking curves) for the conventional MBE and
+MOCATXY grown α-Ga2O3 samples as plotted in Fig. 2(a).
+The rocking curves are measured across the symmetric 30¯30
+peak. The full width at half maxima (FWHM) of 𝜔 quan-
+tifies the out-of-plane mosaic spread of the α-Ga2O3 film.
+For conventionally grown films the out-of-plane crystal dis-
+tribution is ∆𝜔 ≈ 0.55◦ and for MOCATAXY grown films
+it is ∆𝜔 ≈ 0.45◦. The film thicknesses 𝑑 of the conven-
+tionally and MOCATAXY grown films are 𝑑 = 73 nm and
+𝑑 = 127 nm, respectively. Jinno et al., reported that α-Ga2O3
+films are fully relaxed for 𝑑 > 60 nm [5]. Since lattice mis-
+match and relaxation are not impacted by MOCATAXY, it is
+noteworthy that despite the MOCATAXY film being thicker,
+∆𝜔 is substantially smaller compared to what is obtained by
+conventional growth. The same MOCATAXY grown sample
+shown here is studied by STEM and shown in Fig. 4.
+Surface morphologies and root mean square roughnesses
+(𝑅q) are measured by AFM and depicted in Figs. 2(d) and
+2(e). The best surface roughness for conventionally grown α-
+Ga2O3with 𝑑 = 66 nm is 𝑅q = 0.64 nm, while the smoothest
+one for MOCATAXY grown samples with 𝑑 ∼ 270 nm has
+an 𝑅q = 0.94 nm. The larger surface roughness for the MO-
+CATAXY grown sample is likely due to facetting on the top
+surface of the α-Ga2O3 thin film [see Fig. 4(a)]. We specu-
+late that In does not only act as a catalyst but also acts as a
+surface active agent (surfactant) for the growth α-Ga2O3 thin
+films. It is widely understood that In can act as a surfactant
+for the epitaxial growth of GaN-based films [28], and has
+also been observed during the growth of β-Ga2O3 [20] and
+β-(Al, Ga)2O3 [29].
+Depending on the growth conditions
+and growth surface, which can affect the surface diffusion ki-
+netics, surface chemical potentials, and the assessed growth
+mode, the suppression of facetting may be accomplished
+through the use of optimized conditions using In as a surfac-
+tant, enabling a modification in the surface free energies of
+the growing α-Ga2O3 thin film and a change in its growth
+mode [17, 20, 30, 31]. However, surfactant-induced mor-
+phological phase-transitions from 2-dimensional (2D) layer
+growth to 3-dimensional (3D) island growth have also been
+observed during MBE growth [32]. We believe that a simi-
+lar effect occurs for the α-Ga2O3 surfaces studied here when
+In may act as an (anti)surfactant during the growth of these
+films. Note, we have not fully explored all growth regimes
+made accessible through MOCATAXY in this study. Further
+studies may lead to additional improvements in the crystalline
+quality and surface morphologies of the α-Ga2O3 thin films.
+In Figs. 3(a) and 3(b), the impact of 𝜙Ga and 𝑇G, respectively,
+on ∆𝜔 for samples grown by conventional MBE in the O-
+rich regime (blue squares) and in the 𝛤-plateau regime (green
+circles), as well as for samples grown by MOCATAXY (red
+stars), are shown. XRD data and ∆𝜔 are obtained by the
+same methods as described above for Fig. 2. Within the O-
+rich regime at 𝑇G = 640 ◦C, a large ∆𝜔 is observed, Fig. 3(a).
+At higher growth temperatures (i.e. 𝑇G ≥ 660 ◦C), ∆𝜔 are
+similar (or slightly improving) with increasing temperature,
+regardless of growth regime. We speculate that the reason
+the crystal quality improves with 𝑇G, is that there is an in-
+crease in the kinetic energy and a subsequent increase in the
+diffusion length of the adsorbates, allowing the Ga and O
+to reach the proper lattice site. However, if 𝑇G is increased
+too much, a decrease in the surface lifetime of Ga adsorbates
+may occur, resulting in a reduction in the crystalline quality
+of the growing thin films. Using MOCATAXY in the Ga-rich
+regime and fixed 𝑇G, excess Ga may now reduce the needed
+surface diffusion length, improving the crystalline quality of
+the obtained α-Ga2O3 layers. More studies to separate the
+effects of 𝜙Ga and 𝑇G on ∆𝜔 need to be performed, but to the
+best of our knowledge, the obtained ∆𝜔 values are the lowest
+reported in the literature for α-Ga2O3 grown on α-Al2O3.
+Finally, to directly quantify and identify how MOCATAXY
+affects the crystal structure of α-Ga2O3 thin films, high-angle
+annular dark-field STEM (HAADF-STEM) was performed
+along the < 0¯110 > zone axis, and is plotted in Fig. 4. The
+sample shown here is the same as the one shown in Fig. 2(c).
+In Fig. 4(a), a clear contrast differentiates the sapphire sub-
+strate, the epitaxial film (α-Ga2O3), and the protective Au-Pd
+
+5
+Fig 4. HAADF-STEM images showing an overview of Alpha-Ga2O3 film grown on m-plane sapphire. A) The film shows thickness of ___nm with 
+faceting on film surface. Line defects are running from the interface to surface on average every ___nm. Increased brightness at the interface as a result 
+of scattering shows high density of defects. B) Enlarged image of interface show presence of defect due to strain relaxation.
+25 nm
+200 nm
+5 nm
+(a)
+(b)
+!→
+!→
+(c)
+2 0 2 4
+1 0 1 10
+(1011)
+[0001]
+[0110]
+Ga2O3
+Al2O3
+Al2O3
+Ga2O3
+FIG. 4. HAADF-STEM images show an overview of an α-Ga2O3(10¯10) film grown on α-Al2O3(10¯10). (a) The epitaxial film shows
+increased contrast due to misfit dislocations at the film/substrate interface. Threading dislocation propagate through the film and terminating
+at the intersection of its surface periodic faceting. (b) Enlarged image of the film-substrate interface (i.e., the α-Al2O3-α-Ga2O3 interface)
+is shown. Burger circuits are drawn around the edge dislocations. (c) Fast Fourier transform (FFT) of the interface region is shown.
+Diffraction peak separation at (20¯2¯4) and (10¯110) indicate strain relaxation of the α-Ga2O3(10¯10) on α-Al2O3(10¯10).
+sputtered coating. The bright contrast observed at the sub-
+strate/film interface (see Fig. 4(b) and Ref. [26]) is due to
+additional scattering of the electron beam and indicates the
+presence of misfit dislocations. These dislocations arise due
+to the film relaxation caused by strain. A subset of the ob-
+served misfit dislocations propagate and lead to threading
+dislocations. From the contrast variation observed within
+the film [see Fig. 4(a)], an average frequency of one thread-
+ing dislocation every 30 nm laterally along the film/substrate
+interface is observed. While more investigation is needed to
+determine the cause of the faceting and verify the above hy-
+pothesis (e.g., due to the changed growth mode when using
+In-mediated catalysis), it is observed that the threading dislo-
+cations can merge and then continue to propagate toward the
+film surface. These dislocations terminate at the bottom of
+intersecting surface planes, where faceting along the (10¯11)
+plane is observed. The complimentary facet is unidentified
+since the facet is not perpendicular to the beam and tilts out
+of plane. This tilting is detected in Fig. 4(a) by the fading of
+contrast along the surface, in contrast to the sharp change in
+contrast on the (10¯11) plane.
+Figure 4(b) shows an enlarged image of the film/substrate
+interface.
+A pair of edge dislocations is observed and is
+highlighted with their Burgers circuits. This edge dislocation
+pair is observed along the film/substrate interface, and its
+dislocation density is estimated to be 5 × 105 cm−1 (or ∼
+1011 cm−2), i.e., occurring every 20 nm. This is similar to
+what is reported by conventional MBE [5]. To quantify Al/Ga
+inter-diffusion at the interface, a line scan (see S-Fig. 2 [26])
+was performed to quantify the contrast change. An interface
+width of 𝜎 ≈ 0.9 nm was measured from an error function
+fitted to the Al intensity line scan profile (see S-Fig. 2 [26]).
+A fast Fourier transform (FFT), of the interface region shown
+in Fig. 4(b), is displayed in Fig. 4(c). A thin film completely
+strained to the substrate will show a singular diffraction peak.
+However, when the film relaxes its interplanar spacing 𝑑ℎ𝑘𝑙
+changes, resulting in an additional peak, shifted from the sub-
+strate peak. However, shifted peaks in the in-plane direction
+are not visible because the α-Ga2O3 (000¯6) reflection peak is
+approximately 10x weaker than in α-Al2O3. The strain relax-
+ation is observed in the 20¯2¯4 and 10¯110 diffraction peaks of
+α-Ga2O3. The strain relaxation is accomplished by the for-
+mation of edge dislocations at the interface, where the 20¯2¯4
+peak is correlated to the yellow Burgers circuit and the 10¯110
+peak to the cyan Burgers circuit. In addition, no phase separa-
+tion or secondary phases were observed by STEM within the
+α-Ga2O3 film grown by MOCATAXY. However, a bi-layer
+structure from overlapping α-Ga2O3 grains when viewed in
+projection is observed with a slip along the [10¯2¯2] direction
+(see S-Fig. 3 [26]). The presence of this bi-layer structure
+indicates that the film is not single-crystalline. The bi-layer
+structure was confirmed using an ab initio TEM (abTEM)
+simulation [33] which produced a matching HAADF image
+from the crystallographic information framework.
+This TEM investigation of MOCATAXY grown α-Ga2O3
+shows comparable crystal quality to what is measured for
+conventional MBE [5] with regards to edge dislocation den-
+sity and phase purity. We note that the difference in pro-
+jection direction may have prevented imaging of the bi-layer
+structure in this previous report. No faceting of α-Ga2O3
+was observed by conventional MBE when grown on 𝑚-plane
+α-Al2O3 [5, 9].
+CONCLUSION
+Phase-pure α-Ga2O3(10¯10) on α-Al2O3(10¯10) was grown
+using conventional MBE and MOCATAXY with thickness
+up to 𝑑 = 262 nm. We mapped out the 𝛤-dependence on 𝜙Ga
+and 𝑇G and its impact on the crystalline quality and surface
+morphologies. We identified and explored previously inac-
+cessible growth regimes by MOCATAXY, and showed that
+
+6
+it vastly extends the growth regime and improves the out-
+of-plane mosaic spread of the grown α-Ga2O3 films. Using
+In-mediated catalysis, we observe facetting on top of the α-
+Ga2O3(10¯10) layers. This study confirms that this new MBE
+growth mode can be applied to the growth of α-Ga2O3– and
+is not limited to the growth of the β-Ga2O3 and β-(Al,Ga)2O3
+polymorphs. We emphasize more studies are needed to de-
+termine the kinetic parameters that form α-Ga2O3 during
+conventional MBE and MOCATAXY growth, as well as to
+further improve the quality of the grown α-Ga2O3/α-Al2O3
+heterostructures, and to understand the mechanisms leading
+to the surface faceting of α-Ga2O3.
+ACKNOWLEDGEMENTS
+This research is supported by the Air Force Research
+Laboratory-Cornell Center for Epitaxial Solutions (AC-
+CESS), monitored by Dr. Ali Sayir (FA9550-18-1-0529).
+JPM acknowledges the support of a National Science Foun-
+dation Graduate Research Fellowship under Grant No.
+DGE–2139899.
+M. A-O acknowledges financial support
+from the Central Research Development Fund (CRDF) of the
+University of Bremen. This work makes use of PARADIM
+under Cooperative Agreement No.
+DMR-2039380.
+This
+work uses the CCMR and CESI Shared Facilities partly
+sponsored by the NSF MRSEC program (DMR-1719875)
+and MRI DMR-1338010, and the Kavli Institute at Cornell
+(KIC).
+∗ Electronic mail: [email protected]
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+

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+Task formulation for Extracting Social Determinants of Health from Clinical Narratives 
+Manabu Torii, Ian M. Finn, Son Doan, Paul Wang, Elly W. Yang, Daniel S. Zisook 
+ 
+Abstract 
+Objective The 2022 n2c2 NLP Challenge posed identification of social determinants of health 
+(SDOH) in clinical narratives. We present three systems that we developed for the challenge 
+and discuss the distinctive task formulation used in each of the three systems. 
+Materials and Methods The first system identifies target pieces of information independently 
+using machine learning classifiers. The second system uses a large language model (LLM) to 
+extract complete structured outputs per document. The third system extracts candidate 
+phrases using machine learning and identifies target relations with hand-crafted rules. 
+Results The three systems achieved F1 scores of 0.884, 0.831, and 0.663 in the Subtask A of the 
+Challenge, which are ranked third, seventh, and eighth among the 15 participating teams. The 
+review of the extraction results from our systems reveals characteristics of each approach and 
+those of the SODH extraction task.  
+Discussion Phrases and relations annotated in the task is unique and diverse, not conforming to 
+the conventional event extraction task. These annotations are difficult to model with limited 
+training data. The system that extracts information independently, ignoring the annotated 
+relations, achieves the highest F1 score. Meanwhile, LLM with its versatile capability achieves 
+the high F1 score, while respecting the annotated relations. The rule-based system tackling 
+relation extraction obtains the low F1 score, while it is the most explainable approach. 
+Conclusion The F1 scores of the three systems vary in this challenge setting, but each approach 
+has advantages and disadvantages in a practical application. The selection of the approach 
+depends not only on the F1 score but also on the requirements in the application. 
+ 
+ 
+ 
+
+Background and Significance 
+Clinical notes are a rich source of information, containing, among others, patient-reported 
+information and clinicians’ assessments that are not coded in structured records. Automated 
+extraction and coding of information has been widely studied 1. Among different types of 
+information sought in clinical notes, social determinants of health (SDOH) have gained attention 
+in the last several years, due to their significance on one’s health as well as to their unique 
+availability in clinical notes 2,3. In the Track-2 of the 2022 n2c2 NLP Challenge 4,5, extraction of 
+SDOH from clinical notes was posed as a shared task, and a corpus annotated with SDOH was 
+prepared by the challenge organizer. The availability of the annotated corpus would further 
+increase interests in this information extraction task and advance the technology toward real-
+world applications. 
+This paper focuses on three information extraction systems that we developed for our 
+submissions of the Track-2 in the 2022 n2c2 NLP Challenge, while we defer the background of 
+this Challenge task and the review of related studies to the publications by the Challenge 
+organizers3–6. In the corpus prepared for the Challenge, types of annotated phrases are unique 
+and diverse. Relations to be identified among them are difficult to characterize, making the task 
+very different from the conventional event extraction. Each of the three systems we developed 
+employs a different task formulation to tackle this challenge. 
+ 
+Objective 
+Natural language processing (NLP) technology has undergone many changes over the years, 
+especially in the last several years 7. New methods as well as long-standing methods have been 
+evaluated for different clinical NLP tasks in shared-task challenges 8,9. Besides the performance 
+evaluation results, the task formulation considered for each shared-task challenge has 
+contributed to the clinical NLP field, providing the baselines in designing an information 
+extraction system for the same or related task.  Given these backgrounds, two objectives of this 
+paper are as follows: 
+
+1. We describe three systems that we developed for the submissions for the 2022 n2c2 
+NLP Challenge Track-2, which employ both recent and long-standing methods and were 
+ranked high among the participating systems. 
+2. We present three different formulations of the task that we used in our three systems 
+and discuss the motivation, results, and advantages and disadvantages of each 
+approach. 
+ 
+Materials and Methods  
+The Subtask-A of the Track-2, in which we participated, used the Social History Annotation 
+Corpus (SHAC) 3. The data consisted of 1,316, 188, and 373 clinical narrative texts from MIMIC 
+III 10, which were released respectively as the training, development, and test set. During the 
+challenge period, the training and development sets were made available for the participants to 
+develop systems, and the test set was released for the final evaluation. All these data sets were 
+provided as brat annotation files, consisting of narrative text files (.txt) and corresponding 
+annotation files (.ann). Further information of the brat annotation tool can be found in the brat 
+tool paper 11 and on the brat web page 12. 
+In the SHAC corpus, texts are annotated with trigger phrases for five types of SDOH 
+(Alcohol, Tobacco, Drug, Employment, and Living Status) along with their associated argument 
+phrases. A subset of the argument phrases, named labeled arguments, are normalized to 
+predefined labels (e.g., Status is a labeled argument for Alcohol, normalized to one of the three 
+status values: none, current, or past). The rest of the argument phrases, named span-only 
+arguments, do not have labels to normalized to and are “spans only” (e.g., Duration is a span-
+only argument for Alcohol, annotated for the duration of alcohol use, such as “for eight years”). 
+Further information of the corpus, including annotation examples, can be found in the SHAC 
+corpus paper and in the evaluation guideline document 3,6. The evaluation script used in the 
+challenge is provided by the organizer on GitHub 13. 
+
+During our participation in the challenge, we considered three formulations of the task 
+and implemented three systems as described next. We did not explore the use of additional 
+texts or annotations or the augmentation of the provided data. 
+ 
+System 1: Sentence classification and sequence labeling 
+There are many triggers and arguments in the current task. We observed difficult topics in NLP 
+are involved for their detection (e.g., phrase boundary ambiguity; nested phrase annotations; 
+trigger-argument across sentences; one or more annotated phrases per argument type). 
+However, a good fraction of triggers and arguments look easy to identify (e.g., repeatedly 
+annotated unambiguous phrases). Also, the evaluation metric used in the challenge is forgiving 
+(e.g., phrase spans are not required for the labeled argument). Considering these factors, we 
+convert the given task into a set of simpler tasks that can be tackled by common methods. 
+In this approach, an input narrative is first split into sentences using a regular expression 
+pattern, and then, two common methods are applied to each sentence, independently:  
+1. Text classification to identify sentences containing triggers and labeled arguments 
+2. Sequence labeling to extract triggers and span-only arguments in each sentence. 
+There are two key observations behind this approach. First, most of the trigger-argument 
+relations are within a single sentence, and there is usually at most one trigger of the same kind 
+within each sentence, which is also noted in the SHAC corpus paper 3. Second, phrase spans are, 
+in effect, not required in the evaluation of triggers and labeled arguments. That is, labeled 
+arguments are evaluated by the inferred label values only. Triggers are evaluated by the span, 
+but any overlap between the predicted span and the annotated gold span is counted. 
+Therefore, the trigger span is not required in effect if a long enough span is proposed. 
+The two observations lead to a task formulation that, for triggers and labeled 
+arguments, we only need to classify each sentence whether it implies a particular trigger type 
+or a particular labeled argument, e.g., “Does this sentence report a patient’s alcohol abuse?” or 
+“Does this sentence report a patient’s current alcohol abuse?” As for the span-only argument, 
+
+the task needs to be tackled as sequence labeling, specifically the common BIO labeling of 
+tokens 14. A separate model is prepared for each span-only label type and for each trigger type 
+because phrases annotated for span-only arguments and triggers sometimes overlap each 
+other. After triggers and arguments are detected independently, the predictions are merged 
+per sentence. When an argument is predicted by any of the models, the corresponding trigger 
+must be present for it to be reported, and the trigger is additionally predicted, if it is not 
+predicted by the trigger detection methods. 
+For the text classification, a multi-label text classification model was trained using the 
+Hugging Face Transformers library 15, which is used to make binary classification for 28 targets: 
+5 triggers and 23 labeled arguments. The implementation follows a publicly available Jupyter 
+notebook example, “Fine-tune BERT for Multi-label Classification” 16. For the BERT model, 
+Bio_Discharge_Summary_BERT was selected 17, because it seemed to yield slightly 
+better performance than the other model we tested,  BioBERT 18, during the development. 
+For sequence labeling (2), 33 models were trained also using the Hugging Face Transformers 
+library, each of which extracts phrases for a specific trigger and span-only label: 5 triggers and 
+28 span-only labels. The implementation follows the tutorial “Token classification” in the 
+Hugging Face Course 19. For the BERT model, between the two models tested, BioBERT was 
+used. 
+
+ 
+ 
+Figure 1. Few-shotting GPT-J with alcohol narratives 
+To few-shot GPT-J for the social history extraction task, we convert the .ann format of the annotated 
+text into a structured table prompt that maintains the essential content but is more compact and 
+amenable for data capture. The word “unknown” is used in all cases where the .ann file does not 
+have an annotation for the given element.  
+Of note, all information regarding spans is eliminated in the conversion in Figure A. We create a 
+column labeled “Inference” to store categorical annotations. Each E line in the .ann file is translated 
+into a single row in the table prompt, allowing for the possibility of multiple triggers of the same 
+type. 
+Sample GPT-J generator parameters are shown in Figure B. We use “###” to hint to the model when 
+language generation should cease.  
+Figure C shows perfect matching of the model output and the gold .ann representation in prompt 
+format. 
+
+A
+.ann Representation
+GPT-J prompt
+E
+A
+prompt = ""
+Alcohol:"drinking"
+StatusTimeVal:"current"
+Make a table about alcohol use in the following story. Use exact words or phrases from the story.
+Status:"reports"
+She reports drinking 2 alcoholic drinks per month.
+Frequency:"per month"
+I Alcohol I Amount I Duration I Frequency I History I Type I Status I Inference I
+Amount: "2 alcoholic drinks
+I drinking I 2 alcoholic drinks I unknown I per month I unknown I unknown I reports I current I
+###
+E
+A
+Make a table about alcohol use in the following story. Use exact words or phrases from the story.
+Alcohol:"ETOH"
+StatusTimeVal:"none"
+Denies ETOH
+Status:"Denies"
+I Alcohol I Amount I Duration I Frequency I History I Type I Status I Inference |
+I ETOH I unknown I unknown I unknown I unknown I unknown I Denies I none I
+###
+E:
+A
+Make a table about alcohol use in the following story. Use exact words or phrases from the story.
+Alcohol: "alcoholic drinks"
+StatusTimeVal: "current"
+Four to five alcoholic drinks per night.
+Amount: "Four to five alcoholic drinks"
+I Alcohol I Amount I Duration I Frequency I History I Type I Status I Inference I
+Status: "drinks"
+I alcoholic drinks I Four to five alcoholic drinks I unknown I per night I unknown I unknown I drinks I current I
+Frequency: “per night"
+###
+E:
+Make a table about alcohol use in the following story. Use exact words or phrases from the story.
+Alcohol: "Alcohol"
+A
+Status:"no longer drinking"
+StatusTimeVal:“past"
+I Alcohol I Amount I Duration I Frequency I History I Type  Status I Inference I
+History: "in 22 months"
+I Alcohol I unknown I unknown I unknown I in 22 months I unknown I no longer drinking I past I
+###
+?
+A
+Make a table about alcohol use in the following story. Use exact words or phrases from the story.
+?
+h/o EtOH abuse but last drink in 2001.
+I Alcohol I Amount I Duration I Frequency I History I Type I Status I Inference |
+B
+end_sequence="###"
+generator_kwargs = (
+"max_new_tokens':100,
+‘T:,d-do,
+"temperature':.01,
+'clean_up_tokenization_spaces':True,
+'do_sample': True,
+"early_stopping': True,
+"return_full_text': False,
+"pad_token_id':tokenizer.eos_token_id,
+"eos_token_id: int(tokenizer.convert_tokens_to_ids(end_sequence)
+res = generator(prompt, **generator_kwargs)
+print(res)
+c
+:ndno
+:plog
+I EtOH I unknown I unknown I unknown I in 2001 I unknown I h/o I past |System 2: Fine-tuned GPT-J model 
+With the general availability of medium and large size language models we were curious to 
+explore how much of the social history information extraction task could be performed by 
+leveraging the knowledge encoded in an LLM as opposed to layering additional knowledge on 
+top. To this end, we attempted to create a system that performed minimal re-representation of 
+the input data in terms of supplementation with linguistic, structural, or clinical context. A toy 
+example of our thought process is shown in Figure 1. Here we illustrate how few-shotting GPT-J 
+with four brief alcohol related narratives allows the model to correctly generate annotations for 
+an unknown example. Figure 1A demonstrates how we sandwich each narrative with a natural 
+language prompt above and a desired table format below. For the few-shot examples we 
+include data from the E lines (“Event” annotation in brat, i.e., a tuple of phrases) and A lines 
+(“Attribute” annotation in brat, i.e., a label value assigned to a phrase) in the brat .ann files 11 
+but re-formatted to fit the table structure. The rows with gold annotations are placed beneath 
+the header row. We chose the table representation as we suspected it would be more “in 
+distribution” (vs. out of distribution) for GPT-J than other possibilities, including the raw .ann 
+format. In addition, the table format offers flexibility to encompass all E and A information for a 
+given trigger on a single line. For the cases where there are multiple triggers of the same type, 
+we simply add additional rows to the table. 
+The few-shot example shown in Figure 1A was run with parameters indicated in Figure 
+1B. The generated text and gold annotation can be seen in Figure 1C. As formulated, the few-
+shot task is essentially asking the LLM to function as both a question/answer language model 
+(for span extraction) and a few-shot classification model (for categorical assignments). GPT-J 
+performs both tasks admirably in this toy example. 
+While few-shotting demonstrates the power of models like GPT-J to “learn” from a 
+minimal number of examples, the setup is fragile and does not yield high performance across 
+significant numbers of new inferences. The context window for GPT-J does not permit enough 
+few-shot samples to represent the range of annotations for a given social history element. 
+
+Thus, for the actual extraction task we fine-tuned GPT-J using the entirety of the gold .ann files 
+provided. 
+Fine-tuning was performed on a machine with the following specs: 2 V100 32 Gb 
+graphics cards, Intel Xeon 20 core processor, 11Tb of storage and 512 Gb of RAM. The fine-
+tuning python script was written in-house but calls Hugging Face's highly abstracted 
+API. DeepSpeed 20 was used to accomplish offloading as follows: stage 1 shards optimizer states 
+across GPUs, stage 2 adds sharding of gradients, stage 3 adds sharding of model parameters 
+and allows offloading of parameters, weights, and optimizer state. Of note, stage 3 allows 
+offloading to NVMe and CPU + memory. While offloading incurs significant I/O burden, it allows 
+for training arbitrarily large models at the expense of memory, CPU compute, and speed.  
+Just as in the few-shot examples in Figure 1, input to the fine-tuning procedure was 
+provided as single “sandwiches” of natural language prompt, social history narrative, table 
+format, and table rows generated from the annotations in .ann files. Specifically, we 
+incorporated unedited narratives stripped of new lines and span annotations and injected only 
+the “knowledge” that the categorical text is an inference of some sort, and the type of social 
+history data we are looking to generate (in the form of our natural language prompt). In almost 
+every other respect our fine-tuning data is equivalent to using the original files themselves.  
+While we performed a few experiments with different natural language prompts, we do 
+not have data on the effectiveness of our chosen verbiage “Make a table about **** in the 
+following story. Use exact words or phrases from the story.” Anecdotally the choice of prompt 
+did not seem to impact performance significantly and in this use case, and possibly others, the 
+prompt may have been superfluous. The LLM demonstrated considerable ability to memorize 
+the training data, achieving 93-94% F1 score when applied to the training data.  
+We did not generate exact spans from GPT-J, hypothesizing that would be challenging. 
+Due to time constraints, we created some simple heuristics to map the model evaluation text 
+back to the narrative string. We did attempt some experiments where we included an 
+additional word on either side of the gold annotation to try and increase specificity in the 
+eventual map back to the narrative. We do not have results on performance from these 
+
+experiments but anecdotally it seemed to decrease. The loss of information about spans likely 
+resulted in a decreased recall for our effort. 
+ 
+System 3: NLP pipeline reuse 
+In this approach, we regard SDOH information as an event just as the information is annotated 
+and tackle trigger detection, argument detection, and trigger-argument relation extraction. This 
+general framework has been widely used for event extraction 21,22, and subtasks are commonly 
+organized in a pipeline manner, unless they are solved jointly, e.g., System 2. In System 3, we 
+reuse an in-house NLP pipeline built on the UIMA framework 23 to accommodate these subtasks. 
+The pipeline also provides necessary preprocessing, including tokenization, sentence splitting, 
+part-of-speech tagging, and syntactic parsing. The pipeline components integrate different 
+methods and software libraries. For example, for part-of-speech tagging and syntactic parsing, 
+CoreNLP library 24 is used to derive constituent parse trees and dependency graphs. 
+After preprocessing, an existing pipeline component for phrase detection is applied for 
+the extraction of triggers and argument candidates, where trigger phrases are also assigned with 
+the SDOH type. To this end, a sequence labeling model is trained on trigger and argument phrases 
+annotated in the training corpus using Conditional Random Field (CRF) 25. Next, a custom 
+component developed for the current task is applied, which links each detected trigger with 
+argument candidates within the same sentence. For linking, hand-crafted rules are implemented, 
+which are based on the constituent span, the dependency link, or any selected text pattern. Rules 
+were developed following the corpus annotation guidelines and provided examples 3,6 and tested 
+on annotations collected from a few notes in the training set. 
+The existing NLP pipeline, which can provide the system framework and reusable 
+preprocessing components, allowed us to put together this layered system quickly. During our 
+participation in the challenge, however, we could not allocate sufficient time to write rules for 
+many relations and to test them beyond the few examples used the initial development. As 
+reported in the next section, the precision and recall were rather low for this reason. The 
+performance metric reported on this system, therefore, should be interpreted accordingly. 
+
+ 
+Results 
+The three systems were used in our submission of the Subtask-A in the Track-2, where the 
+training, development and test data set were from MIMIC III 10. Table 2 shows the counts of 
+true positives and predicted positives per target type, obtained on the test data using the 
+evaluation script provided by the challenge organizer. As the table rows show, the evaluation 
+script counts triggers and arguments separately, rather than as trigger-argument pairs or 
+trigger-arguments tuples. Then, it computes the final performance metric from the total counts, 
+which are shown in the first row “OVERALL” in Table 2. The span-only arguments are relatively 
+rare, and the performance metric is mostly based on triggers and labeled arguments. The 
+evaluation results in the two tables show the characteristics of each system as well as that of 
+the evaluation metric. 
+System 1 tends to predict more triggers and arguments than the other two systems. 
+That would be attributed to multiple models in the system that independently predict targets 
+without considering trigger-argument relations. The current scoring metric favors independent 
+prediction because, as stated above, triggers and arguments are counted separately toward the 
+scoring. For example, if a trigger is predicted correctly, it is counted as one true positive 
+independent of its arguments; if an argument is predicted correctly, the argument and the 
+associated trigger are both counted. 
+ 
+System 2 achieved a good performance metric, and it may be improved further with a 
+larger model and/or larger data. It is notable that this system generates a complete table, 
+where many relations must be considered together, e.g., trigger-argument, argument-
+argument, and trigger-trigger. The complete relations among triggers and arguments, though 
+restrictive, must help identify consistent answers. For instance, History is a span-only argument 
+type used for a phrase concerning a patient’s last use of substance, e.g., “7 years ago.” 
+Therefore, it is always related to the Status argument, Status=past, in addition to the trigger, 
+and they should be considered together to report coherent outputs. Among the three systems, 
+this system achieved good or the best results for the three History arguments as in Table 2. 
+
+ 
+The overall performance of System 3 is not as high as the other two systems in Table 1, 
+but the trigger extraction performance is close to the other two in Table 2. In fact, the baseline 
+performance of trigger extraction is high in this task because there is a relatively small number 
+of recurrent and unambiguous trigger phrases, such as “ETOH” (Alcohol), “IVDU” (Drug), 
+“Tobacco” (Tobacco), “works” (Employment), and “lives” (Living Status). If a system can 
+memorize those terms, the trigger extraction looks reasonably good. This suggests that the 
+main interest and challenge in this task is argument detection. A major hurdle for System 3 is 
+that there are so many arguments, and it takes time to manually review relations and develop 
+good rules. When a good rule is created, the precision of the extraction can be high, e.g., Drug 
+Status=none or Employment Status=retired. Yet, many rules are needed to boost the recall. 
+ 
+Table 1. The evaluation results of our three systems and the first rank system on the Subtask A 
+test set. There are 3,471 annotated instances (positives).  
+Subtask 
+ 
+True Positives 
+Predicted Positives 
+Precision 
+Recall 
+F1 
+A 
+System 1 
+3,070 
+3,472 
+0.8842 
+0.8845 
+0.8843 
+System 2  
+2,776 
+3,210 
+0.8648 
+0.7998 
+0.8310 
+System 3 
+2,157 
+3,032 
+0.7114 
+0.6214 
+0.6634 
+Rank 1 system 
+N/A 
+N/A 
+0.9093 
+0.9078 
+0.9008 
+B 
+System 1 
+18,376 
+23,261 
+0.7900 
+0.7477 
+0.7683 
+Rank 1 system 
+N/A 
+N/A 
+0.8109 
+0.7703 
+0.7739 
+ 
+ 
+ 
+
+Table 2. The detailed evaluation results of the three systems on the Subtask A test set. In the 
+gold annotation, triggers and labeled arguments are mandatory per “event” and are highlighted 
+in the table. 
+ 
+ 
+ 
+ 
+True Positives 
+Predicted Positives 
+SDOH type 
+argument 
+subtype 
+Positives 
+Sys. 1 
+Sys. 2 
+Sys. 3 
+Sys. 1 
+Sys. 2 
+Sys. 3 
+OVERALL 
+- 
+- 
+3471 
+3070 
+2776 
+2157 
+3472 
+3210 
+3032 
+Alcohol 
+Trigger 
+N/A 
+308 
+302 
+288 
+273 
+310 
+307 
+290 
+  
+Status 
+current 
+110 
+102 
+87 
+90 
+118 
+108 
+224 
+  
+  
+none 
+151 
+144 
+136 
+53 
+148 
+139 
+64 
+  
+  
+past 
+47 
+37 
+37 
+0 
+44 
+60 
+2 
+  
+Amount 
+N/A 
+47 
+32 
+27 
+15 
+45 
+38 
+35 
+  
+Duration 
+N/A 
+6 
+3 
+3 
+0 
+6 
+5 
+7 
+  
+Frequency 
+N/A 
+51 
+36 
+29 
+22 
+48 
+49 
+31 
+  
+History 
+N/A 
+32 
+14 
+16 
+9 
+28 
+26 
+19 
+  
+Type 
+N/A 
+26 
+21 
+16 
+6 
+29 
+23 
+21 
+Drug 
+Trigger 
+N/A 
+189 
+182 
+165 
+166 
+190 
+179 
+178 
+  
+Status 
+current 
+18 
+12 
+11 
+13 
+19 
+21 
+130 
+  
+  
+none 
+153 
+148 
+135 
+47 
+152 
+142 
+48 
+  
+  
+past 
+18 
+11 
+10 
+0 
+15 
+16 
+0 
+  
+Amount 
+N/A 
+2 
+0 
+0 
+0 
+0 
+4 
+2 
+  
+Duration 
+N/A 
+0 
+0 
+0 
+0 
+1 
+1 
+3 
+  
+Frequency 
+N/A 
+6 
+1 
+2 
+0 
+4 
+4 
+3 
+  
+History 
+N/A 
+10 
+6 
+5 
+1 
+15 
+12 
+7 
+  
+Method 
+N/A 
+35 
+20 
+23 
+5 
+23 
+26 
+6 
+  
+Type 
+N/A 
+112 
+90 
+89 
+17 
+115 
+110 
+26 
+Tobacco 
+Trigger 
+N/A 
+321 
+306 
+283 
+280 
+323 
+302 
+306 
+  
+Status 
+current 
+69 
+61 
+44 
+49 
+77 
+61 
+201 
+  
+  
+none 
+137 
+129 
+123 
+57 
+135 
+131 
+85 
+  
+  
+past 
+115 
+93 
+95 
+10 
+104 
+109 
+20 
+  
+Amount 
+N/A 
+105 
+76 
+65 
+47 
+99 
+93 
+71 
+  
+Duration 
+N/A 
+51 
+41 
+34 
+32 
+48 
+45 
+40 
+  
+Frequency 
+N/A 
+36 
+31 
+25 
+20 
+34 
+34 
+28 
+  
+History 
+N/A 
+87 
+57 
+67 
+42 
+83 
+81 
+55 
+  
+Method 
+N/A 
+1 
+0 
+0 
+0 
+0 
+0 
+2 
+  
+Type 
+N/A 
+20 
+17 
+13 
+4 
+22 
+22 
+14 
+
+Employment 
+Trigger 
+N/A 
+168 
+161 
+113 
+135 
+175 
+122 
+157 
+  
+Status 
+employed 
+64 
+57 
+43 
+47 
+67 
+43 
+108 
+  
+  
+homemaker 
+1 
+0 
+0 
+0 
+0 
+0 
+0 
+  
+  
+on_disability 
+10 
+10 
+2 
+0 
+15 
+5 
+2 
+  
+  
+retired 
+38 
+34 
+32 
+33 
+35 
+32 
+35 
+  
+  
+student 
+4 
+1 
+0 
+3 
+1 
+0 
+3 
+  
+  
+unemployed 
+51 
+47 
+34 
+8 
+54 
+42 
+9 
+  
+Duration 
+N/A 
+4 
+1 
+0 
+0 
+3 
+1 
+5 
+  
+History 
+N/A 
+6 
+3 
+0 
+2 
+9 
+3 
+6 
+  
+Type 
+N/A 
+130 
+91 
+56 
+48 
+129 
+89 
+75 
+LivingStatus 
+Trigger 
+N/A 
+242 
+236 
+227 
+228 
+252 
+242 
+243 
+  
+Status 
+current 
+234 
+227 
+220 
+220 
+242 
+236 
+241 
+  
+  
+past 
+8 
+4 
+5 
+2 
+5 
+5 
+2 
+  
+Type 
+alone 
+60 
+59 
+57 
+44 
+62 
+66 
+46 
+  
+  
+homeless 
+4 
+4 
+3 
+0 
+5 
+3 
+0 
+  
+  
+with_family 
+139 
+136 
+131 
+129 
+143 
+137 
+177 
+  
+  
+with_others 
+39 
+25 
+25 
+0 
+34 
+35 
+0 
+  
+Duration 
+N/A 
+4 
+1 
+0 
+0 
+5 
+0 
+1 
+  
+History 
+N/A 
+2 
+1 
+0 
+0 
+1 
+0 
+2 
+ 
+Discussion 
+SDOH information in the SHAC corpus is regarded as an “event,” and it is annotated as a trigger 
+with associated arguments. This annotation framework is widely used in event extraction tasks 
+22. Meanwhile, it is reported that “[v]arious versions of the event extraction task exist, 
+depending on the goal” 26 and “[t]he definition of an event varies in granularity depending on 
+the desired application of event extraction” 27. SDOH annotations in the SHAC corpus are 
+particularly unique, in that they are reports on patients’ conditions, rather than event 
+occurrences 2. Additionally, the annotations include both direct reports (e.g., “past smoker” or 
+“unemployed”) and indirect reports, from which patients’ conditions are inferred (e.g., “He quit 
+smoking” → Smoking Status:past or “former nurse” → Employment Status:unemployed). All 
+these factors make the current task different from the conventional event extraction task. 
+
+In the conventional event extraction task, usually, a trigger is a verb, or its 
+nominalization denoting an event occurrence, and arguments are terms syntactically related to 
+the trigger. However, triggers in the SHAC corpus are a mixture of clues indicative of SDOH 
+reports, including section headers (e.g., “Tobacco history: …”), verbs or derivative nouns used 
+to state habits or status (e.g., “smokes” or “smoker”), and any keywords suggestive of SDOH 
+reports (e.g., “cigarettes” or “ppd” (packs per day)). Then, there are four to seven different 
+kinds of arguments for each of the five SDOH targets. Given many kinds of triggers and 
+arguments, relations between them are diverse and complex. Compared to the conventional 
+event extraction task, it is particularly difficult to characterize relations between triggers and 
+arguments. 
+ 
+To mitigate the challenge, System 1 avoided modeling relations and considered 
+independent information extraction tasks. The advantage of this approach is the ease of the 
+complexity in the relation extraction. There are many methods and techniques applicable to the 
+simplified tasks. The disadvantage is that this approach does not extract phrases and relations 
+as in the corpus annotation guidelines. The system is inherently limited, and it cannot extract 
+two triggers in one sentence or trigger-argument across sentences. 
+ 
+System 2 does not simplify the task and generate complete structured outputs. The 
+advantage of this approach is the complete outputs as well as the single end-to-end model 
+dealing with all the relations simultaneously. The disadvantage is that the model behavior 
+cannot be easily understood or modified because it is a single end-to-end model. Also, a large 
+computing resource is needed for LLM, while that can help improve the performance further 
+and can be considered an advantage. 
+ 
+System 3 is based on trigger-argument relation extraction, conforming to the corpus 
+annotation guidelines. The advantage of this approach is the transparency of the extraction 
+procedure and the interpretability of outputs owing to the pipeline architecture and human-
+readable rules. The disadvantage is that, given many relations in the task, it is time-consuming 
+to analyze them and write good rules. Also, the management of many rules and many pipeline 
+components can be difficult in practice. 
+
+ 
+As discussed above, the three task formulations have different advantages and 
+disadvantages. Notably, though these systems were evaluated in the same challenge, they are 
+not comparable for building an application. For example, if the goal is to automatically populate 
+a structured database with extracted phrases, System 1, which does not extract trigger phrases 
+and labeled argument phrases, is not applicable. Systems 2 and 3 are applicable to such an 
+application, but the user experience as well as the system maintenance effort is vastly different. 
+If users expect explanation for outputs, a rule-based system like System 2 may be necessary 28. 
+It must be crucial to understand users’ needs and expectation in the application 29. 
+ 
+Conclusion 
+In this paper, we describe three information extraction systems that we developed for our 
+participation in the Task-A of the Track-2 in the 2022 n2c2 NLP Challenge, extraction of SDOH 
+from clinical narratives. While the SDOH information is annotated using the event-based 
+annotation framework in the challenge corpus, the meaning of the “trigger” and “argument” is 
+different from the conventional event extraction task. A commonly used approach to event 
+extraction is difficult to apply, due to the diverse and complex relations annotated in this 
+corpus. To overcome this challenge, two alternative task formulations are explored. These 
+approaches have different advantages and disadvantages. The practical utility of the 
+approaches depends on the requirements and expectation in the application. 
+ 
+This paper focuses on SDOH extraction, but the analysis and discussion are applicable to 
+other information extraction tasks in the clinical NLP domain, where target information is often 
+not an “event,” but patients’ conditions, clinicians’ observations and assessments, or various 
+other properties, e.g., severity of a symptom, laterality of an anatomy, a measurement 
+reported for a lab test or a radiographic study, or their combinations. It is desirable if 
+information extraction framework suitable for such targets are investigated further in the 
+clinical NLP domain. 
+ 
+
+Acknowledgments 
+We thank the organizers and the corpus annotators of the 2022 n2c2 NLP Challenge and the 
+MIMIC project for the data used in the study. 
+ 
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+ 
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+Efficient simulation of multielectron dynamics in
+molecules under intense laser pulses:
+Implementation of the multiconfiguration
+time-dependent Hartree-Fock method based on
+the adaptive finite element method
+Yuki Orimo,∗,† Takeshi Sato,†,‡,¶ and Kenichi L. Ishikawa†,‡,¶
+†Department of Nuclear Engineering and Management, Graduate School of Engineering,
+The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
+‡Research Institute for Photon Science and Laser Technology, The University of Tokyo,
+7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033 Japan
+¶Photon Science Center, Graduate School of Engineering, The University of Tokyo, 7-3-1
+Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
+E-mail: [email protected]
+Abstract
+We present an implementation of the multiconfiguration time-dependent Hartree-
+Fock method based on the adaptive finite element method for molecules under intense
+laser pulses. For efficient simulations, orbital functions are propagated by a stable prop-
+agator using the short iterative Arnoldi scheme and our implementation is parallelized
+for distributed memory computing. This is demonstrated by simulating high-harmonic
+generation from a water molecule and achieves a simulation of multielectron dynamics
+with overwhelmingly less computational time, compared to our previous work.
+1
+arXiv:2301.02387v1  [quant-ph]  6 Jan 2023
+
+Keywords
+Ab initio simulation, multielectron dynamics in molecules, intense laser field, TD-MCSCF
+method
+Introduction
+Multielectron dynamics studied in strong-field physics and attosecond science is a com-
+plicated phenomenon, which includes non-perturbative and nonlinear effects, and multiple
+states or paths excited by ultrashort pulses.1–3 Ab initio simulations have important roles
+to understand and predict these physics. Although solving the time-dependent Schr¨odinger
+equation (TDSE) gives an exact description of the dynamics in the non-relativistic regime,
+it is almost impossible to directly solve TDSE for many-body systems due to the exponen-
+tial growth of the computational cost. The time-dependent multiconfiguration self-consistent
+field methods (TD-MCSCF) have been developed to overcome this problem.4–13 In the meth-
+ods, the total wave function is expressed by the configuration interaction (CI) expansion with
+time-dependent orbital functions, whose flexibility effectively reduces the required number of
+configurations. The multiconifguration time-dependent Hartree-Fock (MCTDHF) method5–7
+is the most general approach for fermionic systems. It considers all the possible configurations
+for a given number of orbital functions. As further developed methods, the time-dependent
+complete-active-space self-consistent field method,10 the time-dependent restricted-active-
+space self-consistent field method9 and the time-dependent occupation-restricted multiple
+active-space method13 have also been proposed. They can significantly reduce the num-
+ber of configurations by classifying orbital functions and making restrictions on electronic
+excitation. Today, we can accurately simulate atoms containing several tens of electrons
+under intense/ultrashort laser pulses thanks to an efficient description of wave functions by
+TD-MCSCF methods.14
+However, it is still difficult to handle molecular systems since simple and efficient dis-
+2
+
+cretization of the three-dimensional space such as the polar coordinate for atomic systems
+is not allowed without relying on the symmetries of the systems. One of the elaborated
+discretizations to simulate molecules without prohibitive computational cost is using mul-
+tiresolution grids. The concept of the method is to discretize only a region near nuclei with
+fine grids and the other regions with grids coarse yet sufficiently fine to describe ionizing
+wave packets. We have previously implemented the MCTDHF method based on a multires-
+olution Cartesian grid and successfully computed high-harmonic generation from a water
+molecule.15
+In this study, we further extend our previous work to implement the MCTDHF method
+with a finite element method on an adaptively generated multiresolution mesh (adaptive
+finite element method). As well as our previous implementation, only the center parts of
+the mesh are refined for sharp changes in wave functions and it gradually becomes coarse in
+the outer region such as Fig. 1. We can also easily control the order of accuracy since finite
+element basis functions are used in each cell. Furthermore, we introduce a highly stable
+propagator based on the short iterative Lanczos/Arnoldi propagator16 to address instability
+arising from high spatial resolution.
+Our simulation code is parallelized for distributed
+memory environments, and consequently, achieved over a hundred times faster simulations.
+This paper is organized as follows. In section II, our problem setting is defined and the
+MCTDHF method is briefly reviewed. In section III, we describe our implementation of
+spatial discretization using the adaptive finite element method, the time evolution of wave
+functions with the short iterative Arnoldi propagator, and parallelization. In section IV, we
+show a numerical result of high-harmonic generation from a water molecule. Conclusions
+are given in section V. Hereafter, we use atomic units unless otherwise indicated.
+3
+
+Figure 1: A part of an adaptive finite element mesh for a hydrogen molecule. The red spheres
+show positions of the nuclei and cell colors are electron density.
+Molecular system and the MCTDHF method
+The Hamiltonian of electrons in a molecule under a laser field can be described as follows.
+H =
+�
+i
+H1(ri) + 1
+2
+�
+i̸=j
+H2(ri, rj)
+(1)
+H1(ri) = −1
+2∆i −
+�
+a
+Za
+|ri − ra| − iA(t) · ∇i
+(2)
+H2(ri, rj) =
+1
+|ri − rj|
+(3)
+where ri and ra are the positions of the ith electron and the ath nucleus and Za is the charge
+of the ath nucleus. A(t) = −
+� t
+∞ E(t′)dt′ denotes the vector potential of a laser field applied
+to the simulated systems, where E(t) is the electric field of it.
+Electronic wave functions are modeled by the multiconfiguration time-dependent Hartree-
+Fock (MCTDHF) method.5–7 Here, we just briefly reviews the method and show the equation
+of motions (EOMs). The detailed descriptions and derivation of EOMs can be found in the
+reference.10
+The MCTDHF method expresses a multielectron wave function |Ψ⟩ with a super position
+of all the possible Slater determinants composed of a given time-dependent spatial orbital
+4
+
+set {φp}.
+|Ψ⟩ =
+�
+I
+CI(t) |I⟩
+(4)
+CI(t) is a configuration interaction (CI) coefficient and |I⟩ is an electronic configuration
+(Slater determinant) composed of orbitals. The equation of motion to variationally evolve
+the MCTDHF wave function can be derived from the time-dependent variational principle.17
+The time-dependent variational principle requires that the action integral S[Ψ],
+S[Ψ] =
+� t1
+t0
+dt ⟨Ψ| ˆH − i ∂
+∂t |Ψ⟩ ,
+(5)
+is stationary to an arbitrary infinitesimal wave function variation δΨ,
+δS
+δΨ = 0.
+(6)
+As a solution of the stationary condition (Eq. (6)), the equations of motion (EOMs) for CI
+coefficients and orbitals are given as follows.
+i ˙CI =
+�
+J
+⟨I| ˆH − i ˆX|J⟩ CJ
+(7)
+i | ˙φp⟩ = ˆQ
+�
+ˆH1 |φp⟩ +
+�
+oqrs
+(D−1)o
+pP qs
+or ˆW r
+s |φq⟩
+�
++ i
+�
+q
+|φq⟩ Xq
+p
+(8)
+ˆX is an arbitrary anti-Hermitian operator, which can be determined as
+ˆX =
+�
+pq
+Xp
+q
+�
+σ
+ˆa†
+qσˆapσ,
+(9)
+where apσ(a†
+pσ) is the annihilation (creation) operator for a spatial orbital φp with σ spin
+(up-spin or down-spin), Xp
+q is an arbitrary anti-Hermitian matrix. In this work, we set Xp
+q
+to be zero. ˆQ is a projection operator onto the orthogonal complement of occupied orbitals,
+5
+
+ˆQ = 1 −
+�
+q
+|φq⟩⟨φq| .
+(10)
+D and P are one-body and two-body reduced density matrices, whose matrix elements are
+defined as
+Dp
+q =
+�
+σ
+⟨Ψ|ˆa†
+qσˆapσ|Ψ⟩
+(11)
+P pq
+sr =
+�
+στ
+⟨Ψ|ˆa†
+sσˆa†
+rτˆaqτˆapσ|Ψ⟩ .
+(12)
+ˆW r
+s is the inter-electronic mean-field potential given by
+W r
+s (r) =
+�
+dr′φ∗
+r(r′)φs(r′)
+|r − r′|
+.
+(13)
+Implementation
+This section shows the implementation of our simulation code developed in this work to
+solve Eqs. (7) and (8) as an initial value problem. Simulations of molecular systems require
+efficient spatial discretization so that we can simulate electronic dynamics keeping accuracy
+with realistic computational cost. We employ the adaptive finite element method18,19 for the
+efficient discretization of orbitals based on an open-source finite element library deal.II.20,21
+As described below, while the adaptive finite element method realizes locally high spatial
+resolution, time evolution could be unstable due to it. To stably propagate wave functions
+for a long period, We employ the short iterative Arnoldi propagator. Although the short
+iterative Lanczos propagator is often used in many applications,16,22–24 since the system
+matrix is not Hermitian, the Arnodi algorithm is used instead of the Lanczos algorithm
+in this application. Applying this scheme to all orbitals at once, we have enabled more
+stable time evolution. These numerical computation schemes are described in the rest of
+this section.
+6
+
+Adaptive finite element method
+The adaptive finite element (AFEM) used in this work is an approach to improve the accuracy
+of simulations requiring locally high resolution by using a multiresolution mesh generated
+by local mesh refinement. A finite element mesh is generated by first discretizing the whole
+simulation box with coarse uniform cubic cells, and then dividing these cells into half the size
+in regions requiring higher resolution. We can generate an adaptive multiresolution mesh by
+repeating the second process. Once the multiresolution mesh and cells are generated, most
+of the rest of the processes fall into the usual finite element method.
+The mesh sizes are determined to make an error in each cell, which is given by Kelly’s
+error indicator 25 to estimate the error in each cell from the jump of the gradient of a target
+function, less than a threshold. This work adopts the Coulomb potential of the nuclei in a
+molecule as the target function for the error estimation. We also limit the minimum and
+maximum mesh sizes to avoid generating extremely small and large cells.
+The basis functions located in each cell are direct products of the one-dimensional La-
+grange polynomials passing through the Gauss-Lobatto quadrature points in each cell. The
+quadrature points in each cell are also constructed as the direct product of one-dimensional
+Gauss-Lobatto quadrature points. This basis can be considered to be the three-dimensional
+version of the finite element discrete variable representation (FEDVR) basis.26,27
+Let us define fI,i(r) as the i th basis function in the I th cell, and LI,jx(x), LI,jy(y) and
+LI,jz(z) the (jx, jy, jz) th Lagrange polynomials in each dimension in the I th cell. Then, the
+function fI,i(r) is given by
+fI,i(r) = LI,jx(x)LI,jy(y)LI,jz(z).
+(14)
+These functions are defined only in the I th cell and have zero values in other region than
+that.
+The finite element basis set {bk(r)} is constructed by the basis functions fI,i(r) which
+7
+
+have zero-value on the boundary of each cell and bridged functions that combine two bases
+with nonzero values at the quadrature point shared by two cells on the boundary of adjacent
+cells. The bridged functions are required to ensure the continuity of discretized functions.
+The mesh generation and construction of the basis are carried out by using deal.II functions.
+An arbitrary function h(r) is discretized with this finite element basis as follows.
+h(r) =
+�
+k
+ckbk(r)
+(15)
+ck =
+�
+l
+( ˜
+M −1)k,l
+�
+drbl(r)h(r)
+(16)
+The matrix ˜
+M is the overlap matrix of the basis set {bk(r)}, called the mass matrix in the
+finite element method, defined as
+˜
+Mk,l =
+�
+drbk(r)bl(r).
+(17)
+All the spatial integrals are approximated with Gauss-Lobatto quadrature as follows.
+�
+drh(x, y, z) ≃
+�
+I
+�
+jx,jy,jz
+wx
+I,jxwy
+I,jywz
+I,jzh(xI,jx, yI,jy, zI,jz),
+(18)
+where wd
+I,jd (d = x, y, z) and (xI,jx, yI,jy, zI,jz) are the quadrature weights and points of the
+I th cell.
+Based on this discretization scheme, the equation of motion (Eq. (8)) is converted into a
+matrix-vector equation,
+i ˜
+M ˙cp = (1 − ˜
+M
+�
+q
+cqc†
+q)
+�
+˜H1cp + ˜
+M
+�
+oqrs
+(D−1)o
+pP qs
+or W r
+s ◦ cq
+�
++ i ˜
+M
+�
+q
+cqXq
+p
+(19)
+8
+
+where cp denotes a coefficient vector of orbital φp(r) given by,
+(cp)k =
+�
+drbk(r)φp(r)
+(20)
+and the matrices ˜H1 is defined as the matrix form of the operator ˆH1,
+( ˜H1)k,l =
+�
+drbk(r)H1(r)bl(r).
+(21)
+W r
+s is a coefficient vector of the mean-field potential W r
+s (r) and the element-wise product
+is denoted by “◦”.
+We compute the mean-field potential by solving the following Poisson’s equation, instead
+of directly calculating Eq. (13),
+∆W r
+s (r) = −4πφ∗
+r(r)φs(r)
+(22)
+with a boundary condition
+W r
+s (r)
+���
+r∈Ω =
+�
+dr′φ∗
+r(r′)φs(r′)
+|r − r′|
+,
+(23)
+where Ω denotes the boundary of a simulation box. We solve this equation by the conjugate
+gradient method with algebraic multigrid preconditioning implemented in an open-source
+parallel linear algebra library Trilinos28 interfaced on deal.II.
+Short iterative Arnoldi propagator
+The short iterative Lanczos/Arnoldi propagator is a time evolution method, which approx-
+imates a Hamiltonian in a Krylov subspace by the Lanczos/Arnoldi algorithm and iterates
+short-time propagation of wave functions in the subspace.16 This approach conserves the
+norm of a wave function when a Hamiltonian is Hermitian and enables unconditionally sta-
+9
+
+ble time evolution. It is also possible to use an adaptive time step or a variable Krylov
+subspace dimension based on the error estimation16 However, we cannot straightforwardly
+apply it to the equation of motion of orbitals, since it is only applicable to linear equations.
+Although some applications of the MCSCF methods, where the EOM of orbitals is non-
+linear, use exponential integrators29–31 to enjoy the stability of the short iterative Lanc-
+zos/Arnoldi propagator even only for linear parts of the EOM, in our application, we found
+that the explicit time propagation of the nonlinear parts causes numerical instability prob-
+ably due to the quite fine mesh of AFEM. To avoid this problem, in this work, we propose
+an approach to apply the short iterative Lanczos/Arnoldi propagator by approximately re-
+garding the whole of the EOMs for all orbitals as one linear system.
+The equations of motion for all orbitals (Eq. (8)) can be packed into a matrix-vector
+form, whose elements are operators and ket-vectors.
+i ∂
+∂tφ = ˆGφ
+(24)
+φ =
+�
+��������
+|φ1⟩
+|φ2⟩
+...
+|φn⟩
+�
+��������
+,
+G =
+�
+��������
+ˆG11
+ˆG12
+· · ·
+ˆG1N
+ˆG21
+ˆG22
+· · ·
+ˆG2N
+...
+...
+...
+ˆGN1
+ˆGN2
+· · ·
+ˆGNN
+�
+��������
+(25)
+The matrix element ˆGij is an operator defined as
+ˆGij = δi
+j ˆH1 +
+�
+osr
+(D−1)o
+iP js
+or ˆW r
+s − ⟨φj|
+�
+ˆH1 |φi⟩ +
+�
+oqrs
+(D−1)o
+iP js
+or ˆW r
+s |φq⟩
+�
++ iXj
+i .
+(26)
+The equation (24) is approximately linear if we can assume that orbitals in the operators
+are invariable within a short time ∆t, and then time evolution of orbitals can be described
+as
+φ(t + ∆t) = exp
+�
+−i ˆG∆t
+�
+φ(t).
+(27)
+10
+
+We achieve this time evolution by the short iterative Arnoldi scheme. Although this scheme
+has first-order accuracy since the time-dependency of the operator ˆG in a time step ∆t is
+not considered, it enables highly stable propagation including nonlinear parts and fits our
+implementation based on the AFEM using a fine mesh. The Krylov subspace dimension of
+the Arnoldi algorithm is determined so that errors estimated by the method found in the
+references16,23 are less than a threshold, which is set to be 10−10 in this work. We also adjust
+a time-step size, which is fixed during our simulations, to make the dimension 10-15 at a
+maximum.
+Parallelization
+The spatial discretization and time evolution discussed above are devised to efficiently sim-
+ulate multielectron dynamics in a laser field.
+Nevertheless, computational costs for the
+three-space to describe laser-induced ionization are huge , and distributed memory parallel
+computing is essential. The total number of degrees of freedom (DOF) NDOF in our simula-
+tion can simply be written as NDOF = Norbital × Nspace + NCI, where Norbital, Nspace and NCI
+are the numbers of orbitals, DOF associated with spatial discretization and CI coefficients,
+respectively. Norbital is typically from several to several tens, and Nspace usually increases up
+to several millions. NCI significantly changes depending on a problem since it exponentially
+increases to the numbers of electrons and orbitals. Our strategy to make efficient use of
+many processors in many situations is parallelizing orbital functions with respect to both
+the number of orbitals and the number of degrees of freedom in the AFEM.
+We divide the orbital function set {|φp⟩} by K and create K MPI groups to deal with
+them. Each MPI group has L independent processes that are used to distribute a simulation
+box by using deal.II functions. Distribution of a simulation box and DOFs accompanying it is
+carried out by p4est,20,32 an open-source library to distribute octree structures across multiple
+processors, interfaced to deal.II. This addresses load balancing and optimal distribution of
+the simulation box to reduce MPI communications among the processors (Fig. 2).
+11
+
+Figure 2: An example of a divided simulation box. The color-coded cells correspond the
+distribution to MPI processes.
+Applications
+We simulate high harmonic generation from a water molecule to demonstrate the efficiency
+of our implementation by comparing our previous work.15 For atomic positions of a water
+molecule, two hydrogen atoms of a water molecule are located at (±1.42994, 1.10718, 0) and
+an oxygen atom is located at the origin.
+The laser pulse used in this simulation has a
+wavelength of 2πc/ω = 400nm (c is the speed of light in vacuum) and a peak intensity of
+I0 = 8 × 1014 W/cm2, and is linearly polarized along with x-axis. The pulse duration is 2
+optical cycles with a triangular envelope. The shape of the electric field of the laser pulse is
+defined as, (see also Fig. 3),
+E(t) = E0fenv(t) sin(ωt)
+(28)
+fenv(t) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ωt
+2π
+(0 ≤ ωt ≤ 2π)
+4π − ωt
+2π
+(2π ≤ ωt ≤ 4π)
+,
+(29)
+where E0 is the peak electric field derived from the peak intensity. The time-step size for
+real-time evolution is 0.01 a.u..
+12
+
+Figure 3: The electric field of the laser pulse used in this simulation.
+The simulation box is a cuboid defined within a region [−70, 70]×[−30, 30]×[−30, 30]. We
+apply the exterior complex scaling (ECS) as an absorbing boundary in the outside of a region
+[−35, 35] × [−10, 10] × [−10, 10]. The details of the ECS can be found in the references.33–35
+The finite element mesh is generated to satisfy that the error in each cell is less than
+0.005, which has 6 different sizes between 0.125 a.u. and 4.0 a.u.. At the most distant region
+from the molecule, the largest elements, which are cubes with 4.0 a.u long sides, are used
+to describe sufficiently absorbed orbital functions and the smallest elements, whose edge
+length is 0.125 a.u., are used in the vicinity of the molecule. Figure 4 displays the finite
+element mesh used in this simulation. The finite element basis is constructed from first-
+order Lagrange polynomials, and thus there are 8 quadrature points in a finite element cell.
+While it is possible to dynamically adapt a mesh to time-dependent orbital functions, we
+avoid such approaches due to additional computational costs. This would be helpful to gain
+computational efficiency if our problem was a larger system.
+For the beginning of the simulation, we computed a ground state by imaginary-time
+evolution, whose electronic energy was -76.905 a.u. In figure. 5, we compare our simulation
+result with the previously calculated one. These spectra do not perfectly agree with each
+other since it is extremely difficult to achieve perfect convergence for spatial resolutions in
+3D systems, Nevertheless, overall spectral shapes are quite similar. As well as the previous
+calculations, the simulations with 5 orbitals and 6 orbitals give almost the same spectra.
+13
+
+0.1
+0.0
+-0.1
+0
+25
+50
+75
+100Figure 4: Adaptively generated finite element mesh for a water molecule. The largest element
+is a cube of edge length 4.0 a.u. used to discretize the outer region, and the smallest one is
+a cube of edge length 0.125 a.u. used only in the vicinity of nuclei.
+The simulation using 6 orbitals of present work took 6.5 hours with 240 cores (6 nodes,
+2 Intel Xeon Gold 2.40GHz processors with 20 cores in a node). Remarkably, it is about
+100 times faster than the previous work which took 28 days to finish the simulation. One
+of our achievements is successful distributed parallel computing using the 20 times larger
+resource. In addition to this, at least 5 times acceleration was gained by factors except for
+parallelization. The development of a highly stable propagator mainly contributes to this
+speed-up, which enables time evolution with 4 times as large a time-step size as the previous
+one.
+Conclusion
+We have implemented the MCTDHF method based on the adaptive finite element method
+to simulate multielectron dynamics in molecules under laser fields. A further sophisticated
+discretization is realized by using the multiresolution grid used in our previous implementa-
+tion in the frame of the finite element method. Thanks to the finite element method, we can
+automatically generate an adaptive mesh using Kelly’s error indicator and easily control the
+order of accuracy by changing the polynomial order of basis functions. While locally refined
+meshes enable efficient and accurate simulations, they possibly make time evolution unstable.
+14
+
+140 a.u.
+60 a.u.
+60 a.u.Figure 5: High harmonic spectra of a water molecule exposed to a laser pulse with a wave-
+length of 400nm and a peak intensity of 8 × 1014 W/cm2. (a) The spectrum taken from
+Ref.15 The data is normalized for the maximum to be unity. (b) The spectra computed by
+the present work.
+We developed a more stable propagator based on the short iterative Arnoldi scheme than
+exponential integrators. This propagator evolves all orbital functions together as a vector
+by using the short iterative Arnoldi scheme. In addition, our simulation code is parallelized
+for distributed memory computing, which handles both the orbital set and spatial degrees
+of freedom in parallel.
+We have applied the present implementation to a simulation of high-harmonic generation
+from a water molecule in an intense visible laser pulse to compare with our previous work,15
+and obtained the spectra showing a good agreement with overwhelmingly less computational
+time. Parallelization has made the greatest contribution to this reduction in computation
+time, and in this study, we were able to successfully use 20 times larger computational
+resources than in the past. It is also important to note that we were able to use a 4 times
+larger time-step size thanks to the stable propagator.
+This study prepared the adaptive mesh based on the discretization error of the Coulomb
+15
+
+(a)
+()
+6 orbitals
+6 orbitals
+10-1
+10-1
+5 orbitals
+Intensity (a.u.)
+Intensity (a.u.)
+10-3
+10-3
+10-5
+10-5
+10-7
+10-7
+10-9
+10-9
+10
+20
+30
+0
+10
+20
+0
+30
+Harmonic order
+Harmonic orderpotential of the nuclei, therefore the mesh is fixed during simulations, but it is possible
+to dynamically adapt the mesh to a wave function or nuclear positions at each time step.
+We consider that it brings efficiency when a larger simulation box is needed or when the
+nuclei can move. In future works, we will present ab initio simulations of more complicated
+molecular systems and simulations considering nuclear dynamics in a combination of this
+development and more advanced theories such as the TD-ORMAS method13 and the time-
+dependent coupled cluster theory.36
+Data availability
+The data and source code used in this study are available upon reasonable request.
+Competing interests
+The authors declare there are no competing interests.
+Funding information
+This research was supported in part by a Grant-in-Aid for Scientific Research (Grants No.
+JP19H00869, No. JP21K18903, and No. JP22H05025) and a Grant-in-Aid for Early-Career
+Scientists (Grant No. JP22K14616) from the Ministry of Education, Culture, Sports, Sci-
+ence and Technology (MEXT) of Japan.
+This research was also partially supported by
+JST CREST (Grant No. JPMJCR15N1) and by MEXT Quantum Leap Flagship Program
+(MEXT Q-LEAP) Grant Number JPMXS0118067246.
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+19
+

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+arXiv:2301.08621v1  [quant-ph]  20 Jan 2023
+Improved Real-time Post-Processing for Qantum Random Number Generators
+Qian Li,1 Xiaoming Sun,1 Xingjian Zhang,2 and Hongyi Zhou1, ∗
+1State Key Lab of Processors, Institute of Computing Technology,
+Chinese Academy of Sciences, 100190, Beijing, China.
+2Center for Qantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
+Randomness extraction is a key problem in cryptography and theoretical computer science. With the recent
+rapid development of quantum cryptography, quantum-proof randomness extraction has also been widely
+studied, addressing the security issues in the presence of a quantum adversary. In contrast with conventional
+quantum-proof randomness extractors characterizing the input raw data as min-entropy sources, we find
+that the input raw data generated by a large class of trusted-device quantum random number generators
+can be characterized as the so-called reverse block source. Tis fact enables us to design improved extractors.
+Specifically, we propose two novel quantum-proof randomness extractors for reverse block sources that realize
+real-time block-wise extraction. In comparison with the general min-entropy randomness extractors, our
+designs achieve a significantly higher extraction speed and a longer output data length with the same seed
+length. In addition, they enjoy the property of online algorithms, which process the raw data on the fly without
+waiting for the entire input raw data to be available. Tese features make our designs an adequate choice for
+the real-time post-processing of practical quantum random number generators. Applying our extractors to the
+raw data of the fastest known quantum random number generator, we achieve a simulated extraction speed
+as high as 374 Gbps.
+∗ [email protected]
+
+I.
+INTRODUCTION
+Randomness extraction aims at distilling uniform randomness from a weak random source [1], which is widely
+applied ranging from cryptography to distributed algorithms. Recently, quantum cryptography [2] has been devel-
+oped rapidly, whose security is guaranteed by the fundamental principle of quantum mechanics [3, 4]. Compared
+with its classical counterpart, the unique characteristic of intrinsic randomness enables quantum cryptography with
+the feasibility of secure communication regardless of the computation power of the eavesdroppers. Randomness
+extraction also serves as a key step, privacy amplification [5, 6], in the post-processing of quantum cryptography,
+eliminating the side information possessed by a quantum adversary.
+Randomness extraction is realized by various extractors that are mainly composed of a family of hashing func-
+tions. In quantum cryptography, only quantum-proof extractors can provide information-theoretic security in the
+presence of a quantum adversary [7]. Among all the quantum-proof extractors, Trevisan’s extractor [8, 9] and the
+Toeplitz-hashing extractor [10] are two of the most popular choices. Both of them are strong extractors [11], which
+means that the extracted randomness is independent of the seed. Trevisan’s extractor requires a seed length of only
+polylogarithmic scaling of the input length, which is much lower than the linear scaling for Toeplitz-hashing ex-
+tractor. On the other hand, the output speed is much lower than that of the Toeplitz-hashing extractor [12]. Te
+Toeplitz-hashing extractor is constructed from a cyclic matrix, which is easy to implement and can be accelerated
+by the Fast Fourier Transformation (FFT) [13].
+In the practical implementation of an extractor, there are some subtle practical issues. One is the security of the
+block-wise post-processing adopted by most implementations of quantum random number generators (QRNGs). Te
+seed used in the extractor is a limited resource. To save the seed and accelerate the extraction speed, the raw data
+is divided into multiple small blocks. Ten a small length of seed is repeatedly used for each block. Tis method
+assumes the small blocks of the raw data to be mutually independent. However, this is not satisfied by general input
+raw data and results in security issues. Another problem is that the current extraction speed is much lower than
+the raw data generation speed in most QRNG implementations [14]. Te real-time generation speed of the extracted
+quantum random numbers is restricted by the extraction speed in the post-processing, which becomes a botleneck
+in the applications of quantum random numbers in quantum communication tasks. Finally, for some commercial
+QRNGs where the seed cannot be updated, the output data length is limited by the seed length. We want to extract
+as many random numbers as possible with a limited seed length.
+To deal with the practical issues above, we consider properties of some specific randomness sources beyond the
+conventional min-entropy source characterization. In this work, we design two novel quantum-proof randomness
+extraction algorithms for a large class of QRNGs where the raw data can be described as the reverse block source
+[15]. Our results are inspired by the extractors designed in [16, 17] for block sources and Santha-Vazirani sources,
+respectively. Both of our two extractors are online algorithms: as the input raw data arrives in real time, the two
+extractors proceed on-the-fly input raw data piece-by-piece, where the processing is independent of the data in the
+future. In fact, our two extractors are block-wise extractors. Tat is, they partition the input raw data into blocks
+serially in the time order the input arrives, and then apply a min-entropy extractor to each block. Moreover, suppose
+the input raw data is of length 푁, then our first extractor requires a seed length of 푂(log 푁) and takes equipartition
+of the block lengths, and our second extractor requires only a seed length of 푂(1) and incremental block lengths.
+Compared to the first extractor, the second one scarifies the extraction speed while enjoys a seed length independent
+of the input length. As a result, this extractor can deal with infinite raw data, without the need of updating the seed
+and determining the raw data length prior to extraction. For both extractors, the output length is a constant fraction
+of the min-entropy of the raw data, which indicate that our extractors are quite efficient. To show the performances
+of the extractors, we make a simulation estimating the output speed. It turns out that the extraction speed is adequate
+for the post-processing of the fastest known implementations of QRNGs [18–20].
+II.
+PRELIMINARIES
+Troughout the paper, we use capital and lowercase leters to represent random variables and their assignments,
+respectively. We use 푈푚 to represent the perfectly uniform random variable on 푚-bit strings and 휌푈푚 to represent
+the 푚-dimensional maximally mixed state.
+Definition II.1 (Conditional min-entropy). Let 푌 be a classical random variable that takes value 푦 with probability
+2
+
+푝푦 and E be a quantum system. Te state of the composite system can be writen as 휌푌 E = �
+푦 푝푦|푦⟩⟨푦| ⊗ 휌푦
+E, where
+{|푦⟩}푦 forms an orthonormal basis. Te conditional min-entropy of 푌 given E is 퐻min(푌 |E)휌푌 E = − log2 푝guess(푌 |E),
+where 푝guess(푌 |E) is the maximum average probability of guessing 푌 given the quantum system E. Tat is,
+푝guess(푌 |E) = max
+{퐸푦
+E }푦
+��
+푦
+푝푦Tr
+�
+퐸푦
+E휌푦
+E
+��
+,
+(1)
+where the maximization is taken over all positive operator-valued measures (POVMs) {퐸푦
+E}푦 on E.
+When system 푌 is decoupled from E, where 휌푌 E = �
+푦 푝푦|푦⟩⟨푦| ⊗ 휌E, the conditional min-entropy of 푌 given E
+reduces to the classical min-entropy, 퐻min(푌) = − log2 max푦 푝푦. When 휌푌 E is clear from the context, we will denote
+the conditional min-entropy as 퐻min(푌 |E) for brevity.
+In this paper, we call the raw data generated by a QRNG as a random source. A general random source is the
+min-entropy source, where the conditional min-entropy is lower bounded.
+Definition II.2 (Min-entropy quantum-proof extractor). A function Ext : {0, 1}푛 × {0, 1}푑 → {0, 1}푚 is a (훿푛,휖)
+min-entropy quantum-proof extractor, if for every random source 푌 and quantum system E satisfying 퐻min(푌 |E) ≥ 훿푛,
+we have
+1
+2 ∥휌Ext(푌,푆) E − 휌푈푚 ⊗ 휌E∥ ≤ 휖,
+(2)
+where 푆 is called the seed, which is a perfectly uniform random variable on 푑-bit strings independent of the system 푌 E
+and ∥ · ∥ denotes the trace norm defined by ∥퐴∥ = Tr
+√
+퐴†퐴. An extractor Ext is said to be strong if
+1
+2 ∥휌Ext(푌,푆)푆 E − 휌푈푚 ⊗ 휌푈푑 ⊗ 휌E∥ ≤ 휖.
+(3)
+We call the concatenation of the output string of a strong extractor with the seed as an expansion, denoted as the tuple
+(Ext(푦,푠),푠).
+It is straightforward to see that an expansion is a standard (훿푛,휖) min-entropy quantum-proof extractor. If the
+output of an extractor satisfies Eq. (2) or (3), we say that the output is 휖-close to a uniform distribution.
+A widely used randomness extractor is the Toeplitz-hashing extractor.
+Definition II.3 (Toeplitz-hashing extractor). A 푢 × 푛 matrix 푇 is a Toeplitz matrix if 푇 푖 푗 = 푇 푖+1,푗+1 = 푠푗−푖 for all
+푖 = 1, · · · ,푢 − 1 and 푗 = 1, · · · ,푛 − 1. A Toeplitz matrix over the finite field 퐺퐹 (2), 푇푠, can be specified by a bit string
+푠 = (푠1−푢,푠2−푢, · · · ,푠푛−1) of length 푢 + 푛 − 1.Given any 푛,푑 ∈ N+ where 푑 ≥ 푛, define the Toeplitz-hashing extractor
+Ext푛,푑
+푇
+: {0, 1}푛 × {0, 1}푑 → {0, 1}푑−푛+1 as Ext푛,푑
+푇 (푦,푠) = 푇푠 · 푦, and define the expanded Toeplitz-hashing extractor
+Ext푛,푑
+푇 ′ : {0, 1}푛 × {0, 1}푑 → {0, 1}2푑−푛+1 as Ext푛,푑
+푇 ′ (푦,푠) = (푇푠 ·푦,푠), where the matrix product operation · is calculated
+over the field 퐺퐹 (2).
+Since {푇푠 · 푦|푠 ∈ {0, 1}푢+푛−1} is a family of pairwise independent hashing functions [21, 22], according to the
+quantum Lefover Hash Lemma [23], we can prove that the Toeplitz-hashing extractor is a min-entropy quantum-
+proof strong extractor.
+Lemma II.4 ([23]). For every 푛 ∈ N+ and 훿 > 0, Ext푛,푑
+푇
+is a (훿푛, 휖) min-entropy quantum-proof strong extractor, where
+휖 = 2−(훿푛+푛−푑−1)/2. Equivalently, Ext푛,푑
+푇 ′ is a (훿푛,휖) min-entropy quantum-proof extractor.
+Note that the output of Ext푛,푑
+푇 ′ has 2푑 − 푛 + 1 bits, where the last 푑 bits form the seed. We remark that though the
+Toeplitz-hashing extractor Ext푛
+푇 is strong, the expanded Toeplitz-hashing extractor Ext푛
+푇 ′ is not.
+III.
+MAIN RESULT
+We use 푋 = 푋1푋2 · · ·푋푁 ∈ ({0, 1}푏)푁 to denote the raw data generated by a QRNG, which contains 푁 samples
+each of 푏 bits. For a set 퐼 ⊂ N+, we write 푋퐼 for the restriction of 푋 to the samples determined by 퐼. For example, if
+퐼 = {2, 3, 5}, then 푋퐼 = 푋2푋3푋5. We use E to denote the quantum system possessed by the quantum adversary.
+3
+
+A.
+Reverse block source
+Intuitively, for a given randomness extractor, there is a trade-off between its performance and the generality of the
+random sources it applies to. Te more special random sources the extractor works for, the beter performance the
+extractor may achieve. In this paper, the notion of reverse block sources, which are more special than the min-entropy
+sources, plays a critical role in the sense that (i) raw data of a large class of QRNGs can be described as a reverse
+block source and (ii) prety good quantum-proof extractors for reverse block sources exist. A quantum version of
+the reverse block source is defined below.
+Definition III.1 (Reverse block source, adapted from Definition 1 in [15]). A string of random variables 푋 =
+푋1 · · ·푋푁 ∈ ({0, 1}푏)푁 is a (푏, 푁,훿)-reverse block source given a quantum system E if for every 1 ≤ 푖 ≤ 푁 and every
+푥푖+1,푥푖+2, · · · ,푥푁 ,
+퐻min(푋푖|푋푖+1 = 푥푖+1, · · · ,푋푁 = 푥푁, E) ≥ 훿 · 푏.
+(4)
+As shown later in Sec. III B, we introduce the reverse block source for the convenience of designing an online
+algorithm. A reverse block source can be intuitively understood as a time-reversed block source 퐻min(푋푖|푋푖−1 =
+푥푖−1, · · · ,푋1 = 푥1, E) where a new sample can not be completely predicted by the samples that already exists, i.e.,
+the net randomness increment is non-zero. For a reverse block source, these specific samples are from the future. Ac-
+tually, for QRNGs where the raw data are mutually independent, such as the ones based on single photon detection
+[24–26], vacuum fluctuations [27–29], and photon arrival time [30–32], the min-entropy source automatically satis-
+fies Eq. (4), where ∀푥푖+1, · · · ,푥푁 , 퐻min(푋푖|E) = 퐻min(푋푖|푋푖+1 = 푥푖+1, · · · ,푋푁 = 푥푁, E), hence is also a reverse block
+source. For QRNGs with correlated raw data, one can construct appropriate physical models to check whether Eq. (4)
+is satisfied. In Appendix A, we take the fastest implementation, the one based on phase fluctuation of spontaneous
+emission [18, 19], as an example to prove Eq. (4).
+We also consider a smoothed version of the reverse block source. For 푋 = 푋1 · · ·푋푁 ∈ ({0, 1}푏)푁 , Denote the
+underlying joint quantum state of the random source over the systems 푋 and E as 휌푋 E. We call 푋 a (푏, 푁,훿, 휖s)-
+smoothed reverse block source, if there exists a state 휌∗
+푋 E that is 휀푠-close to 휌푋 E,
+1
+2 ∥휌∗
+푋 E − 휌푋 E∥ ≤ 휀푠,
+(5)
+such that 휌∗
+푋 E is a reverse block source. Te smoothed reverse block source is in the same spirit of the smooth
+conditional entropy [33]. Te motivation of introducing a smoothed version is to exclude singular points or region
+in a probability distribution, which will help extract more randomness.
+Here we remark that we mainly consider the trusted-device QRNGs where the extraction speed is a botleneck. For
+QRNGs with a higher security level, such as the semi-device-independent QRNGs and device-independent QRNGs,
+the reverse block source property is not satisfied in general. Tese types of QRNGs requires fewer assumptions and
+characterizations on the devices at the expense of relatively low randomness generation rates of raw data. Ten their
+real-time randomness generation rates are not limited by the extraction speed.
+B.
+Extractors for reverse block sources
+In this section, we design two online quantum-proof extractors that can both extract a constant fraction of the
+min-entropy from reverse block sources. Te two extractors both proceed in the following fashion: partition the
+input raw data into blocks, and apply a min-entropy quantum-proof extractor to each block using part of the output
+of the previous block as the seed. Te basic building block of our designs is family of (훿푛,휖푛) min-entropy quantum-
+proof extractors, denoted by Ext푔 = {Ext푛
+푔 : 푛 ∈ N+}. Here, the subscript of Ext푔 means “gadget”. Let 푑푛 and 푚푛
+denote the seed length and output length of Ext푛
+푔, respectively, i.e., Ext푛
+푔 : {0, 1}푛 × {0, 1}푑푛 → {0, 1}푚푛. We require
+the gadget Ext푛
+푔 to satisfy: (i) 휖푛 is exponentially small, for instance, 휖푛 = 2−훿푛/4, which aims at that the summation
+�∞
+푛=1 휖푛 converges; and (ii) 푚푛 − 푑푛 = Ω(훿푛), which means that Ext푛
+푔 extracts a constant fraction of min-entropy
+from the raw data. As an explicit construction, Ext푛
+푔 can be specified as the expanded Toeplitz-hashing extractor
+Ext푛,푑푛
+푇 ′
+where 푑푛 = (1 +훿/2)푛 − 1, then 푚푛 −푑푛 = 훿푛/2, 푚푛 = (1 +훿)푛 − 1, and 휖푛 = 2−훿푛/4. In the rest of this paper,
+4
+
+we abbreviate Ext푛,푑푛
+푇
+and Ext푛,푑푛
+푇 ′
+where 푑푛 = (1 + 훿/2)푛 − 1 to Ext푛
+푇 and Ext푛
+푇 ′, respectively. Besides, to simplify
+our discussion, we assume 휖푛 = 2−훿푛/4, and the analysis for other exponentially small values of 휖푛 is similar.
+Te two extractors are described in Algorithms 1 and 2, respectively. Given an input raw data of length 푁, the
+first extractor, named Ext푒푞
+푟푏푠, evenly partitions the input raw data into blocks each of size 푂(log 푁) and requires
+푂(log 푁) random bits as the initial seed. In particular, if the min-entropy quantum-proof extractor Ext푔 in use is the
+expansion of a strong extractor such as the expanded Toeplitz-hashing extractor, then Ext푒푞
+푟푏푠 degenerates exactly
+to the following naive extractor: partition the input raw data into equal-sized blocks and apply the corresponding
+strong extractor to each block separately with the same seed. Te second extractor, named Ext푛푒푞
+푟푏푠 , is inspired by
+the extractor that can extract randomness from Santha-Vazirani sources using a seed of constant length [17]. It uses
+only 푂(1) random bits as the initial seed and requires incremental block lengths. Compared to the first extractor,
+the second one is less hardware-friendly and sacrifices the extraction speed in general. On the other hand, it enjoys
+the property that the seed length is independent of the input length. As a result, this extractor can deal with infinite
+raw data, without the need of determining the raw data length prior to extraction. In other words, one does not need
+to update the seed in practical implementations. We remark that the initial seed is indispensable because there does
+not exist any nontrivial deterministic extractor for reverse block sources. Te proof is presented in Appendix B.
+Algorithm 1: Ext푒푞
+푟푏푠
+1 Input: 푏 ∈ N+ and 0 < 휖, 훿 < 1. A string 푥 = 푥1, 푥2, · · · , 푥푁 sampled from a (푏, 푁,훿)-reverse block source;
+2 Let 푖 := 1 and 푛 :=
+�
+4
+훿푏 · log
+�
+푁
+휖
+��
+;
+3 Sample a uniform random bit string 푠 (1) of length 푑푏푛;
+4 for ℓ = 1 to 푁 /푛 do
+5
+Let 퐼ℓ := [푖,푖 + 푛 − 1];
+6
+Compute 푧 (ℓ) := Ext푏푛
+푔 (푥퐼ℓ ,푠 (ℓ));
+7
+Let 푖 := 푖 + 푛;
+8
+Cut 푧 (ℓ) into two substrings, denoted by 푟 (ℓ) and 푠 (ℓ+1), of size 푚푏푛 − 푑푏푛 and 푑푏푛 respectively;
+9
+Output 푟 (ℓ).
+Algorithm 2: Ext푛푒푞
+푟푏푠
+1 Input: 푏 ∈ N+ and 0 < 훿 < 1. A string 푥 = 푥1,푥2, · · · sampled from a (푏, ∞,훿)-reverse block source;
+2 Parameter: 푛1, Δ ∈ N+;
+3 Let 푖 := 1;
+4 Sample a uniform random bit string 푠 (1) of length 푑푏푛1;
+5 for ℓ = 1 to ∞ do
+6
+Let 퐼ℓ := [푖,푖 + 푛ℓ − 1];
+7
+Compute 푧 (ℓ) := Ext푏푛ℓ
+푔
+(푥퐼ℓ ,푠 (ℓ));
+8
+Let 푖 := 푖 + 푛ℓ and 푛ℓ+1 := 푛ℓ + Δ;
+9
+Cut 푧 (ℓ) into two substrings, denoted by 푟 (ℓ) and 푠 (ℓ+1), of size 푚푏푛ℓ − 푑푏푛ℓ+1 and 푑푏푛ℓ+1 respectivelya;
+10
+Output 푟 (ℓ).
+a We should choose the parameters 푛1 and Δ properly to have 푚푏푛ℓ − 푑푏푛ℓ+1 ≥ 1.
+Note that if we impose Δ = 0 and let 푛1 =
+� 4
+훿푏 · log � 푁
+휖
+��
+, then Ext푛푒푞
+푟푏푠 becomes Ext푒푞
+푟푏푠. We first analyze the second
+extractor Ext푛푒푞
+푟푏푠 .
+Teorem III.2. Te extractor Ext푛푒푞
+푟푏푠 satisfies the following properties:
+(a) It uses a seed of 푑푏푛1 length.
+5
+
+FIG. 1: Illustration of Ext푛푒푞
+푟푏푠 .
+(b) For any 푘 ∈ N, we have
+1
+2 ∥휌푟 (1)◦푟 (2)◦···◦푟 (푘) E − 휌푈휂푘 ⊗ 휌E∥ ≤
+푘
+�
+ℓ=1
+2−훿푏푛ℓ/4 <
+2−훿푏푛1/4
+1 − 2−훿푏Δ/4,
+(6)
+where 휂푘 = �푘
+ℓ=1(푚푏푛ℓ − 푑푏푛ℓ+1).
+Proof. Te nontrivial part is Part (b). For convenience of presentation, we use 퐼푘:∞ to represent 퐼푘 ∪ 퐼푘+1 ∪ · · · ∪ 퐼∞.
+In fact, we will prove that for any 푘 ∈ N it has
+1
+2
+���휌푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1) ◦푋퐼푘+1:∞ E − ��푈휂푘 +푑푏푛푘+1 ⊗ 휌푋퐼푘+1:∞ E
+��� ≤
+푘
+�
+ℓ=1
+2−훿푏푛ℓ/4,
+(7)
+which implies Part (b) immediately.
+Te proof is by an induction on 푘. Te base case when 푘 = 0 is trivial. Te induction proceeds as follows. Suppose
+Eq. (7) is true. Ten due to the contractivity of trace-preserving quantum operations, we have
+1
+2
+����휌푟 (1)◦푟 (2)◦···◦푟 (푘)◦Ext푔(푋퐼푘+1,푠 (푘+1))◦푋퐼푘+2:∞ E − 휌푈휂푘 ⊗ 휌Ext푔
+�
+푋퐼푘+1,푈푑푏푛푘+1
+�
+◦푋퐼푘+2:∞ E
+���� ≤
+푘
+�
+ℓ=1
+2−훿푏푛ℓ/4.
+(8)
+On the other hand, by Definition III.1, for any assignment 푥퐼푘+2:∞ of 푋퐼푘+2:∞, we have
+퐻min(푋퐼푘+1 | 푋퐼푘+2:∞ = 푥퐼푘+2:∞, E) ≥ 훿푏푛푘+1.
+(9)
+Ten, recalling that Ext푔 is a min-entropy quantum-proof extractor, it follows that
+1
+2
+����휌Ext푔
+�
+푋퐼푘+1,푈푑푏푛푘+1
+�
+◦푋퐼푘+2:∞ E − 휌푈푚푏푛푘+1 ⊗ 휌푋퐼푘+2:∞ E
+���� ≤ 2−훿푏푛푘+1/4.
+(10)
+Tus,
+1
+2
+����휌푈휂푘 ⊗ 휌Ext푔
+�
+푋퐼푘+1,푈푑푏푛푘+1
+�
+◦푋퐼푘+2:∞ E − 휌푈휂푘 ⊗ 휌푈푚푏푛푘+1 ⊗ 휌푋퐼푘+2:∞ E
+���� ≤ 2−훿푏푛푘+1/4.
+(11)
+Finally, combining inequalities (8) and (11) and applying the triangle inequality, we conclude that
+1
+2
+���휌푟 (1)◦푟 (2)◦···◦푟 (푘+1)◦푠 (푘+2)◦푋퐼푘+2:∞ E − 휌푈휂푘+1+푑푏푛푘+2 ⊗ 휌푋퐼푘+2:∞ E
+��� ≤
+푘+1
+�
+ℓ=1
+2−훿푏푛ℓ/4.
+(12)
+By using the summation formula for the geometric progression, the above inequality is further upper bounded by
+2−훿푏푛1/4
+1−2−훿푏Δ/4 .
+□
+6
+
+n1
+n1 + ( - 1)
+n1 + △
+x
+X1
+X
+Xe
+5(1
+Ext
+Ext
+Ext,
+6
+9
+9
+r(1)
+r(2)
+r(t)As can be seen from the proof, the parameter Δ of Ext푛푒푞
+푟푏푠 must be strictly positive, since the upper bound
+�푘
+ℓ=1 2−훿푏푛ℓ/4 on the error converges if and only if Δ > 0. Teorem III.2 implies that the (infinitely long) output
+string 푟 (1) ◦ 푟 (2) ◦ · · · can be arbitrarily close to the uniform distribution by choosing a sufficiently large constant
+푛1. As an explicit construction, suppose Ext푛푒푞
+푟푏푠 adopts the expanded Toeplitz-hashing extractor Ext푛
+푇 ′ as the gadget
+Ext푛
+푔, where 푑푛 = (1 +훿/2)푛 − 1 and 푚푛 = (1 +훿)푛 − 1. We further set Δ = 1. We require that 푛1 ≥ 4/훿 + 1 such that
+푚푏푛ℓ −푑푏푛ℓ+1 = 푏(훿푛ℓ/2 − 1 −훿/2) ≥ 1 for any ℓ. Ten Ext푛푒푞
+푟푏푠 extracts (1 +훿)푏(푛1 + ℓ − 1) − 1 random bits from the
+ℓ-th block 푥퐼ℓ , outputs the first 푚푏푛ℓ − 푑푏푛ℓ+1 ≈ 훿푏푛ℓ/2 bits, and then uses the last (1 + 훿/2)푏(푛1 + ℓ) − 1 bits as the
+seed of the next block.
+Via a similar argument as in Teorem III.2, we have the following result for Ext푒푞
+푟푏푠.
+Teorem III.3. Te extractor Ext푒푞
+푟푏푠 uses 푑푏푛 random bits as a seed and outputs a string 푟 (1) ◦ 푟 (2) ◦ · · · ◦ 푟 (푁/푛)
+satisfying that
+1
+2 ∥휌푟 (1)◦푟 (2)◦···◦푟 (푁 /푛)◦E − 휌푈휂 ⊗ 휌E∥ ≤ 푁
+푛 · 2−훿푏푛/4 ≤ 휖,
+(13)
+where 휂 := 푁
+푛 · (푚푏푛 − 푑푏푛).
+In particular, suppose Ext푒푞
+푟푏푠 adopts Ext푛
+푇 ′ as the gadget Ext푛
+푔. Ten Ext푒푞
+푟푏푠 extracts (1+훿)푏푛−1 random bits from
+the ℓ-th block 푥퐼ℓ and outputs the first 푚푏푛 −푑푏푛 = 훿푏푛/2 bits. Te last 푑푏푛 = (1+훿/2)푏푛 −1 bits, which is exactly the
+seed used in this block, will be reused as the seed in the next block. Terefore, Ext푒푞
+푟푏푠 uses 푑푏푛 ≈ (4/훿 + 2) log (푁/휖)
+random bits as seed, and outputs (푁/푛) · (푚푏푛 − 푑푏푛) ≈ 훿푏푁/2 bits in total. Tough Ext푒푞
+푟푏푠 uses more seed than
+Ext푛푒푞
+푟푏푠 , it is more hardware-friendly and can achieve much higher extraction speed.
+Corollary III.4. Suppose the random sources in Algorithms 1 and 2 are replaced with the (푏, 푁,훿,휖s) and (푏, ∞,훿,휖s)-
+smoothed reverse block sources, respectively. By using the same processing procedures, the output state of the extractor
+Ext푛푒푞
+푟푏푠 satisfies
+1
+2 ∥휌푟 (1)◦푟 (2)◦···◦푟 (푘) E − 휌푈휂푘 ⊗ 휌E∥ <
+2−훿푏푛1/4
+1 − 2−훿푏Δ/4 + 2휖푠,
+(14)
+and the output state of the extractor Ext푒푞
+푟푏푠 satisfies
+1
+2 ∥휌푟 (1)◦푟 (2)◦···◦푟 (푁 /푛)◦E − 휌푈휂 ⊗ 휌E∥ ≤ 휖 + 2휖푠.
+(15)
+Proof. We prove the smoothed version of Algorithm 2, and the proof for the smoothed version of Algorithm 1 follows
+essentially the same procedures. For brevity, denote
+2−훿푏푛1/4
+1−2−훿푏Δ/4 := 휖. According to the definition of the smoothed
+reverse block source, there exists a state 휌∗
+푋 E that is 휖푠-close to the real output state 휌푋 E such that 휌∗
+푋 E determines
+a (푏, ∞,훿) reverse block source. Using the result in Tm. III.2,
+1
+2
+���휌푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1)◦푋퐼푘+1:∞ E − 휌푈휂푘 +푑푏푛푘+1 ⊗ 휌푋퐼푘+1:∞ E
+���
+≤1
+2
+���휌푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1)◦푋퐼푘+1:∞ E − 휌∗
+푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1) ◦푋퐼푘+1:∞ E
+���
++ 1
+2
+���휌∗
+푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1) ◦푋퐼푘+1:∞ E − 휌푈휂푘 +푑푏푛푘+1 ⊗ 휌∗
+푋퐼푘+1:∞ E
+���
++ 1
+2
+���휌푈휂푘 +푑푏푛푘+1 ⊗ 휌∗
+푋퐼푘+1:∞ E − 휌푈휂푘 +푑푏푛푘+1 ⊗ 휌푋퐼푘+1:∞ E
+���
+≤휖s + 휖 + 휖s = 휖 + 2휖s,
+(16)
+where the first inequality is due to the contractivity of trace-preserving quantum operations while the last inequality
+comes from the fact that the purified distance is an upper bound of the trace distance.
+□
+Tis result implies that the output for a smoothed reverse block source can also be arbitrarily close to a uniform
+distributed sequence.
+7
+
+IV.
+SIMULATIONS OF THE REAL-TIME RANDOMNESS GENERATION RATE
+In this section, we make a simulation estimating the extraction speed of the first extractor Ext푒푞
+푟푏푠 implemented
+in the Xilinx Kintex-7 XC7K480T Field Programmable Gate Array (FPGA), a common application in industry. Here,
+we adopt the expanded Toeplitz-hashing extractor Ext푛
+푇 ′ as the gadget Ext푛
+푔. We use the raw data from Ref. [20] as
+the random source, which is a reverse block source with parameters 푏 = 8 and 훿 = 0.85. We consider a raw data
+length of 푁 = 251 bits and a total security parameter 휖 = 10−10, which means the final output data is 10−10-close to
+a uniform distribution. Correspondingly, the block length is 푛 =
+� 4
+훿푏 · log � 푁
+휖
+��
+= 50.
+According to the discussion afer Teorem III.3, Ext푒푞
+푟푏푠 will use a seed of (1 + 훿/2)푏푛 − 1 = 569 bits and output
+about 0.85 PB random bits. Precisely, Ext푒푞
+푟푏푠 simply divides the input raw data into blocks with length 50, where
+each block contains 푏 × 푛 = 400 bits, and multiplies a 170 × 400 Toeplitz matrix by a 400-dimension vector over
+퐺퐹 (2). Since the addition 푎 + 푏 and multiplication 푎 · 푏 over 퐺퐹 (2) are exactly 푎 ⊕ 푏 and 푎 ∧ 푏, respectively, which
+are both basic logical operations, the matrix multiplication involves 170 × 400 = 68000 ‘∧’ operations and 68000 ‘⊕’
+operations.
+Te parameters of the Xilinx Kintex-7 XC7K480T FPGA are as follows. Te clock rate is set to be 200 MHz; the
+number of Look-Up-Tables (LUTs) is 3 × 105; each LUT can perform 5 basic logical operations simultaneously. To
+make full use of the FPGA, we can perform the matrix multiplications of ⌊3 × 105 × 5/(2 ∗ 68000)⌋ = 11 blocks in
+parallel. Terefore, the extraction speed of Ext푒푞
+푟푏푠 is 200 × 106 × 11 × 170 = 374 Gbps, which is improved by one
+order of magnitude compared to the state-of-the-art result [34]. As a result, the extraction speed is not a botleneck
+for the high-speed QRNGs any more; hence our online extraction is adequate for the post-processing of the fastest
+known implementation of QRNGs [18, 19].
+V.
+CONCLUSION
+In conclusion, we design two novel quantum randomness extractors based on the reverse-block-source property
+that is satisfied by a large class of trusted-device QRNGs. Tese results provide theoretical supports to the current
+real-time block-wise post-processing widely applied in experiments and industry. Te first extractor improves the
+real-time extraction speed while the second one can extract infinite raw data with only a constant seed length. In
+particular, the first extractor is easy to be implemented in a FPGA. Te real-time extraction speed with a common
+FPGA is high enough for the real-time post-processing of the fastest known QRNGs.
+For future work, it is interesting to explore other properties beyond the general min-entropy source to improve
+the post-processing. Te improvement may come from boosting the extraction speed or saving the seed length. On
+the other hand, randomness extraction with an imperfect seed or even without seed is also a practical and promising
+direction. An interesting open question is whether randomness can still be extracted online without the seed for
+certain non-trivial random sources.
+ACKNOWLEDGMENTS
+We thank Y. Nie and B. Bai for enlightening discussions, and Salil Vadhan for telling us the details of the extractor
+which extracts randomness from Santha-Vazirani sources using a seed of constant length. Tis work was supported
+in part by the National Natural Science Foundation of China Grants No. 61832003, 61872334, 61801459, 62002229,
+1217040781, and the Strategic Priority Research Program of Chinese Academy of Sciences Grant No. XDB28000000.
+Appendix A: QRNG raw data as a reverse block source
+We focus on the QRNG based on measuring the phase fluctuation of a laser, which is the fastest and most widely
+applied implementation. Te details of the QRNG design and randomness quantification are given in Ref. [35]. Here,
+we make a brief summary. A laser wave with a random phase 휙(푡) passes through an interferometer with time delay
+휏푙. Afer the interference, the random phase fluctuation is then transformed into an intensity fluctuation and can be
+sampled by an analog-to-digital converter (ADC) with a sampling frequency 1/휏푠. Te laser intensity fluctuation is
+8
+
+transformed into voltage signal 푉 in ADC. When the sampled voltage signal falls in some interval of the ADC, it will
+generate a sequence of corresponding random numbers. Te sequence length determines the resolution of the ADC.
+For example, an 8-bit ADC will be divided into 28 intervals. We illustrate the physical seting in Fig. 2.
+FIG. 2: Typical seting of a QRNG based on phase fluctuation. MZI: Mach-Zehnder interferometer; PD: photo
+detector; ADC: analog-to-digital converter.
+According to Ref. [35], the voltage will be proportional to the phase difference Δ휙(휏) = 휙(푡 + 휏) − 휙(푡), i.e.,
+푉 = 푘Δ휙(휏푙), where 푘 is a constant. We assume that the spontaneous emission leads to a differential random
+phase characterized by a Gaussian white noise in time domain 퐹sp(푡), whose expectation and variance are given by
+E[퐹sp(푡)] = 0 and Var(퐹sp(푡)) = 휎2, respectively. Te phase difference comes from the integration of the differential
+random phase,
+Δ휙(휏) =
+∫ 푡0+휏
+푡=푡0
+퐹sp(푡)푑푡.
+(A1)
+Ten, Δ휙(휏) follows a Gaussian distribution 퐺(0,
+�
+휎2휏푙) with zero mean and variance 휏휎2 due to the property of
+Gaussian white noise. Te voltage also follows a Gaussian distribution, 퐺(0,
+�
+푘2휎2휏푙). When 휏푠 > 휏푙, the raw data
+generated by each sample will be independent. In the asymptotic limit, the conditional min-entropy per sample is
+given by [35]
+퐻min(푉푖|E) ≥ − log2
+�
+max
+푗
+∫
+푉푖 ∈푆푗
+퐺(0,
+�
+푘2휎2휏푙)푑푉푖
+�
+,
+(A2)
+where 푉푖 is the 푖-th sample and 푆푗 is the 푗-th interval of the analog-to-digital converter (ADC). While when 휏푠 ≤
+휏푙, there will be correlations between samples and randomness quantification will be different. Without loss of
+generality, we consider 휏푙 = 푞휏푠 with some integer 푞, as shown in Fig. 3. Comparing the adjacent two samples, for
+FIG. 3: Illustration of the phase interference.
+example, the first two samples, we have
+Sample1 :
+Δ휙1(휏푙) = 휙(푡 = (푞휏푠)) − 휙(푡 = 0) =
+∫ 푡=휏푠
+푡=0
+퐹sp(푡)푑푡 +
+∫ 푡=푞휏푠
+푡=휏푠
+퐹sp(푡)푑푡,
+Sample2 :
+Δ휙2(휏푙) = 휙(푡 = (푞 + 1)휏푠) − 휙(푡 = 휏푠) =
+∫ 푡=푞휏푠
+푡=휏푠
+퐹sp(푡)푑푡 +
+∫ 푡=(푞+1)휏푠
+푡=푞휏푠
+퐹sp(푡)푑푡.
+(A3)
+9
+
+Ti
+中1 Φ2 
+Φ3
+Φn-2 Φn-1 Φn Φn+1
+t
+TsMZI
+Laser
+PD
+ADCSuppose a 푏-bit ADC is applied in the experiment. Ten 푏-bit raw data can be generated per sample, with a min-
+entropy lower bound
+퐻min(푋푖|E) ≥ − log2
+�
+max
+푗
+∫
+푉푖 ∈푆푗
+퐺(0,
+�
+푘2휎2휏푙)푑푉푖
+�
+,
+(A4)
+where 푋푖 ∈ {0, 1}푏 is the sequence of raw random numbers corresponding to the voltage signal 푉푖.
+Te conditional min-entropy 퐻min(푋푖|푋푖+1 = 푥푖+1, · · · , 푋∞ = 푥∞, E) will be less than 퐻min(푉푖). Te net increment
+randomness comes from the integration of random phase in one time bin of an interval 휏푠, which corresponds to an
+effective variance of the voltage
+Var(푉)eff = Var(푉)휏푠
+휏푙
+.
+(A5)
+Ten we have
+퐻min(푋푖|푋푖+1 = 푥푖+1, · · · , 푋∞ = 푥∞, E) ≥ − log2
+�
+max
+푗
+∫
+푉푖 ∈푆푗
+퐺(0,
+�
+푘2휎2휏푠)푑푉푗
+�
+:= 훿푏,
+(A6)
+which forms a (푏, ∞,훿)-reverse block source.
+Appendix B: Initial seed is indispensable for reverse block sources
+Te following theorem claims that there does not exist any deterministic extractor that can extract even one bit
+of almost uniformly-distributed random number from every reverse block source. Te proof is essentially the same
+as that for Santha-Vazirani sources [36, 37].
+Teorem B.1. For all 푏, 푁 ∈ N+ and 0 < 훿 < 1, and any function Ext : {0, 1}푏푁 → {0, 1}, there exists a (푏, 푁,훿)-
+reverse block source 푋 such that either P[Ext(푋) = 1] ≥ 2−훿 or P[Ext(푋) = 1] ≤ 1 − 2−훿.
+Proof. Because |Ext−1(0)| + |Ext−1(1)| = 2푏푁, either |Ext−1(0)| ≥ 2푏푁−1 or |Ext−1(1)| ≥ 2푏푁−1. Without loss of
+generality, let us assume that |Ext−1(1)| ≥ 2푏푁−1. Pick an arbitrary subset 푆 of Ext−1(1) with |푆| = 2푏푁−1. Consider
+the source 푋 that is uniformly distributed on 푆 with probability 2−훿 and is uniformly distributed on {0, 1}푏푁 \푆 with
+probability 1 − 2−훿. It is easy to check that P[Ext(푋) = 1] ≥ 2−훿 > 1/2.
+In addition, we have that P[푋 = 푥]/P[푋 = 푥 ′] ≤ (2−훿)/(1 − 2−훿) for any 푥,푥 ′ ∈ {0, 1}푏푁 . Ten by Definition III.1,
+it is straightforward to check that 푋 is a (푏, 푁,훿)-reverse block source.
+□
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+11
+

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+Randomization advice and ambiguity aversion∗
+Christoph Kuzmics†
+Brian W. Rogers‡
+Xiannong Zhang§
+January 10, 2023
+Abstract
+We design and implement lab experiments to evaluate the normative
+appeal of behavior arising from models of ambiguity-averse preferences. We
+report two main empirical findings. First, we demonstrate that behavior
+reflects an incomplete understanding of the problem, providing evidence
+that subjects do not act on the basis of preferences alone. Second, additional
+clarification of the decision making environment pushes subjects’ choices in
+the direction of ambiguity aversion models, regardless of whether or not the
+choices are also consistent with subjective expected utility, supporting the
+position that subjects find such behavior normatively appealing.
+JEL codes: C91, D81
+Keywords: Knightian uncertainty, subjective expected utility, ambiguity aversion,
+lab experiment
+∗We thank Michael Greinecker, John Nachbar, Paulo Natenzon, Andrea Prat, Todd Sarver,
+Tomasz Strzalecki, Peter Wakker and Jonathan Weinstein for insightful comments. The authors
+gratefully acknowledge the funding from the Weidenbaum Center on the Economy, Government,
+and Public Policy at Washington University in St. Louis.
+†University of Graz, Austria, [email protected]
+‡Washington University in St. Louis, U.S.A., [email protected]
+§Corresponding author, Washington University in St. Louis, 1 Brookings Dr. St. Louis,
+Missouri, U.S.A. (63130), [email protected]
+1
+arXiv:2301.03304v1  [econ.TH]  9 Jan 2023
+
+1
+Introduction
+Many economic decisions are made under uncertainty that cannot be readily quan-
+tified by objective probabilities. Consider saving decisions, that is investing money
+into bonds or stocks in the presence of inflation uncertainties and general uncer-
+tainties about the economic future. Even in the absence of wars and pandemics
+most people find it hard to attach probabilities to the relevant possible events, let
+alone agree on such an assessment.
+The classical paradigm of rational decision making under such uncertainty is
+subjective expected utility (SEU), underpinned by the axiomatic foundation of
+Savage [1954]. The experimental designs of Ellsberg [1961], later implemented in
+many studies, see e.g., the survey of Trautmann and Van De Kuilen [2015], have
+challenged the SEU paradigm. This challenge was mostly on positive, that is,
+empirical, grounds in that these experiments found that many people behave in a
+way that is inconsistent with subjective expected utility. The fact, however, that
+this behavior has been so robust in lab experiments and that many researchers have
+developed axiomatic foundations for alternative models of decision making that
+admit such behavior, see references below, may be received as posing a normative
+challenge to SEU in postulating that a broader set of preference models could be
+considered normatively appealing.
+In this paper we aim to test the normative appeal of these alternative models
+of decision making under uncertainty. We use the term “normative” in the sense of
+Ellsberg’s 1962 PhD thesis, see Ellsberg [2001, pages 22-26], and as in the “subjec-
+tive” definition of rationality given by Gilboa [2012, p. 5]: We consider a decision
+normatively appealing (to the decision maker) if the decision maker (still) makes
+this choice after thorough reflection. In all of our treatments, subjects are provided
+with a complete, and fairly standard, description of all payoff-relevant aspects of
+the environment. We operationalize this reflection in the lab by providing subjects
+with supplementary descriptions that, while not payoff-relevant, emphasize certain
+ways to think about the environment. The descriptions are provided in the form
+of short videos that subjects watch before the elicitation of their choice.
+Our findings are relevant to all models of ambiguity aversion that are monotone.
+That is, if one act is better than another act in all states, then the inferior act
+2
+
+cannot be chosen when the superior act is available. We call such models classical
+ambiguity aversion (CAA) models.1
+The experimental environment is directly inspired by the hedging argument of
+Raiffa [1961] in the context of the Ellsberg [1961] two-color urn experiment. A
+risky urn contains 49 White and 51 Red balls.2 An ambiguous urn also contains
+100 balls, each of which is either Green or Yellow, but nothing more is known
+about the composition of the ambiguous urn.
+The decision-maker (DM) must
+choose one of three actions, after which the experimenter draws one ball from each
+urn Together, these determine the consequence for the DM, which is either “Win”
+or “Lose”, with Win being strictly preferred to Lose. We call the choices bets for
+White, Green, and Yellow. Each bet Wins if the experimenter draws a ball of the
+corresponding color, and loses otherwise. Ellsberg’s key insight is that Bet White,
+while commonly chosen, is incompatible with SEU.3
+In this context, the Raiffa [1961] hedge against ambiguity works as follows.
+The DM flips a fair coin and bets Green if the coin lands on heads and bets Yellow
+if the coin lands on tails. This randomized action provides an (objective) winning
+probability of 50%, regardless of the color of the drawn ball from the ambiguous
+urn. As this is higher than the 49% winning probability from betting White, this
+action strictly dominates the Ellsberg choice.4
+1This category includes most preference models reviewed in the recent survey of Machina
+and Siniscalchi [2014], such as the maxmin expected utility model of Gilboa and Schmeidler
+[1989], the Choquet expected utility model of Schmeidler [1989], the smooth ambiguity model of
+Klibanoff et al. [2005], the variational and multiplier preference models of Maccheroni et al. [2006]
+and Hansen and Sargent [2001], confidence function preferences of Chateauneuf and Faro [2009],
+uncertainty aversion preferences of Cerreia-Vioglio et al. [2011], and the incomplete preference
+model of Bewley [2002].
+2This variation from 50/50 avoids identification complications that can arise from indifference.
+3If Bet White is chosen, it must be weakly preferred to Bet Yellow, so that the DM’s subjective
+probability of a yellow ball being drawn from the ambiguous urn must be at most 49%. But then
+the subjective probability of a green ball being drawn from the ambiguous urn must be at least
+51%, so that Bet Green is strictly preferred to Bet White, a contradiction.
+4Kuzmics [2017], appealing to results in classical statistical decision theory – in particular the
+complete class theorem of Wald [1947], has shown that a DM who can randomize over choices and
+commit to following the realized prescription of this randomization can never make choices that
+are inconsistent with SEU and at the same time consistent with CAA in any decision problem.
+See also Bade [2015], Oechssler and Roomets [2014], Azrieli et al. [2018], Baillon et al. [2022a],
+and Baillon et al. [2022b] for similar results and arguments along these lines. One can, in fact,
+regard ambiguity aversion as a preference for randomization, see e.g., Eichberger and Kelsey
+[1996] and Epstein and Schneider [2010].
+3
+
+In light of the Raiffa [1961] argument, what are the possible explanations for
+a classically ambiguity-averse DM to nonetheless bet on White in this experiment
+even though it is dominated? First, it is possible, even likely, that designing a
+random choice does not occur to subjects as an option. Second, even if a subject
+recognizes such a possibility, it is possible that they cannot, or choose not to,
+go through the required construction and reasoning that would allow them to
+see that betting on White is dominated.5 But suppose now that a subject does
+recognize the possibility and understands the argument.
+A third possibility is
+that the subject has no access to a suitable randomization device, nor thinks they
+can simulate one. So suppose the subjects does have a fair coin. The fourth and
+final explanation is that the subject lacks the ability to commit to the randomized
+action. Once the coin flip realizes, the subject could revisit their choice, and if they
+are ambiguity averse they will want to flip the coin again, and again ad infinitum,
+with one possible outcome that they bet on White in the end after all.
+Classical ambiguity aversion models do not allow us to delve into the reasons
+behind the choice of White in the presence of the Raiffa [1961] argument, as these
+preference models are axiomatized for preferences over the space of all pure acts,
+see e.g., Seo [2009], and not over the set of all mixed acts as in Anscombe and
+Aumann [1963]. This means, however, that whether or not a classically ambiguity
+averse subject who understands the environment may choose to bet on White
+depends simply on whether the Raiffa [1961] hedge (including the commitment to
+its outcome) is provided as a pure choice or not. If it is given as a pure choice the
+subject cannot choose to bet on White. If it is not given, they can.
+Motivated by these considerations, our experimental treatments provide differ-
+ential supplementary observations that focus on the hedging argument. All of the
+supplementary observations are given after a (common to all treatments) standard
+and complete description of the environment, and before the elicitation of the sub-
+ject’s choice. If the standard and complete description of the environment suffices
+to impart a full understanding of the consequences of each possible action, then
+5The argument is based on the reduction of compound lotteries and as Halevy [2007] found
+many subjects who make Ellsberg choices also fail to reduce compound lotteries. This failure,
+however, can hardly be seen as normatively appealing - it is simply a mathematical mistake. Ab-
+dellaoui et al. [2015] has found a much weaker association between displayed ambiguity aversion
+and a failure to reduce compound lotteries.
+4
+
+the treatment effects of the supplementary observations will be null. Indeed, there
+is no marginal payoff-relevant information in the supplementary observations.
+To the contrary, our first main finding is that the commentary significantly
+changes behavior. Since the commentary does not affect a subject’s underlying
+preferences, it changes behavior through modifying a subjects’ understanding of
+the environment. This is only possible if the subject’s understanding after the stan-
+dard description was incomplete or erroneous. We conclude that many subjects
+indeed have an incomplete or erroneous understanding after being provided with a
+standard and complete description of the classical Ellsberg two-urn environment,
+so that directly applying the revealed preference toolkit to such data may not be
+appropriate. Instead, a given choice may reflect an incomplete comprehension of
+the environment, and therefore be viewed as a mistake, or the result of confusion,
+rather than as a manifestation of the subject’s preference.
+Our second main finding concerns the direction of the effects of the supplemen-
+tal observations. In this regard, and to the extent we can infer preferences from
+the treatments with commentary, our data supports the broader normative appeal
+of ambiguity aversion models over SEU in the following sense: When the Ellsberg
+choice (bet White) is compatible with ambiguity averse preferences but not with
+SEU, the supplemental hedging observations increase the frequency of Ellsberg be-
+havior; when the Ellsberg choice is incompatible with ambiguity aversion (and so
+also with SEU), the same observations decrease the frequency of Ellsberg behavior.
+The paper proceeds as follows: The experimental design is given in Section 2.
+The results are given in Section 3. Section 4 offers a discussion of these results and
+Section 5 provides a brief survey of related literature before Section 6 concludes.
+Experimental instructions are presented in the Appendix.
+2
+Experimental Design
+The slight variation of the Ellsberg two-color urn experiment, outlined above,
+serves as the baseline and control. We then vary the supplemental observations
+that subjects receive about the decision-making environment.
+Our treatments
+differ from the baseline along two dimensions. First, in addition to the standard
+options of betting on White, Yellow, or Green, in some treatments subjects are
+5
+
+presented with an additional fourth option, which executes a bet on either Green
+or Yellow according to the outcome of a fair coin to be tossed by a third party
+after the balls are drawn from the urns - the Raiffa hedge. Note that this option
+also serves as a commitment device for randomization, as the bet will be executed
+by the experimenter on the subject’s behalf. Recall that the compatibility of the
+Ellsberg choice (bet White) with classical ambiguity aversion models hinges on the
+presence or absence of this option.
+Second, after the environment is fully described as transparently as possible, in
+some treatments subjects watch short videos before making their (single) choice.
+The videos, while all factually correct, emphasize different aspects of the con-
+sequences of using the randomization device, in ways that we hypothesize may
+change some subjects’ understanding of the merits of the Raiffa hedge.
+In the main treatment, the coin flip bet is included as an option, and subjects
+are presented with a single video (denoted V 1 and available here) containing sup-
+plemental observations that describe the hedging argument of Raiffa [1961].6 It
+describes the outcome of betting on the coin conditional on the outcome of the
+ball drawn from the ambiguous urn. It states that the winning probability using
+that option is 50% in either case (green ball or yellow ball drawn), and concludes
+by reminding the subject that betting on White wins with probability 49%.
+This video, as well as our videos, does not explicitly advocate for any particular
+choice, so that it contains observations rather than advice. The transcripts of the
+videos are read by an anonymous (to subjects) third party to avoid a perception
+that the experimenters are giving implicit advice.
+Partly to control for a possible experimenter demand effect, we designed a
+second video (denoted V 2 and available here), in which the structure of the argu-
+ment and the language is symmetric to the first video. It describes the outcome
+of betting on the coin conditional on the outcome of the coin flip. It states that
+no known winning probability can be specified in either case (heads or tails), and
+concludes by reminding the subject that betting on White wins with probability
+49%. Again, it does not advocate for any particular choice.
+We ran treatments utilizing exclusively this second video, as well as treatments
+in which subjects were presented with both videos before eliciting their choice (in
+6Transcripts of all videos are included in the appendix for an offline audience.
+6
+
+both orders; there were no order effects).7
+As we want to understand the effect of the supplementary observations inde-
+pendently from the effect of presenting the hedging device as an explicit option, we
+ran a parallel set of treatments with similar videos but where the available options
+were simply bets for White, Green, or Yellow, as in the baseline case, without the
+coin flip option. In these treatments, the Ellsberg choice remains compatible with
+CAA models. We varied the videos slightly to accommodate the different choice
+set. First, as there was no explicit coin, before showing either V 1 or V 2, we showed
+a preliminary video (denoted V 0 and available here) in which it was explained that
+a subject could imagine a virtual coin toss, and then bet on Green/Yellow accord-
+ing to the outcome. Second, in videos V 1 and V 2 the coin toss was referred to as
+a virtual coin toss. We refer to the treatment with the explicit hedge/commitment
+option as “Coin” and those without it as “No Coin” treatments.
+3
+Results
+Table 1 summarizes the main findings.
+W
+G
+Y
+W
+G
+Y
+C
+Baseline
+45%
+41%
+14%
+Baseline
+37%
+26%
+11%
+26%
+(26/58)
+(24/58)
+(8/58)
+(20/54)
+(14/54)
+(6/54)
+(14/54)
+V1
+27%
+55%
+18%
+V1
+29%
+2%
+2%
+67%
+(12/44)
+(24/44)
+(8/44)
+(14/48)
+(1/48)
+(1/48)
+(32/48)
+V2
+-
+-
+-
+V2
+41%
+9%
+9%
+41%
+(18/44)
+(4/44)
+(4/44)
+(18/44)
+V1+V2
+58%
+28%
+13%
+V1+V2
+23%
+13%
+9%
+55%
+(35/60)
+(17/60)
+(8/60)
+(13/56)
+(7/56)
+(5/56)
+(31/56)
+Table 1:
+Summary of data (left: without a randomization device; right: explicit
+randomization option).
+7In sessions using both videos, we showed one video first, and the other video next.
+We
+did this in both orders. Then after both videos were shown, the subjects were provided with
+additional time to revisit any portions of either or both of the two videos before proceeding to
+enter their bet. During this time subjects could pause, rewind, and switch between videos as
+they liked. We thereby tried to minimize possible order effects and, indeed, there is no evidence
+that the order of the videos has any effect, and so we have pooled that data in our analysis.
+7
+
+We summarize the key findings from this data in the following three results.
+Result 1 If preferences alone dictate choices, the supplementary observations con-
+tained in the videos should have no effect on behavior. For the No Coin (Coin,
+resp.) treatments, pooling the data for Green and Yellow (being conservative), the
+p-value for the null hypothesis that choice frequencies are the same in the baseline
+and the V 1 video treatment is 0.067 (< 0.001, resp.)8, and for the null hypothesis
+that choice frequencies are the same in the baseline and the neutral V 1 + V 2 video
+treatment is 0.142 (0.007, resp.).
+Result 2 The p-value for the null hypothesis that neutral video observations (V 1+
+V 2) does not decrease the choice of White relative to the baseline is ≥ 0.5 for the
+No Coin treatments, and 0.057 for the Coin treatments.
+Result 3 Without supplemental observations there is no significant difference in
+the frequency of Bet White between the Coin and No Coin treatment, p-value 0.402.
+With supplemental observations (V 1 + V 2) there is a significant difference in the
+frequency of Bet White between the Coin and No Coin treatment, p-value < 0.001.
+4
+Discussion
+There is firm evidence that behavior across treatments is not a pure consequence
+of underlying preferences combined with a complete understanding of the environ-
+ment. The observations contained in the videos cannot change preferences as they
+do not change any of the payoff-relevant considerations. Rather, any differences
+in behavior must come from differences in the subjects’ understanding.
+Suppose we adopt the view that choices after studying both videos indeed re-
+veal preferences, since subjects may have a more complete understanding of the
+environment and the consequences of their choices after considering the observa-
+tions contained therein. Even then, 23% of subjects, i.e., those who chose Bet
+White in the Coin treatment, have preferences different from any CAA model.
+The remaining 77% of subjects make choices in the Coin treatment that can be
+explained by CAA as well as SEU.
+8All statistical tests performed in this paper are likelihood ratio tests.
+8
+
+However, the No Coin treatments provide an interesting contrast. In these
+treatments, Bet White is undominated and the supplemental observations increase
+the frequency of Bet White relative to the baseline description. If, again, we view
+the choices after studying both videos as revealing preferences, the choice of Bet
+White made by 58% of subjects is inconsistent with SEU but is consistent with
+CAA models.
+Together, these findings suggest that CAA models have broader normative ap-
+peal than (the narrower theory of) SEU despite their descriptive problems in some
+environments. Of note, we asked subjects (in a non-incentivized post-experiment
+questionnaire) if their preference became more or less clear after watching the
+videos. In the No Coin treatments 17% (27 out of 162) of subjects reported that
+their preferences became “less clear” after watching the videos, which is much
+higher than the 3% (6 out of 202) reporting the same in the Coin treatment
+(p < 0.001), calling into question the presumption that behavior directly reveals
+preferences, especially in the No Coin treatments. One interpretation is that many
+subjects find monotonicity to be a normatively appealing property, yet lack the
+sophistication to identify its consequences.
+We conclude this discussion by considering preferences that may depend on
+the timing of the resolution of uncertainty, as in the models of Seo [2009], Saito
+[2015], and Ke and Zhang [2020]. Such models are not classical, as they are not
+monotone.9 In the Coin treatments subjects were (truthfully) told, as part of the
+baseline description of the environment, that the coin flip would be executed after
+the balls were drawn from the urns and revealed. Thus, the choice of Bet White
+in the Coin treatments (37% in the baseline and 23% after both videos) is even
+inconsistent with these more flexible models.
+Roughly, one could categorize our subjects as follows. There is one group of
+subjects (≥ 23%) who make choices inconsistent with all models of ambiguity
+aversion. There is a second group of subjects (≤ 58% - 23% = 35%) who make
+choices consistent with ambiguity aversion but not with SEU. The remainder make
+choices consistent with SEU. Subjects in the second group would have a demand
+for randomization devices as they cannot, to their satisfaction, create and commit
+9Motivated by the state separability embedded in monotonicity, Bommier [2017] provides a
+model where monotonicity is relaxed.
+9
+
+to randomized choices themselves.
+5
+Related Literature
+A number of papers have studied the consistency of subjects’ choices across decision
+problems. These include Binmore et al. [2012], Stahl [2014], Voorhoeve et al. [2016]
+and Crockett et al. [2019].
+This literature finds, on the whole, that relatively
+few subjects make consistent choices, and those who do tend to be ambiguity-
+neutral. The lack of consistency can be interpreted as evidence against people
+choosing according to a clear preference. However, ambiguity-averse DMs may
+find inconsistent choices to be a useful hedge against ambiguity, see e.g., Kuzmics
+[2017] and Azrieli et al. [2018] for more general arguments.10 Our single choice
+design is immune to such problems. This is why we constrained our design to a
+single incentivized elicitation per subject, even at the cost of forgoing the ability
+to conduct within-subject analyses across treatments.
+Spears [2009], Dominiak and Schnedler [2011], and Oechssler et al. [2016] study
+experiments in which subjects are given the Raiffa hedge as an option, similarly to
+our baseline Coin treatment (without supplemental commentary). Generally, they
+find very few subjects choosing this option, with more subjects instead choosing a
+dominated option. This too is evidence against CAA models. They also find that
+subjects do not care about the timing of the resolution of uncertainty, evidence
+even against the non-CAA models of Seo [2009], Saito [2015], and Ke and Zhang
+[2020]. Our focus is on the possible effects of providing explicit descriptions of the
+Raiffa hedge. We thus add to these findings that such observations significantly
+influence behavior, and does so in directions that support the appeal of CAA
+models.
+Finally, several studies test, in various ways, the normative appeal of ambigu-
+ity aversion preference models.11 The closest to our design is that of Slovic and
+Tversky [1974], who give subjects written advice for and against Allais [1953] and
+10This problem persists under many different preference elicitation schemes.
+See also e.g.,
+Baillon et al. [2014], Bade [2015], Oechssler and Roomets [2015], which builds on earlier work
+on eliciting non-expected utility preferences under objective uncertainty by e.g., Karni and Safra
+[1987].
+11Al-Najjar and Weinstein [2011] provide normative arguments against ambiguity aversion.
+10
+
+Ellsberg [1961] choices. However, their advice is built around the independence
+axiom, and so concerns a quite distinct domain. Jabarian and Lazarus [2022] also
+study the effect of a form of advice on subjects’ decisions in a framework with
+ambiguity aversion. Their framework involves two independent draws from the
+same two-color ambiguous urn (and two draws from a 50-50 risky urn) in which
+many subjects make dominated choices, similarly also to Yang and Yao [2017] and
+Kuzmics et al. [2022]. Subjects win if they draw two balls of the same color from
+the urn that they choose, making betting on the ambiguous urn a (weakly) domi-
+nant choice - as the more extreme the ball distribution in the ambiguous urn the
+higher the chance of drawing two balls of the same color. Jabarian and Lazarus
+[2022] have treatments in which subjects are given additional decision problems
+that should help them understand the mechanism why a choice is dominated. They
+find that while subjects do seem to understand the mechanism, they, nevertheless,
+do not seem to transfer this knowledge to the original problem in which they make
+dominated choices regardless. Finally, Keller et al. [2007], Trautmann et al. [2008],
+Charness et al. [2013], and Keck et al. [2014] study decision problems with am-
+biguity in groups (or under peer observation) and find, on the whole, that group
+discussion and related phenomena tend to lead to more ambiguity-neutral choices.
+6
+Conclusion
+We have subjected classical preference models of ambiguity aversion models to
+tests of their normative appeal with experiments that stay close to the original
+Ellsberg (two-color urn) design.
+We find that subjects’ choices are affected by payoff-irrelevant commentary.
+This implies that at least one of the two treatments, without or with commentary,
+does not allow the full revelation of subjects’ preferences.
+At least some of our subjects do seem to see a certain normative appeal in
+the kind of behavior prescribed by classical models of ambiguity aversion and, in
+particular, the monotonicity axiom. Giving subjects access to additional commen-
+tary, in the form of short video clips, results in behavior that is significantly more
+consistent with these models.
+The nature of this normative appeal suggests that people, after sufficient re-
+11
+
+flection, would have a demand for the ability to commit to randomized choices,
+a demand which one would surmise should be observable. It would be interest-
+ing to identify instruments outside the lab, in the various areas of application of
+ambiguity aversion models, that could serve to satisfy this demand.
+We also find that our subjects lack a complete and perfect understanding of
+their decision environment and how their choices map into final outcomes, in spite
+of the fact that we did our best to describe the environment completely and accu-
+rately. If this is the case, then there is room for further descriptions to influence
+behavior. We have shown that this is indeed readily observable, using the rela-
+tively weak instrument of short video clips providing commentary on the hedging
+argument of Raiffa.
+This means that in classical designs, it may be necessary
+to view a given choice as arising from a combination of preferences and how the
+subject understands the environment, where the second channel is non-trivial. Ac-
+cordingly, a given choice may not provide direct evidence for or against any given
+preference model.
+A
+Experimental Design
+A.1
+Experiment details
+The experimental sessions took place in April and May of 2018, and February of
+2020. The experiment was conducted at the Experimental and Behavioral Eco-
+nomics Laboratory (EBEL) at University of California, Santa Barbara. There are
+two waves of data collection. In the first wave, 176 students participated in 10
+sessions and the average session length was 60 minutes. In the second wave, 213
+students participated in 12 sessions and the average session length was 60 minutes.
+In all sessions, subjects answered exactly one incentivized question, which was re-
+lated to guessing the color of a ball. If the guess was correct, the subject received
+10 USD, and 0 otherwise. The show-up fee for all sessions was 5 dollars. At the
+end of each session we conducted a short questionnaire. The questions were not
+incentivized, but we emphasized that answering these questions would be help-
+ful for our research. The experiment was programmed using z-Tree [Fischbacher,
+2007]. See Figure 2 for a screen shot.
+12
+
+A.2
+Physical environment
+In all sessions, the urns and states were implemented using two cardboard boxes
+and colored ping-pong balls. During the experiment (and in what follows), we refer
+to the two containers as Box A and Box B. A photo of the boxes can be found in
+Figure 1 (a). The protocols we used were guided by the desire to be as clear and
+transparent as possible. Box A contained 49 white and 51 red balls. The balls were
+displayed in clear plastic tubes at the beginning of the experiment so that subjects
+could easily see that there were two more red than white balls. Photos of the tubes
+are included as Figure 1 (b). After showing the balls to subjects, they were poured
+into Box A. On the other hand, it was important that the exact contents of Box
+B were unknown. We therefore informed subjects that Box B contained 100 balls,
+each of which was either green or yellow, but we were intentionally not telling them
+anything further about the contents. Box B was shaken so that it was credible that
+it contained the same number of balls as Box A. After this presentation, subjects
+were told that they could inspect all the boxes and balls at the conclusion of the
+experiment if they so desired.
+In each session, one subject was randomly selected to act as a monitor. The
+monitor was the person who physically conducted all draws of balls and displayed
+their colors to the other subjects, as well as coin flips, as relevant.
+a
+b
+Figure 1: Boxes
+13
+
+A
+BA.3
+A Screen Shot
+Figure 2:
+Second experiment: Video review.
+After seeing videos and before
+making their choices, subjects had the chance to re-visit all videos they watched
+before.
+A.4
+Questionnaire
+Table 2 shows the additional questions that we asked at the end of the experiment.
+14
+
+Click here to start the first video
+Start
+Click here to start the second video
+Click here to start the third video
+Start
+Start
+Nowyou canreviewall threevideos
+You have enough time to watch the full videos more than two times
+Proceed to next stageQuestions asked in all groups
+Gender (Male, Female, Prefer not to tell)
+Major
+How many Green balls do you think there are in Box B?
+How many Yellow balls do you think there are in Box B?
+Was there any part of the experiment that was unclear?
+After watching the videos, my preference over choices was:
+(More clear, Less clear, Unchanged, I don’t know)
+Questions asked when V1 and V2 are both presented
+Which video do you think is more compelling? (V1, V2, Equal)
+Questions asked in when option Coin is not available
+Do you find it difficult to simulate a coin toss in you head? (Yes, No)
+Questions asked when subjects chose White ball in Box A
+Why did you choose White?
+Questions asked when subjects chose Green or Yellow ball in Box B
+Why did you choose Green or Yellow?
+Questions asked in no-video treatments
+Why did you recommend this to your friend?
+Table 2: List of questions asked in the questionnaire
+In the battery of sessions for the experiment, different treatments required dif-
+ferent questions. The first block lists the questions that we asked in all treatments.
+The second block lists questions that are asked when both videos are presented.
+We denote by V1 the video in favor of the hedging argument and by V2 the video
+with the counter-argument. The third block lists questions that are asked when
+subjects are offered only three options and must implement the randomization
+with a virtual coin on their own. The fourth and fifth blocks list questions contin-
+gent on subjects’ choices. In the treatments in which videos are not shown to the
+subjects before their incentivized choices, we showed the video before the ques-
+tionnaire and asked for their “recommendation” in the questionnaire. We further
+asked for their reasoning. This is listed in the last block.
+A.5
+Instructions
+We attached the instruction of the most comprehensive treatment. In this treat-
+ment, we provided the subjects with 4 options and both videos. Instructions for
+15
+
+all the other treatments are written in a similar fashion.
+Instructions
+Welcome to the experiment! Please take a seat as directed. Please wait for
+instructions and do not touch the computer until you are instructed to do so.
+Please put away and silence all personal belongings, especially your phone. We
+need your full attention for the entire experiment.
+Adjust your chair so that
+you can see the screen in the front of the room.
+The experiment you will be
+participating in today is an experiment in decision making. At the end of the
+experiment you will be paid for your participation in cash.
+Each of you may
+earn different amounts. The amount you earn depends on your decisions and on
+chance. You will be using the computer for the experiment, and all decisions will
+be made through the computer. DO NOT socialize or talk during the experiment.
+All instructions and descriptions that you will be given in this experiment are
+factually accurate. According to the policy of this lab, at no point will we attempt
+to deceive you in any way. Your payment today will include a $5 show up fee.
+One of you will be randomly selected to act as a monitor. The monitor will be
+paid a fixed amount for the experiment. The monitor will assist us in running
+the experiment and verifying the procedures. If you have any questions about the
+description of the experiment, raise your hand and your question will be answered
+out loud so everyone can hear. We will not answer any questions about how you
+“should” make your choices.
+As I said before, do not use the computer until
+you are asked to do so. When it is time to use the computer, please follow the
+instructions precisely.
+We will now explain the experiment. There are two containers on the table that
+we will refer to as Box A and Box B. This is Box A. The Box is empty. Box A will
+contain 100 ping pong balls. Each of the balls in Box A will be either White, like
+this, or Red, like this. Specifically, Box A will contain exactly 49 White balls and
+51 Red balls, for a total of 100 balls. You don’t have to remember these numbers.
+When it is time to make a decision, we will remind you of these numbers. We
+have counted and displayed the balls in these tubes to make it easier to show the
+contents of Box A. There are 25 white balls in this tube and 24 in this tube, for
+a total of 49 white balls. There are 25 red balls in this tube and 26 red balls in
+16
+
+this tube, for a total of 51. We will now pour these balls into Box A and shake
+it to mix the balls together. This is Box B. We have already filled Box B with
+100 ping pong balls. Each ball is either Green, like this, or Yellow, like this. We
+will not reveal the exact numbers of Green and Yellow balls. Instead, you know
+only that there are 100 balls in total, consisting of some combination of Green and
+Yellow balls. We will now shake Box B to mix the balls up. At the end of the
+experiment, you will have an opportunity to inspect the Boxes and ping pong balls
+if you wish. In a few moments we will ask the Monitor to draw one ball from each
+Box for everyone to observe. You will be asked to choose from several options that
+correspond to guessing the color of a ball that the Monitor draws. If your guess
+matches the result, you will receive 10 dollars in additional to the show up fee. If
+your guess does not match, you will receive 0 dollars in addition to the show up
+fee.
+We will now start the experiment. On the computer desktop you will find a
+green icon named zleaf. Double click it now. Now there should be a welcome
+screen. Type your name and click the OK button in the welcome screen. One of
+you has been randomly selected by the software to serve as the monitor. Please
+raise your hand if you are the monitor. Could you please click the OK button
+on your screen and come to the front? Now your screen should have changed to
+“Please listen to the instructions”. Please leave it like that and do not click OK.
+In a few moments the Monitor is going to draw one ball from Box A and one ball
+from Box B. We are going to ask you to bet on the outcome of those draws.
+Specifically, you will be able to place one of 4 bets. Let me explain three of
+these options first. You can bet on White (from Box A), Green (from Box B) or
+Yellow (from Box B). Notice that you cannot bet on Red. If you bet on the White
+ball from Box A, then your payoff is not related to the draw from Box B. In other
+words, if the monitor draws a White ball from Box A, you win. If the monitor
+draws a Red ball, you lose. Similarly, if you choose Green or Yellow, your payoff
+only depends on the draw from Box B. For the fourth option, your bet will depend
+on the outcome of a coin flip. The monitor will flip a coin like this. If the coin
+lands on Heads, then we will set your bet to Green. If the coin lands on tails,
+we will set your bet to Yellow. To repeat, we will set your bet to either Green
+or Yellow, depending on the coin flip result. Again, you don’t have to write this
+17
+
+down, since we will remind you about all the options when it is time to make your
+choice.
+Before you make you decision, we are going to provide you with some comments
+on the experiment contained in 2 short videos. The videos are in total about 5
+minutes long. After the videos, you will make your choice by selecting one of the
+four options. We will proceed like this: You will first watch the two videos. Then,
+you will have a chance to review the videos if you like. During the review session,
+you can pause or rewind the videos. There will be enough time to watch both
+videos more than two times in the review session. After the review session, we are
+going to ask for your choice. After you enter your decision, please wait for others
+to finish. There will not be any further instructions until all of you make your
+decisions. Please follow the instructions on the screen and focus on the videos as
+much as possible. If you finish early, please remain quiet since others may still be
+watching. Now please put on the headphones provided at your desk and watch
+the videos. Once you are ready, please click OK.
+The monitor is now going to draw the balls. Please look away and draw a ball
+from Box A and show it to everyone. The color is [REALIZED COLOR]. Please
+put the ball back. We will write down the result on the blackboard. Now please
+look away and draw a ball from Box B and show it to everyone. The color is
+[REALIZED COLOR]. Please put the ball back. We will write down the result
+on the blackboard. Please toss the coin and announce the result. The result is
+[REALIZED SIDE]. Please put the coin down. We will write down the result on
+the blackboard. Now please return to your seat and enter these results into your
+computer screen, accompanied by an Experimenter. You can now see the outcome
+and your earnings on the screen. If you have questions about your payoff, please
+raise your hand.
+We will now conduct a short questionnaire. Please wait for the questionnaire
+to start. The monitor doesn’t have to fill the questionnaire. Please complete the
+questionnaire. Please be as specific as you can in your responses. Answering the
+question is helpful to our research, but your responses are entirely voluntary. After
+you finish, please wait for others. We will call you to the front by your participant
+ID to be paid before leaving. Thank you very much for your participation. This
+concludes the experiment. We will now begin calling you to the front to be paid
+18
+
+before leaving.
+A.6
+Video scripts
+A.6.1
+Names and notations
+Recall that we denote the video that explains the Raiffa hedging argument, used
+in our main treatment, by V1 and its counter argument by V2. In the treatments
+without an explicit coin flip option, the instructions do not describe a coin. In-
+stead, we show a short video introducing the hedging idea through the use of an
+“imaginary coin.” We call this video V0. V0 is neither in favor of hedging nor
+against hedging. It merely states the idea of conditioning one’s bet on the outcome
+of a virtual coin flip. We then slightly modified V1 and V2 by changing “the coin
+flip option” to “the imaginary coin.” For more details, please see the script below.
+To summarize, we have in total five distinct videos, listed in the table below,
+where “p” stands for “physical coin” and “v” for “virtual coin.”
+V1p
+V2p
+V0
+V1v
+V2v
+In favor of hedging?
+Yes
+No
+n/a
+Yes
+No
+Against hedging?
+No
+Yes
+n/a
+No
+Yes
+Describe hedging using a real coin?
+Yes
+Yes
+n/a
+No
+No
+Describe hedging using a virtual coin?
+No
+No
+n/a
+Yes
+Yes
+Description of a virtual coin?
+n/a
+n/a
+Yes
+n/a
+n/a
+A.6.2
+V0 Script
+Recall that your three options are to choose: a White Ball from Box A, a Green
+Ball from Box B, or a Yellow Ball from Box B. Let me suggest a new method for
+choosing how to bet. To use this method, you need to create a random event.
+So, imagine you have a coin and you can flip it. The coin lands on Heads with
+probability 50% and on Tails with probability 50%. Before the toss, you plan to
+bet on a Green Ball from Box B if the coin lands on Heads, and on a Yellow Ball
+from Box B if the coin lands on Tails. Using this rule, you will not bet on a White
+Ball from Box A. To summarize, you first imagine the outcome of the coin flip.
+Then you choose Green Ball from Box B if the coin lands on Heads and you choose
+19
+
+Yellow Ball from Box B if the coin lands on Tails. Click to view
+A.6.3
+V1p Script
+Recall that Box A contains 49 white balls and 51 red balls. So, you will win with
+probability 49% if you choose the “White Ball from Box A.” Let me describe the
+outcome when you choose the “Coin flip for green/yellow ball.” Recall that Box
+B contains an unknown combination of 100 Green and Yellow balls. So when the
+Monitor draws a ball from Box B there are two possible cases: the ball can either
+be Green, or it can be Yellow. Suppose the ball happens to be Green. Now, when
+the monitor flips the coin, it will land either on Heads or on Tails. Each case is
+equally likely: the probability of Heads is 50% and the probability of Tails is 50%.
+If the coin lands on Heads, you would bet on Green and win. If the coin lands
+on Tails, you would bet on Yellow and lose. So, what we have observed is that if
+the ball from Box B happens to be Green, you would win with probability 50%.
+Now suppose that the ball from Box B happens to be Yellow. As before, when
+the monitor flips the coin, it will land either on Heads or on Tails. Each case is
+equally likely: the probability of Heads is 50% and the probability of Tails is 50%.
+If the coin lands on Heads, you would bet on Green and lose. If the coin lands on
+Tails, you would bet on Yellow and win. So, what we have observed now is that if
+the ball from Box B happens to be Yellow, you would again win with probability
+50%. To summarize, if you choose the option “Coin flip for green/yellow ball”
+you will win with probability 50% whether the ball from Box B is green or yellow.
+Therefore, it does not matter how many of the balls are green and how many are
+yellow, since you will win with probability 50% in either case. By betting instead
+on a White Ball from Box A, you will win with probability 49%. Click to view
+A.6.4
+V2p Script
+Recall that Box A contains 49 white balls and 51 red balls. So, you will win with
+probability 49% if you choose the “White Ball from Box A.” Let me describe the
+outcome when you choose the “Coin flip for green/yellow ball.” If you choose this
+option, there are two possibilities: when the monitor flips the coin, it will land
+either on Heads or on Tails. Each case is equally likely: the probability of Heads
+20
+
+is 50% and the probability of Tails is 50%. Suppose the coin happens to land on
+Heads. In this case, you would be betting on a Green Ball from Box B. The chance
+that betting on a Green Ball from Box B wins depends on how many green balls
+are in Box B. Since you are not told how many green balls are in Box B, your
+probability of winning is uncertain. So, what we have observed is that if the coin
+lands on Heads, your probability of winning is uncertain. Now suppose the coin
+happens to land on Tails. In this case, you would be betting on a Yellow Ball from
+Box B. The chance that betting on a Yellow Ball from Box B wins depends on
+how many yellow balls are in Box B. Since you are not told how many yellow balls
+are in Box B, your probability of winning is again uncertain. Click to view
+A.6.5
+V1v Script
+Recall that Box A contains 49 white balls and 51 red balls. So, you will win with
+probability 49% if you choose the “White Ball from Box A.” Let me describe the
+outcome when you choose the method based on the coin flip I described before.
+Recall that Box B contains an unknown combination of 100 Green and Yellow
+balls. So when the Monitor draws a ball from Box B there are two possible cases:
+the ball can either be Green, or it can be Yellow. Suppose the ball happens to
+be Green. Now, when you imagine flipping the coin, it will land either on Heads
+or on Tails. Each case is equally likely: the probability of Heads is 50% and the
+probability of Tails is 50%. If the coin lands on Heads, you would bet on Green
+and win. If the coin lands on Tails, you would bet on Yellow and lose. So, what
+we have observed is that if the ball from Box B happens to be Green, you would
+win with probability 50%. Now suppose that the ball from Box B happens to be
+Yellow. As before, when you imagine flipping the coin, it will land either on Heads
+or on Tails. Each case is equally likely: the probability of Heads is 50% and the
+probability of Tails is 50%. If the coin lands on Heads, you would bet on Green
+and lose. If the coin lands on Tails, you would bet on Yellow and win. So, what we
+have observed now is that if the ball from Box B happens to be Yellow, you would
+again win with probability 50%. To summarize, if you use the method based on
+the coin flip, you will win with probability 50% whether the ball from Box B is
+green or yellow. Therefore, it does not matter how many of the balls are green
+21
+
+and how many are yellow, since you will win with probability 50% in either case.
+By betting instead on a White Ball from Box A, you have a known probability of
+winning of 49%. Click to view
+A.6.6
+V2v Script
+Recall that Box A contains 49 white balls and 51 red balls. So, you will win with
+probability 49% if you choose the “White Ball from Box A.” Let me describe the
+outcome when you choose the method based on the coin flip I described before.
+If you use this method, there are two possibilities: when you imagine flipping the
+coin, it will land either on Heads or on Tails. Each case is equally likely: the
+probability of Heads is 50% and the probability of Tails is 50%. Suppose the coin
+happens to land on Heads. In this case, you would be betting on a Green Ball
+from Box B. The chance that betting on a Green Ball from Box B wins depends on
+how many green balls are in Box B. Since you are not told how many green balls
+are in Box B, your probability of winning is uncertain. So, what we have observed
+is that if your coin lands on Heads, your probability of winning is uncertain. Now
+suppose your coin happens to land on Tails. In this case, you would be betting on
+a Yellow Ball from Box B. The chance that betting on a Yellow Ball from Box B
+wins depends on how many yellow balls are in Box B. Since you are not told how
+many yellow balls are in Box B, your probability of winning is again uncertain.
+So, what we have now observed is that if your coin lands on Tails, your probability
+of winning is also uncertain. To summarize, if you use the method based on the
+coin flip, your probability of winning is uncertain if the coin lands on Heads and
+it is also uncertain if the coin lands on Tails. By betting instead on a White Ball
+from Box A, you have a known probability of winning of 49%. Click to view
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+

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+ 
+1 
+Dimethylamino terminated ferroelectric nematogens revealing high 
+permittivity 
+ 
+Martin Cigl, Natalia Podoliak, Tomáš Landovský, Dalibor Repček, Petr Kužel, 
+and Vladimíra Novotná* 
+ 
+Institute of Physics of the Czech Academy of Sciences, Na Slovance 2, Prague, Czech 
+Republic 
+ 
+Abstract 
+Since the recent discoveries, ferroelectric nematics became of upmost interest due to their 
+outstanding ferroelectric properties. In this work, we prepared a series of polar molecules 
+revealing a ferroelectric nematic phase (NF) with a very high dielectric constant (>104). A new 
+motif, which differs from previously reported molecular structures, was optimized to support 
+the NF phase. For all homologues the NF phase was observed directly on the cooling from the 
+isotropic phase and ferroelectric behaviour was investigated by dielectric spectroscopy, second 
+harmonic generation, polarization current measurements and by analysis of textures in the 
+polarized light. The presented materials combine ferroelectricity with giant permittivity in a 
+fluid media at room temperatures, so they appear to be extremely attractive. Polarity of 
+molecules with the strong susceptibility to the electric field represent high potential for various 
+applications in energy-efficient memory devices or capacitors.  
+ 
+1. 
+Introduction 
+In thermotropic liquid crystals (LCs) molecules can self-assemble and create intermediate 
+phases (mesophases) in a certain temperature range between liquid and crystalline phases [1], 
+combining the fluidity of liquids with the anisotropy characteristic for crystals. Anisotropic 
+properties of LC medium manifest itself as a result of the anisotropic shape of (partially) 
+ordered constituent molecules. A large variety of phases and structures can be observed in LCs, 
+which are susceptible to external field and boundary conditions. Many LC phases reveal a large 
+electro-optical response, which became a background of mass-production technological 
+applications (monitors, sensors, etc.). First, the ferroelectricity in LCs was associated with 
+chirality of the constituent molecules and only a tilted smectic phase formed by chiral rod-like 
+molecules [1] was considered to feature ferroelectricity (FE) and/or antiferroelectricity (AF). 
+With the discovery of bent-core materials [2], it was found that non-chiral mesogens may also 
+form FE and AF phases as the close packing and hindered rotation can lead to the structural 
+chirality. Nevertheless, due to a higher viscosity, smectic phases never reached such broad 
+application range as nematics. 
+Recent discoveries stimulated renewed intensive progress in the field of nematic liquid 
+crystals. For a conventional nematic phase, the director orientations n and –n are 
+indistinguishable due to the thermal fluctuations, so they form only non-polar phases. However, 
+
+ 
+2 
+as far back as in 1918, Max Born [3] predicted a possibility of a ferroelectric fluid, in which all 
+the dipoles point in the same direction. In such a nematic ferroelectric state (NF), the dipole 
+moments μ should be strong enough such that the dipole-dipole interactions overwhelm the 
+thermal fluctuations. In 2017, a real breakthrough was announced in the development of LCs, 
+as the first two ferroelectric nematics (denoted RM734 and DIO) were reported simultaneously 
+by two research teams [4-6]. Both materials reveal extremely high longitudinal dipole moments 
+(about 10 D), anomalously huge dielectric anisotropy Δε, and a spontaneous polarisation of 
+about 4 μC/cm2, which is an order of magnitude higher than the previously reported values in 
+other ferroelectric LC phases. Recently, these materials have been intensively studied [7-17]. 
+Mandle at al. [9] synthesised a homologue series relevant to the molecular structure of RM734 
+and analysed the mesomorphic properties and tendencies leading to the NF phase. The 
+compounds have been intensively studied by Ljubljana researchers [10-12] and by the Boulder 
+group [13,14]. The existence of ferroelectric domains with a different macroscopic orientation 
+of the dipoles in the absence of electric field was reported [10-14]. Details of polar nature of 
+self-assembly, evolution of topological objects and analysis of their character [12,17,18] are 
+under intensive research progress. Currently, the research is focused on the preparation and 
+characterisation of new compounds. Machine learning procedures were applied to predict ideal 
+conditions for the NF phase presence, including a dipole moment value, aspect ratio, length of 
+the molecule as well as the dipolar angle [19]. In spite of the fact that these conditions are rather 
+restrictive, development in the designing of prosperous molecular structures was promoted. 
+At the moment, microscopic organisation of the polar molecules and the mechanism of 
+the phase transition to the ferroelectric nematic phase undergo intensive research and 
+stimulating debates. A theoretical description of the ferroelectric nematic phase has been 
+proposed [20,21], and chiral analogues of highly polar molecules were developed recently [22]. 
+Additionally, a possibility of oligomer synthesis was shown [23] and new phases and effects 
+introduced. In any case, the ferroelectric properties combined with the giant permittivity in a 
+fluid media represent an attractive rapidly developing subject. Since the discovery of NF phase, 
+the ongoing research is mostly concentrated on the design of new molecular structures. Up to 
+now, the library of NF materials is strictly limited to a couple of general structures possessing a 
+suitable aspect ratio and a large enough dipole moment, which develops due to the effective 
+electron donating and withdrawing groups within the molecules.  
+In this contribution, we demonstrate newly designed structural motif (see Fig.1). In 
+contrast to the previously reported molecular designs [3-5,8-19], which utilise an oxygen-based 
+electron donating group, we synthesised a series possessing a more efficient nitrogen electron 
+donating group in the terminal part of the aromatic system. Such a design yields higher dipole 
+moment along the long molecular axis compared to other published materials. To modify the 
+lateral interactions, which are strong in highly polar systems, we introduced a lateral alkyl chain 
+with varied number of carbon atoms from 1 to 6. Based on these considerations, we synthesised 
+a series of compounds (Fig. 1) which exhibit the NF phase directly below the isotropic liquid 
+on cooling. By tuning the lateral substitution, we shifted the temperature interval of NF down 
+to the room temperature (RT), at which it may eventually relax to a stable glassy state preserving 
+the ferroelectric behaviour. 
+ 
+
+ 
+3 
+  
+Fig. 1. Chemical formula of compounds NFn with n = 1 - 6. 
+ 
+2. 
+Materials and methods 
+Chemical formula of the studied compounds is presented in Fig. 1. Synthesis of materials 
+started from commercial 4-aminosalicylic acid (1, see Scheme 1). Its amino group was 
+protected by acetylation and the carboxylic group was protected by alkylative esterification by 
+methyl iodide, so as neither of the two groups interfere with the alkylation of phenolic hydroxyl. 
+Protected derivative 2 was then alkylated by 1-bromoalkanes to get a series of alkyl homologues 
+3-n. In the next steps, the acetyl group was cleaved by acidic hydrolysis under mild conditions 
+and the liberated amino group was alkylated by dimethyl sulphate yielding the key intermediate, 
+acid 4-n. The lowest alkyl homologue (4-1) was synthesised directly from acid 1 by alkylation 
+with the excess of dimethyl sulphate. The second part of the molecular core was synthesised 
+from 4-hydroxybenzoic acid (5), which was protected by the reaction with 3,4-dihydro-2H-
+pyrane and reacted with 4-nitrophenol in a DCC-mediated esterification. The protected 
+hydroxyl group was then liberated by the treatment with p-toluenesulfonic acid. The final step 
+of the synthesis was esterification of acids 4-n with phenol 6 mediated by EDC.  
+Differential scanning calorimetry (DSC) measurements were performed to acquire 
+thermal properties. For electro-optical studies, a polarising optical microscope was used, 
+equipped with a heating/cooling stage. Details about the compound characterisation and 
+experimental apparatus are in Supplemental file. 
+ 
+Scheme 1. 
+Synthesis of the studied polar nematogens denoted NFn with n varying from 1 
+to 6.  
+
+ 
+4 
+3. 
+Results 
+We studied all newly synthesised homologues by DSC and observed textures and their 
+changes in polarising microscope to assess the phase behaviour. We performed DSC 
+measurements in a broad temperature range. We established the melting point (m.p.) from the 
+first heating run, during which we observed a direct transformation from the crystalline to the 
+isotropic (Iso.) phase. After the first heating of the fresh sample, we followed with a cooling 
+run from the Iso phase down to -25°C. On the cooling run, the compounds transformed to the 
+liquid crystalline state at a significantly lower temperature, Tiso. Under the polarising 
+microscope, we observed characteristic textures in the LC state, which were previously ascribed 
+to the ferroelectric nematic phase, NF [12-18]. In the following description, the properties of NF 
+phase are systematically uncovered. 
+The analysed DSC data are summarised in Table 1. Compounds NF1, NF2, and NF3 
+did not crystallise during the cooling run, however, they crystallised during the subsequent 
+heating. These homologues revealed the ferroelectric nematic phase only during the cooling of 
+the sample; temperature stabilisation or heating of the sample caused the crystallisation. The 
+homologue NF4 revealed the shortest temperature range of the NF phase and crystallised at 
+about 74°C. On the other hand, the longest homologues NF5 and NF6 did not crystallise during 
+the DSC measurements at all. For these homologues, the NF phase persisted during the second 
+and third cooling-heating DSC cycles. The stability of the NF phase for these two homologues 
+was confirmed during electro-optical measurements: the NF phase was stable for several hours 
+at RT. The DSC thermograph for the homologue NF6 is demonstrated in Fig. 2. For the first 
+heating of the sample, the melting point (m.p.) was established; for the second heating run, the 
+NF phase melted at a temperature corresponding to Tiso. A glassy transition was clearly 
+distinguishable and its temperature, Tg, was determined from the onset calculated at a half heat 
+capacity, cp, see Table 1. Glassy properties and ability to form fibres from the melted compound 
+NF5 is demonstrated in Supplemental file (Fig. S2).  
+ 
+Fig. 2. DSC thermograph detected for NF6 during the first and second heating and cooling 
+runs.  
+
+8
+- the first heating
+- the second heating
+6
+- the cooling
+Heat flow ( mW)
+ glassy state
+N.
+Iso.
+-20
+0
+20
+40
+60
+80
+100
+120
+T(C) 
+5 
+Table 1. 
+Calorimetric data taken from DSC measurements: melting point, m.p., detected 
+at the first heating run, the NF-Iso phase transition temperature, Tiso, and the glassy transition 
+temperature, Tg. All temperatures are presented in °C, and enthalpy changes, H, in J/g, are in 
+square brackets at the corresponding temperatures.  
+ 
+m.p. 
+H (J/g) 
+Tiso (C) 
+H (J/g) 
+Tg 
+H (J/g) 
+NF1 
+188 [+98.6] 
+170 [-2.73] 
+24 [+0.47] 
+NF2 
+150 [+71.3 
+136 [-7.59] 
+30 [+0.42] 
+NF3 
+156 [+73.8] 
+116 [-7.13] 
+15 [+0.27] 
+NF4 
+144 [+80.2] 
+96 [-6.11] 
+- 
+NF5 
+120 [+55.1] 
+82 [-4.86] 
+-9 [+0.44] 
+NF6 
+104 [+51.4] 
+65 [-3.33] 
+4 [+0.28] 
+ 
+In the polarising microscope, we observed various textural features in different 
+commercial or home-made cells. There are two basic geometries for rod-shaped liquid 
+crystalline molecules: in HG cells, the molecules are oriented along the cell surface, and in the 
+HT cell, a homeotropic anchoring ensured molecular orientation perpendicular to this direction. 
+In the HG cell, the alignment is provided by rubbed polyimide layers with a small pretilt to 
+arrange defect-free textures. The pretilt results in nonzero polar surface energy as was pointed 
+out by Chen et al. [14]. Two kinds of HG cells were available, with parallel (HG-P) or 
+antiparallel (HG-A) rubbing directions on opposite glass surfaces. 
+Let us start with HG-A cells and compare the results for various cell thicknesses. In this 
+geometry, we observed two kinds of domains. The texture in 5m HG-A cell for the studied 
+homologue NF6 is shown in Fig. 3. The dominating type of domains are twisted domains, which 
+were described for the NF phase in literature [12]. The twisted domains are recognisable when 
+slightly uncrossing the analyser from the crossed position. Another type of domains can be 
+observed in less extensive areas of the HG-A samples. In the upper right part of Fig. 3, we found 
+“red-colour” domains with characteristic borderline approximately parallel to the rubbing 
+direction. The red colour was typical for these domains in 5 m HG-A cell, see Fig. S2-S5 in 
+Supplemental for other homologues. Extinction position in these domains are not easy to be 
+established and the colour of these domains changes when rotating the sample with respect to 
+the polariser position. We did not find twisted domains in HG-P cells with parallel alignment. 
+In this geometry, we observed homogeneously aligned area as well as “red” domains, as is 
+demonstrated for NF6 in Fig. S6 in Supplemental file.  
+We concentrate on twisted domains, which are very frequent in the HG-A geometry. 
+We found that for very thin HG-A sample, the twisted domain can be extended for a large area 
+by quick cooling from the isotropic phase (with a rate 20 K/min). Rather big twisted domains 
+separated by a zig-zag borderline are demonstrated in Fig. 4(a) for NF6 in 1.6 m HG-A cell. 
+One can see that the borderline between the twisted domains is oriented approximately 
+perpendicularly to the rubbing direction. When we turn the analyser from the crossed position 
+by an angle of ~ 20 degrees, we clearly observe two kinds of domains (see insets in Fig. 4); the 
+sense of twist is opposite for two neighbouring domains and they are separated by 2 
+
+ 
+6 
+disclination line. In the paper by Sebastian [12], similar domains were observed for another 
+type of ferroelectric nematogen and designated “sierra-domains”. In our particular case, these 
+twisted domains reveal sharper contour and can be renamed as “shark-domains”. For the 
+homologue NF5, the twisted domains are demonstrated in Fig. S4 in Supplemental. Schematic 
+picture of molecular twist between surfaces with antiparallel alignment is shown in Fig. 4(b).  
+ 
+Fig. 3. 
+Microphotograph of NF6 homologue in 5 m HG-A cell. The red arrow (R) 
+marks the rubbing direction, the white arrows show the polariser (P) / analyser (A) directions. 
+ 
+
+R
+50um 
+7 
+Fig. 4. 
+Textures of NF6 in 1.6 m HG-A cell under a polarizing microscope (a) between 
+crossed polarisers, the red arrow marks the rubbing direction, R, the orientation of the analyser 
+(A) and the polariser (P) is schematically shown by white arrows. In the figure (b) there is a 
+schematic arrangement of molecules in neighbouring twisted domains between glass surfaces 
+with antiparallel rubbing. The part of the figure (a) marked by white lines is shown in (c) and 
+(d) when A is rotated by an angle of about 20 degrees counterclockwise or clockwise from the 
+crossed position. 
+ 
+ 
+An application of an electric field in HG geometry led to a rather complex effect. The 
+colour of twisted domains slightly changed under the applied electric field and additional stripes 
+appeared across the twisted domain structure approximately parallel to the rubbing direction. 
+As the application of the field in the HG cell supplied only limited information and a detail 
+analysis is rather problematic, we studied HT cells under applied bias. In this geometry, the 
+applied electric field is approximately parallel to the molecular dipole moment and we can 
+observe a rearrangement of molecules. In Fig. 5, we demonstrate the HT texture for homologue 
+NF6 with and without applied electric field of about 5 V/m. In the upper part of Fig. 5, an area 
+without electrode is observed. When the field is switched on (Fig. 5(b)), the molecules reorient 
+along the field and the texture under the electrode area becomes black. After switching the 
+electric field off, the HT texture turns back to a lighter type, similar to the virgin texture (Fig. 
+5(a)), within several seconds.  
+
+(a)
+(b)
+R
+100 μm
+c)
+(d) 
+8 
+We investigated the switching properties of the studied compounds in HT geometry. 
+Due to electrostatic interactions, the results are influenced both by the cell geometry and by the 
+character of the aligning layer. For n=1-4, the homologues NFn reveal strong vitrification and 
+an increase in viscosity when approaching the glass transition temperature Tg. This temperature 
+is relatively high and the samples feature a higher conductivity, which limited our studies for 
+these homologues. On the contrary, homologues NF5 and NF6 could be subjected to the applied 
+field for a longer time (several hours); the polarisation was measured repeatedly and the results 
+were reproducible. At the room temperature, these two homologues stay in LC phase for a long 
+time and they start to crystallise only after several hours.  
+For homologue NF5, the temperature dependence of the polarisation is presented in Fig. 
+6(a). The polarisation values are calculated by the time-integration of a switching current 
+profile. In Fig. 6(b), the switching current is plotted versus the applied electric field at a 
+frequency 10 Hz and at temperature 52 °C. For both homologues NF5 and NF6, we detected a 
+continuous increase in polarisation values on cooling process in the NF phase. A coexistence of 
+the Iso and NF phases was checked under the polarising microscope and it was observed only 
+in a narrow temperature interval of about 2 °C. The decrease in polarisation values shown in 
+Fig. 6(a) is connected with an increase in switching time. Such a slowing-down of molecular 
+dynamics is connected with an increase in the sample viscosity.  
+ 
+Fig. 5. 
+Texture of NF6 in 5 m thick HT cell, (a) without electric field and (b) under 
+applied electric field of about 5 V/m perpendicular to the cell. The orientation of polarisers 
+and of the applied electric field are marked by black symbols for illustration. Upper part of the 
+figure shows an area without electrodes. 
+ 
+To analyse the effect of the applied field in different cell geometries, we prepared a 
+home-made gap-cell with in-plane electrodes. Two glass slides were separated by copper 35 
+m thick ribbons, with a gap distance of about 1 mm. In this cell, the domains disappeared 
+
+(a)
+(b)
+No electrodes
+Electrode-edge
+E
+X
+P
+100μm 
+9 
+under the applied electric field as all the molecules were aligned along the applied electric field. 
+After the switching-off of the external electric field, the domain structure was partially 
+reconstructed in several seconds. Microphotographs can be found in Supplemental file (Fig. 
+S8). Unfortunately, the thickness of 35 m was rather large to reach homogeneous alignment 
+through the whole cell thickness. Additionally, we are aware that the applied electric field was 
+not homogeneously distributed. Technological tasks of the cell preparation and detailed 
+analysis of the defects in electric field are still under work.  
+ 
+Fig. 6. 
+(a) The temperature dependence of the polarisation of NF5, which was 
+calculated from the polarisation current. (b) The current profile at a temperature T=52 °C is 
+demonstrated with a triangular profile of the applied electric field at a frequency 10 Hz. 
+ 
+ 
+We measured the dielectric spectra of all compounds in a temperature range from the 
+isotropic liquid to RT in order to study the molecular dynamics. The applied measuring field 
+was smaller than 0.01 V/m (higher probing fields could influence the dielectric measurements 
+
+(a)
+4
+(μC/cm²
+3
+2
+P
+0
+40
+45
+50
+55
+60
+65
+70
+T(°C)
+(b)
+10
+4
+current (arb. units)
+5
+2
+E
+(V/μm)
+0
+0
+-5
+-2
+-10
+-4
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10 
+10 
+as the studied compounds are really sensitive to external fields). In the ferroelectric NF phase, 
+we found one distinct quite strong relaxation mode appearing at the Iso-NF phase transition on 
+cooling and remaining visible down to RT. On the other hand, when the sample is in the 
+crystalline state, this mode is not present and the permittivity is low (10). We demonstrate 
+three-dimensional plots of the real, ’, and imaginary, ’’, parts of permittivity versus frequency 
+and temperature, T, for compound NF5 in Fig. 7. For homologues NF2, NF3 and NF6, the 3D-
+plots of permittivity are shown in Supplemental file, Figs. S9-S11. All the presented dielectric 
+data were obtained in 12 m thick cells with gold electrodes and no surfactant layers.  
+We encountered a disturbing effect of surfactant, similarly as it was mentioned in 
+previous works dealing with dielectric spectroscopy of the NF phase [16]. For such a type of 
+polar phase, it was reported that the polymer layers effectively influence the permittivity 
+measurements. Due to a non-conductive character of polymer layers on the cell surfaces, there 
+is a barrier which causes a spatial variation of the charge and influences the measured effective 
+permittivity values. We fitted the dielectric data to the Cole-Cole formula (see Supplemental 
+file for the details) to obtain information about the dielectric strength, , and the relaxation 
+frequency, fr. We detected large only slightly temperature dependent values of  up to 15103. 
+In contrast, the relaxation frequency decreases within the whole temperature range of the NF 
+phase on cooling and follows the Arrhenius law. Such behaviour is documented in Fig. 8 for 
+homologue NF6, which followed Arrhenius behaviour ideally and the activation energy, Ea, 
+was calculated to be 102 kJ/mol. For other compounds, the linearity of fr in logarithmic scale 
+(versus 1/T in absolute temperature scale) was confirmed only far from the Iso-NF phase 
+transition (see Fig. S12 in Supplemental file). Non-homogeneity of molecular alignment and/or 
+influence of electrodes should be taken into consideration to explain the deviation from 
+Arrhenius law. 
+
+ 
+11 
+ 
+Fig. 7. 
+3D-plot of (a) real, ’, and (b) imaginary, ’’, parts of the permittivity versus the 
+frequency and the temperature, T, for compound NF5. Dielectric measurements were performed 
+in 12 m cell with gold electrodes and no surfactant layer. 
+
+16 
+14 
+12
+10
+8
+3
+6
+4
+30
+2
+40
+50
+0
+60
+101
+102
+70
+103
+frequency (Hz)
+104
+80
+105
+106
+6
+(103)
+2
+30
+40
+50
+101
+102
+60
+103
+70
+frequency (Hz)
+104
+105
+80
+106 
+12 
+ 
+Fig. 8. 
+Temperature dependences of (a) the dielectric strength, , and relaxation 
+frequency, fr, for NF6 in 12 m cell without surfactant layer. In the inset fr is presented in the 
+logarithmic scale versus reciprocal temperature, 1/T, in Kelvins and the activation energy, EA, 
+was established from the slope. 
+ 
+Strong polar character of the NF phase was proved by SHG measurements. The SHG 
+experiments were carried out in transmission configuration according the scheme described in 
+Supplemental file. We utilised HG cells and SHG measurement results are presented for 
+compounds NF5 and NF6 in Fig. 9. On cooling the sample from the isotropic phase, the SHG 
+signal abruptly grows from zero value at the transition temperature to NF phase. With ongoing 
+temperature decrease, the SHG intensity slows down its increase, reaches the maximum and 
+slowly starts to decrease. All of this happens within the NF phase, where we would expect a 
+gradual increase in the SHG signal upon cooling. Moreover, even for the weakest applied 
+intensity of the fundamental laser beam, a small drop in SHG intensity was detected in 
+subsequent measuring runs at the same temperature. From this it follows that the decrease in 
+the SHG signal upon cooling may be explained by partial decomposition of our samples caused 
+by rather strong intensity of the pulse laser beam.  
+X-ray scattering experiments confirmed nematic character of the observed mesophase. 
+Nematic phase is characterised by the long-range orientational order and only broad diffuse 
+peaks of low intensity can be detected. For homologue NF5, the signal at small scattering angles 
+is rather wide and can be fitted with two signals, with maxima corresponding to 18.8 Å and 
+10.5 Å at T=75°C, 22.5 Å and 10.4 Å at T=30°C. As the length of molecules, l, can be 
+approximately established as l~20.9 Å, the peak at the small scattering angle matches perfectly 
+to the long dimension of the molecules. The peak at a wide-angle region has also a very broad 
+
+口
+60
+12
+■
+4
+E,=102 kJ/mol
+口
+(zH)
+口
+40
+(10°
+8
+口
+(zH)
+2
+3V
+口
+1.3
+1.4
+1.5
+1.6
+20
+4
+10" RT (Jmoll
+0
+0
+40
+45
+50
+55
+60
+T(C) 
+13 
+profile with the maximum corresponding to 4.4 Å for all measuring temperatures, and it 
+corresponds to an average distance between the molecules.  
+ 
+Fig. 9. 
+SHG signals for NF5 and NF6 in HG cells  
+ 
+4. 
+Conclusions 
+We proposed a new structural modification of highly polar molecules self-assembling 
+and forming the ferroelectric nematic phase NF. All the prepared compounds exhibit a direct 
+phase transition to the ferroelectric nematic phase on cooling from the isotropic phase. In the 
+presented homologue series, a prolongation of a side-chain resulted in the NF phase persistence 
+down to the room temperatures and stability for at least several hours. Ferroelectric character 
+of the nematic phase was proven by several experimental techniques. Characteristic textural 
+features for the ferroelectric nematics were observed in several sample geometries. The 
+ferroelectric switching process was detected and the polarisation was calculated from the 
+measured polarisation current. The values of polarisation were found to increase continuously 
+on cooling from the isotropic phase, reaching up to 4 C/cm2. For all the studied homologues, 
+the dielectric studies show a strong polar mode characteristic for the NF phase, disappearing in 
+the isotropic or crystalline phases. The dielectric strength of this mode exceeds values of about 
+15103, which is the maximum reached for the NF phase up to now. Nevertheless, the 
+characterisation of the defects in the NF phase and the role of the electrodes are not yet 
+completely solved and will need a deeper insight. The dipole moment of the molecules was 
+calculated and established to be about 14 D, which is larger than the value reported for DIO or 
+RM734.  
+The discovery of ferroelectricity for nematics opened new opportunities in the liquid 
+crystal research and generally in the field of condensed matter. The NF phase represents a highly 
+polar structure responsive to very small applied fields and it features a variety of new effects 
+
+0.6
+NF5
+0.4
+NF6
+0.2
+0.0
+30
+40
+50
+60
+70
+80
+90
+T (C) 
+14 
+induced by the confining surfaces. Generally, the application potential of the NF phase is 
+immense and not yet completely explored. Our particular room-temperature-stable soft phase 
+exhibits huge dielectric constant and can be important in future for the development of memory 
+devices, capacitors and actuators. 
+ 
+Disclosure statement 
+No potential conflict of interest was reported by the authors. 
+ 
+Acknowledgments 
+Authors acknowledge project MAGNELIQ, that received funding from the European Union’s 
+Horizon 2020 research and innovation programme under grant agreement No 899285; and 
+project 22-16499S from the Czech Science Foundation. V.N. is grateful to Damian Pociecha 
+and Ewa Gorecka from Warsaw University for their help with x-ray measurements. 
+ 
+References 
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+[2] 
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+[3] 
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+[4] 
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+[7] 
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+[8] 
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+2021. 
+ 
+ 
+
+1 
+ 
+Supplemental information 
+ 
+Dimethylamino terminated ferroelectric nematogens revealing high 
+permittivity 
+ 
+Martin Cigl, Natalia Podoliak, Tomáš Landovský, Dalibor Repček, Petr Kužel, 
+and Vladimíra Novotná* 
+ 
+Institute of Physics of the Czech Academy of Sciences, Na Slovance 2, Prague, Czech 
+Republic 
+ 
+Contents 
+1. 
+Synthesis and compound characterisation 
+1.1. 
+General synthesis 
+1.2. 
+Synthetic procedures 
+1.3. 
+Equipment and apparatus 
+2. 
+Mesomorphic properties 
+2.2. 
+Textures 
+2.3. 
+Dielectric spectroscopy and electro-optical properties 
+ 
+ 
+1. 
+Syntheses and compound characterisation  
+1.1.  
+General synthesis 
+All starting materials and reagents were purchased from Sigma-Aldrich, Acros Organics or 
+Lach:Ner. All solvents used for the synthesis were “p.a.” grade.  Tetrahydrofuran was further 
+distilled from calcium hydride to obtain sufficiently dry solvent. 1H NMR spectra were 
+recorded on Varian VNMRS300 instrument; deuteriochloroform (CDCl3) and 
+hexadeuteriodimethyl sulfoxide (DMSO-d6) were used as solvents and the signals of the 
+solvent served as an internal standard. Chemical shifts () are given in ppm and J values are 
+given in Hz. Elemental analyses were carried out on Elementar vario EL III instrument. The 
+purity of all final compounds was checked by HPLC analysis (high-pressure pump ECOM 
+Alpha; column WATREX Biospher Si 100, 250 × 4 mm, 5 m; detector WATREX UVD 
+250) and were found to be >99.8 %. Column chromatography was carried out using Merck 
+Kieselgel 60 (60−100 μm). 
+
+2 
+ 
+Synthesis of materials started from commercial 4-aminosalicylic acid (1, see Scheme 
+1). Its amino group was protected by acetylation and the carboxylic group was protected by 
+alkylative esterification by methyl iodide, so as neither of the two groups interfere with the 
+alkylation of phenolic hydroxyl. Protected derivative 2 was then alkylated by 1-
+bromoalkanes to get a series of alkyl homologues 3-n. In the next steps, the acetyl group 
+was cleaved by acidic hydrolysis under mild conditions and the liberated amino group was 
+alkylated by dimethyl sulphate yielding the key intermediate, acid 4-n. The lowest alkyl 
+homologue (4-1) was synthesised directly from acid 1 by alkylation with the excess of 
+dimethyl sulphate. The second part of the molecular core was synthesised from 4-
+hydroxybenzoic acid (5), which was protected by the reaction with 3,4-dihydro-2H-pyrane 
+and reacted with 4-nitrophenol in a DCC-mediated esterification. The protected hydroxyl 
+group was then liberated by the treatment with p-toluenesulfonic acid. The final step of the 
+synthesis was esterification of acids 4-n with phenol 6 mediated by EDC.  
+ 
+ 
+Scheme 1. 
+Synthetic procedures for the preparation of target compounds NFn. 
+ 
+1.2. 
+Synthetic procedures 
+Methyl 4-acetamido-2-hydroxybenzoate (2) 
+Acetic hydride (30 mL, 0.31 mol) was added dropwise to the suspension of powdered 4-
+aminosalicylic acid (20.0 g, 0.13 mol) in acetonitrile (250 mL). The reaction mixture was 
+stirred for 2 h and the resulting suspension filtered. The filter cake was washed by the small 
+amount of acetonitrile to remove the residue of acetic acid and dried in a vacuum dryer at 
+40 °C. Yield 24.77 g (96 %). 
+
+3 
+ 
+Dry 4-acetamidosalicylic acid (24.0 g, 0.12 mol) was dissolved in DMF. Powdered KHCO3 
+was added with stirring, resulting in CO2 evolution. Then methyl iodide was added dropwise 
+and the reaction mixture stirred for 6 h under anhydrous conditions (CaCl2 tube). The 
+resulting suspension was poured into water and neutralised with concentrated HCl. White 
+precipitate was filtered off and crystallised from 50% aqueous methanol. Yield 24.43 g 
+(95 %). 1H NMR (DMSO-d6) : 10.23 (1 H, s), 7.71 (1 H, d, J=8.8 Hz), 7.37 (1 H, d, J=1.8 
+Hz), 7.05 (1 H, dd, J=8.8, 2.3 Hz), 3.85 (3 H, s), 2.07 (3 H, s). 
+ 
+General procedure for alkylation of benzoate 2 
+Benzoate 2 was dissolved in dry DMF and powdered K2CO3 and KI (omitted if iodoalkane 
+was used) were added with stirring. Mixture was heated to 50 °C and 1-bromoalkane was 
+added. Reaction was stirred at 50 °C under anhydrous conditions (CaCl2 tube) for 10 h. The 
+cooled resulting mixture was poured into cold water, neutralised with concentrated HCl and 
+the precipitated product was filtered off and crystallised from ethanol.  
+Methyl 4-acetamido-2-ethoxybenzoate (3-1) 
+The reaction of benzoate 2 (5.0 g, 23.90 mmol) with ethyl iodide (5.45 g, 39.32 mmol) in 
+the presence of K2CO3 (5.0 g, 36.18 mmol) in dry DMF (50 mL) yielded 4.83 g (85 %) of 
+3a. 1H NMR (CDCl3) : 11.25 (1 H, br. s.), 8.36 (1 H, d, J=2.3 Hz), 7.94 (1 H, d, J=9.4 Hz), 
+6.58 (1 H, dd, J=9.1, 2.6 Hz), 4.11 (2 H, q, J=7.0 Hz), 3.89 (3 H, s), 2.23 (3 H, s), 1.42 (3 
+H, t, J=7.0 Hz). 
+Methyl 4-acetamido-2-propoxybenzoate (3-2) 
+The reaction of benzoate 2 (5.0 g, 23.90 mmol) with 1-bromopropane (8.71 g, 70.82 mmol) 
+in the presence of K2CO3 (10.0 g, 72.36 mmol) in dry DMF (60 mL) yielded 3.53 g (58 %) 
+of 3b. 1H NMR (CDCl3) :10.95 (1 H, s), 7.79 (1 H, d, J=8.8 Hz), 7.60 (1 H, d, J=2.3 Hz), 
+6.81 (1 H, dd, J=8.8, 2.3 Hz), 4.09 (2 H, t, J=6.5 Hz), 1.81 - 2.05 (2 H, m), 1.10 (3 H, t, 
+J=7.3 Hz). 
+Methyl 4-acetamido-2-butoxybenzoate (3-c) 
+The reaction of benzoate 2 (5.0 g, 23.90 mmol) with 1-bromobutane (4.53 g, 32.40 mmol) 
+in the presence of K2CO3 (5.0 g, 36.18 mmol) in dry DMF (50 mL) yielded 5.42 g (86 %) 
+of 3c. 1H NMR (CHLOROFORM-d)  ppm 10.85 (1 H, d, J=8.8 Hz), 7.77 (1 H, s), 7.60 (1 
+H, d, J=2.3 Hz 6.79 (1 H, dd, J=8.8, 2.3 Hz), 4.04 (2 H, t, J=6.5 Hz), 3.86 (2 H, s), 2.19 (3 
+H, s), 1.72 - 1.93 (2 H, m), 1.42 - 1.61 (2 H, m), 0.97 (3 H, t, J=7.3 Hz). 
+Methyl 4-acetamido-2-(pentyloxy)benzoate (3-d) 
+The reaction of benzoate 2 (10.0 g, 47.80 mmol) with 1-iodopentane (19.88 g, 98.37 mmol) 
+in the presence of K2CO3 (15.0 g, 0.11 mol) in dry DMF (100 mL) yielded 9.81 g (75 %) of 
+3d. 1H NMR (CDCl3) :10.86 (1 H, s), 7.80 (1 H, d, J=8.8 Hz), 7.58 (1 H, d, J=2.3 Hz), 6.79 
+(1 H, dd, J=8.8, 2.3 Hz), 4.01 (2 H, t, J=6.7 Hz), 1.77 - 2.03 (2 H, m), 1.31 - 1.57 (4 H, m), 
+0.91 (3 H, t, J=7.3 Hz). 
+Methyl 4-acetamido-2-(hexyloxy)benzoate (3-e) 
+The reaction of benzoate 2 (10.0 g, 47.80 mmol) with 1-bromohexane (15.78 g, 
+95.60 mmol) in the presence of K2CO3 (15.0 g, 0.11 mol) in dry DMF (100 mL) yielded 
+
+4 
+ 
+9.73 g (69 %) of 3e. 1H NMR (CDCl3) : 10.83 (1 H, s), 7.79 (1 H, d, J=8.8 Hz), 7.59 (1 H, 
+d, J=2.3 Hz), 6.80 (1 H, dd, J=8.8, 2.3 Hz), 4.02 (2 H, t, J=6.7 Hz), 3.86 (3 H, s), 2.19 (3 H, 
+s), 1.71 - 1.91 (2 H, m), 1.42 - 1.56 (2 H, m), 1.20 - 1.41 (4 H, m), 0.91 (3 H, t, J=7.3 Hz). 
+ 
+General procedure for deacetylation of amino group 
+Methyl 4-acetamido-2-(alkoxy)benzoate 3 was dissolved in methanol at 50 °C and the 
+concentrated H2SO4 was carefully added dropwise. The reaction mixture was stirred at 50 
+°C for 30 min and then poured into cold water and neutralised with NaOH. The neutral 
+dispersion of the product in water was extracted with ethyl acetate, combined organic layers 
+were washed with water and brine. After drying with anhydrous MgSO4, the solvent was 
+removed on rotary evaporator and the residue was purified by column chromatography on 
+silica gel to yield methyl 2-(alkoxy)-4-aminobenzoate as an intermediate. 
+NOTE: The presence of water in the deacetylation reaction (e.g. use of diluted H2SO4) leads 
+to considerable amounts of decarboxylation byproduct. 
+ 
+General procedure for methylation of amino group 
+Methyl 2-(alkoxy)-4-aminobenzoate was dissolved in DMSO and powdered K2CO3 was 
+added with stirring. A mixture was heated to 50 °C and dimethyl sulphate was added 
+dropwise. The reaction mixture was stirred at the same temperature and under anhydrous 
+conditions (CaCl2 tube) overnight. The progress of the reaction was monitored using TLC 
+(CH2Cl2-acetone 95 : 5). The resulting mixture was filtered and solid Na2S was added to the 
+filtrate. The mixture was stirred for 4 h at room temperature and then poured into water. 
+After 30 min of standing, the solution was neutralised with the concentrated acetic acid and 
+the precipitated product was collected by filtration. A crude product was purified by column 
+chromatography on silica gel and crystallised from methanol. 
+4-(Dimethylamino)-2-methoxybenzoic acid (4-1) 
+Compound 4a was synthesised by direct methylation of acid 1 using the general procedure 
+for methylation of amino group described above. Reaction of benzoic acid 1 (10.0 g, 
+65.30 mmol) with dimethyl sulphate (42.45 g, 0.33 mol) in the presence of K2CO3 (50.0 g, 
+0.36 mol) in DMSO (150 mL) and subsequent treatment with Na2S (16.0 g, 0.21 mol) 
+yielded 10.01 g (79 %) of 4-1. 
+4-(Dimethylamino)-2-ethoxybenzoic acid (4-2) 
+Following the general procedure, starting from methyl 4-acetamido-2-ethoxybenzoate 3-1 
+(4.80 g, 20.23 mmol), which was deacylated using H2SO4 (5.0 mL, 96%) in methanol 
+(50 mL). The free amine was methylated using dimethyl sulphate (10.52 g, 80.90 mmol) and 
+K2CO3 (12.0 g, 86.82 mmol) in DMSO (50 mL) followed by the treatment with Na2S 
+(1.60 g, 20.50 mmol) yielded 2.29 g (54 %) of 4-2. 1H NMR (CDCl3) : 10.71 (1 H, br. s.), 
+7.99 (1 H, d, J=9.4 Hz), 6.37 (1 H, dd, J=8.8, 2.3 Hz), 6.10 (1 H, d, J=2.3 Hz), 4.29 (2 H, q, 
+J=7.0 Hz), 3.05 (6 H, s), 1.55 (3 H, t, J=7.0 Hz). 
+4-(Dimethylamino)-2-propoxybenzoic acid (4-3) 
+Using the described general procedure: 4-acetamido-2-propoxybenzoate 3-2 (3.50 g, 
+13.93 mmol) was deacylated using H2SO4 (3.5 mL, 96%) in methanol (40 mL). Liberated 
+
+5 
+ 
+amine was methylated using dimethyl sulphate (7.24 g, 55.68 mmol) and K2CO3 (7.70 g, 
+55.71 mmol) in DMSO (50 mL) followed by the treatment with Na2S (1.10 g, 14.09 mmol) 
+yielded 1.91 g (61 %) of 4-3. 1H NMR (CDCl3) : 10.73 (1 H, br. s.), 8.01 (1 H, d, J=9.4 
+Hz), 6.42 (1 H, dd, J=9.1, 2.1 Hz), 6.19 (1 H, d, J=1.8 Hz), 4.19 (2 H, t, J=6.5 Hz), 3.07 (6 
+H, s), 1.95 (2 H, sext.,  7.2 Hz), 1.10 (3 H, t, J=7.3 Hz) 
+2-Butoxy-4-(dimethylamino)benzoic acid (4-4) 
+Using the general deacetylation and alkylation protocol: 4-acetamido-2-butoxybenzoate       
+3-3 (5.30 g, 19.98 mmol) was deacylated using H2SO4 (5.0 mL, 96%) in methanol (50 mL). 
+Liberated amine was methylated using dimethyl sulphate (10.40 g, 79.98 mmol) and K2CO3 
+(11.50 g, 83.21 mmol) in DMSO (50 mL) followed by the treatment with Na2S (1.60 g, 
+20.50 mmol) yielded 2.42 g (51 %) of 4-4. 1H NMR (CDCl3) : 10.71 (1 H, br. s.), 8.00 (1 
+H, d, J=8.8 Hz), 6.40 (1 H, dd, J=8.8, 2.3 Hz), 6.17 (1 H, d, J=2.3 Hz), 4.21 (2 H, t, J=6.7 
+Hz), 3.06 (6 H, s), 1.85 - 1.94 (2 H, m), 1.43 - 1.58 (2 H, m), 0.92 (3 H, , t, J=7.3 Hz). 
+4-(Dimethylamino)-2-(pentyloxy)benzoic acid (4-5) 
+Using the above mentioned general protocols: 4-acetamido-2-(pentyloxy)benzoate 3-4 
+(9.50 g, 34.76 mmol) was deacylated using H2SO4 (5.0 mL, 96%) in methanol (150 mL). 
+Liberated amine was methylated using dimethyl sulphate (27.12 g, 0.21 mol) and K2CO3 
+(29.0 g, 0.21 mol) in DMSO (150 mL) followed by the treatment with Na2S (10.90 g, 
+0.14 mol) yielded 5.41 g (62 %) of 4-3. 1H NMR (CDCl3) : 10.70 (1 H, br. s.), 8.00 (1 H, 
+d, J=8.8 Hz), 6.41 (1 H, dd, J=8.8, 2.3 Hz), 6.18 (1 H, d, J=2.3 Hz), 4.21 (2 H, t, J=6.7 Hz), 
+3.06 (6 H, s), 1.82 - 1.99 (2 H, m), 1.31 - 1.55 (4 H, m), 0.94 (3 H, , t, J=7.3 Hz). 
+4-(Dimethylamino)-2-(hexyloxy)benzoic acid (4-6) 
+Using the above mentioned general protocols: 4-acetamido-2-(hexyloxy)benzoate 3-3 
+(9.50 g, 32.38 mmol) was deacylated using H2SO4 (5.0 mL, 96%) in methanol (150 mL). 
+Liberated amine was methylated using dimethyl sulphate (21.10 g, 0.16 mol) and K2CO3 
+(23.0 g, 0.17 mol) in DMSO (150 mL) followed by the treatment with Na2S (7.20 g, 
+0.10 mol) yielded 4.82 g (56 %) of 4-6. 1H NMR (CDCl3) : 10.72 (1 H, br. s.), 7.99 (1 H, 
+d, J=8.8 Hz), 6.37 (1 H, dd, J=8.8, 2.3 Hz), 6.11 (1 H, d, J=2.3 Hz), 4.20 (2 H, t, J=6.7 Hz), 
+3.06 (6 H, s), 1.85 - 1.99 (2 H, m), 1.21 - 1.51 (6 H, m), 0.91 (3 H, , t, J=7.3 Hz). 
+4-Nitrophenyl 4-hydroxybenzoate (6) 
+3,4-Dihydro-2H-pyrane (14.30 g, 0.17 mol) was added dropwise to the suspension of 4-
+hydroxybenzoic acid (13.80 g, 0.10 mol) in diethylether (200 ml). The reaction mixture was 
+stirred overnight under anhydrous conditions (CaCl2 tube) and then filtered. The filter cake 
+contained the majority of desired 4-((tetrahydro-2H-pyran-2-yl)oxy)benzoic acid. The 
+filtrate was vigorously stirred with aqueous NaOH (80 ml, 10%) for 30 min, and then the 
+aqueous layer was separated and neutralised by HCl. The pH was further adjusted to ca. 4 
+using acetic acid. The precipitated solid was collected, washed with cold water and dried 
+under vacuum and finally combined with the dry portion obtained from the filter cake. 
+ 
+4-((Tetrahydro-2H-pyran-2-yl)oxy)benzoic acid (31.35 g, 0.14 mol) and 4-nitrophenol 
+(19.60 g, 0.14 mol) were dissolved in dry THF (250 mL) and cooled to ca. 10 °C. Then N,N´-
+dicyclohexylcarbodiimide (DCC, 30.60 g, 0.15 mol ) and 4-(dimethylamino)-pyridine 
+(DMAP, 5.60 g, 46.22 mmol) were added and the reaction mixture was stirred under 
+
+6 
+ 
+anhydrous conditions for 12 h. The precipitated N,N´-dicyclohexylurea was filtered off and 
+the filtrate diluted with ethyl acetate (100 mL). The resulting solution was washed with 
+diluted HCl (100 mL, 1 : 15), then with water and the solvents were removed on rotary 
+evaporator. The solid residue was dissolved in CHCl3-methanol mixture (1 : 1) and 
+toluenesulfonic acid (4.0 g, 23.22 mmol) was added. The reaction mixture was stirred at 
+45 °C for 1 h and then evaporated to dryness on rotary evaporator. A crude product was 
+crystallised from acetone. Yield 28.71 g (66 %). 1H NMR (DMSO-d6)  8.33 (2 H, d, J=8.6 
+Hz), 8.08 (2 H, d, J=8.8 Hz), 7.58 (2 H, d, J=8.6 Hz), 7.20 (2 H, d, J=8.8 Hz), 5.62 – 5.65 
+(1 H, m), 3.63 - 3.75 (1 H, m), 3.48 - 3.60 (1 H, m), 1.21 - 1.97 (6 H, m). 
+ 
+General procedure for EDC-mediated esterification 
+2-(Alkoxy)-4-(dimethylamino)benzoic acid 4-n and 4-nitrophenyl 4-hydroxybenzoate (6) 
+ were suspended in dry dichloromethane (50 ml) and cooled to 2 – 8 °C in ice-water bath. 
+Then 
+N-(3-dimethylaminopropyl)-N′-ethylcarbodiimide 
+hydrochloride 
+(EDC) 
+and 
+4-(N,N-dimethylamino)pyridine (DMAP) (0.1 g, 0.82 mmol) were added. The reaction 
+mixture was stirred for 2 hours under anhydrous conditions and the temperature was let rise 
+as ice in the cooling bath melted. The resulting solution diluted with CH2Cl2 and washed 
+with water and brine. Organic layer was dried over anhydrous magnesium sulphate and 
+evaporated on the rotary evaporator. The residue was purified by column chromatography 
+on silica gel in CH2Cl2-acetone eluent and recrystallised from acetone. 
+4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-methoxybenzoate (NF1) 
+4-(Dimethylamino)-2-methoxybenzoic acid (4-1, 78.1 mg, 0.40 mmol) was esterified with 
+4-nitrophenyl 4-hydroxybenzoate (6, 104.5 mg, 0.40 mmol) using EDC (81 mg, 0.42 mmol) 
+and DMAP (51.0 mg, 0.42 mmol) in dichloromethane (2.0 mL) as described in general 
+procedure. Yield 92.5 mg (53 %). 1H NMR (CDCl3) : 8.33 (2 H, d, J=8.8 Hz), 8.24 (2 H, 
+d, J=8.8 Hz), 8.03 (1 H, d, J=9.4 Hz), 7.41 (4 H, dd, J=14.1, 8.8 Hz), 6.33 (1 H, dd, J=8.8, 
+2.3 Hz), 6.16 (1 H, d, J=2.3 Hz), 3.95 (3 H, s), 3.11 (6 H, s). Anal. calcd. for C23H20N2O7: 
+C 63.30, H 4.62, N 6.42; found C 63.86, H 4.68, N 6.47 %. 
+4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-ethoxybenzoate (NF2) 
+4-(Dimethylamino)-2-methoxybenzoic acid (4-1)  
+The reaction of 4-(dimethylamino)-2-ethoxybenzoic acid (4-2, 1.0 g, 4.78 mmol) was 
+esterified with 4-nitrophenyl 4-hydroxybenzoate (6, 1.24 g, 4.78 mmol) using EDC (1.0 g, 
+5.16 mmol) and DMAP (0.29 g, 2.39 mmol) in dichloromethane (30 mL) yielded 1.03 g 
+(48 %). 1H NMR (CDCl3) : 8.33 (2 H, d, J=9.4 Hz), 8.24 (2 H, d, J=8.2 Hz), 8.01 (1 H, d, 
+J=9.4 Hz), 7.33 - 7.48 (4 H, m), 6.32 (1 H, dd, J=8.8, 2.3 Hz), 6.16 (1 H, d, J=1.8 Hz), 4.15 
+(2 H, d, J=7.0 Hz), 3.08 (6 H, s), 1.49 (3 H, t, J=7.0 Hz). 13C{H} NMR (CDCl3) : 163.72 
+(s), 163.10 (s), 162.29 (s), 156.46 (s), 155.73 (s), 155.24 (s), 145.31 (s), 134.35 (s), 131.79 
+(s), 125.24 (s), 125.03 (s), 122.67 (s), 122.62 (s), 104.59 (s), 103.90 (s), 95.65 (s), 64.41 (s), 
+40.11 (s), 14.77 (s). Anal. calcd. for C24H22N2O7: C 64.00, H 4.92, N 6.11; found C 63.87, 
+H 4.98, N 6.11 %. 
+4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-propoxybenzoate (NF3) 
+Starting from 4-(dimethylamino)-2-propoxybenzoic acid (4-3, 1.25 g, 5.78 mmol) and 4-
+nitrophenyl 4-hydroxybenzoate (6, 1.50 g, 5.78 mmol) with EDC (1.18 g, 6.03 mmol) and 
+
+7 
+ 
+DMAP (0.68 g, 5.61 mmol) in dichloromethane (50 mL) yielded 1.36 g (51 %). 1H NMR 
+(CDCl3) : 8.34 (2 H, d, J=8.8 Hz), 8.24 (2 H, d, J=8.8 Hz), 8.00 (1 H, d, J=9.4 Hz), 7.33 - 
+7.50 (4 H, m), 6.32 (1 H, dd, J=8.8, 2.3 Hz), 6.15 (1 H, d, J=2.3 Hz), 4.04 (2 H, t, J=6.5 Hz), 
+3.09 (6 H, s), 1.81 - 1.98 (2 H, m), 1.08 (3 H, t, J=7.3 Hz). 13C{H} NMR (CDCl3) : 163.73 
+(s), 163.29 (s), 162.34 (s), 156.54 (s), 155.74 (s), 155.27 (s), 145.36 (s), 134.46 (s), 131.84 
+(s), 125.26 (s), 125.04 (s), 122.65 (s), 122.60 (s), 104.66 (s), 103.84 (s), 95.45 (s), 70.18 (s), 
+40.11 (s), 22.64 (s), 10.68 (s). Anal. calcd. for C25H24N2O7: C 64.65, H 5.21, N 6.03; found 
+C 64.56, H 5.19, N 5.98 %. 
+4-[(4-Nitrophenoxy)carbonyl]phenyl 2-butoxy-4-(dimethylamino)benzoate (NF4) 
+Esterification of 2-butoxy-4-(dimethylamino)benzoic acid (4-4, 2.0 g, 8.43 mmol) with 4-
+nitrophenyl 4-hydroxybenzoate (6, 2.50 g, 9.64 mmol) using EDC (2.0 g, 10.22 mmol) and 
+DMAP (0.58 g, 4.78 mmol) in dichloromethane (70 mL) yielded 2.31 g (51 %). 1H NMR 
+(CDCl3) : 8.34 (2 H, d, J=9.4 Hz), 8.24 (2 H, d, J=8.2 Hz), 8.00 (1 H, d, J=8.8 Hz), 7.31 - 
+7.50 (4 H, m), 6.32 (1 H, dd, J=8.8, 2.3 Hz), 6.16 (1 H, d, J=1.8 Hz), 4.08 (2 H, t, J=6.5 Hz), 
+3.09 (6 H, s), 1.74 - 1.96 (2 H, m), 1.42 - 1.66 (2 H, m), 0.95 (3 H, t, J=7.3 Hz). 13C{H} 
+NMR (CDCl3) : 163.73 (s), 163.33 (s), 162.30 (s), 156.52 (s), 155.74 (s), 155.25 (s), 145.34 
+(s), 134.47 (s), 131.82 (s), 125.25 (s), 125.02 (s), 122.66 (s), 122.60 (s), 104.65 (s), 103.83 
+(s), 95.43 (s), 68.35 (s), 40.15 (s), 31.30 (s), 19.25 (s), 13.85 (s). Anal. calcd. for C26H26N2O7:  
+C 65.26, H 5.48, N 5.85; found C 65.15, H 5.16, N 5.80 %. 
+4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-(pentyloxy)benzoate (NF5) 
+Following the general procedure above 4-(dimethylamino)-2-propoxybenzoic acid (4-3, 
+2.0 g, 7.96 mmol) and 4-nitrophenyl 4-hydroxybenzoate (6, 2.06 g, 7.94 mmol) were 
+reacted in the presence of EDC (1.68 g, 8.59 mmol) and DMAP (0.50 g, 4.13 mmol) in 
+dichloromethane (70 mL) yielded 1.76 g (45 %). 1H NMR (CDCl3) : 8.33 (2 H, d, J=9.2 
+Hz), 8.24 (2 H, d, J=8.6 Hz), 7.99 (1 H, d, J=9.2 Hz), 7.32 - 7.49 (4 H, m), 6.32 (1 H, dd, 
+J=8.9, 2.3 Hz), 6.15 (1 H, d, J=2.0 Hz), 4.07 (2 H, t, J=6.6 Hz), 3.08 (6 H, s), 1.79 - 1.94 (2 
+H, m), 1.26 - 1.55 (4 H, m), 0.83 - 0.94 (3 H, m). 13C{H} NMR (CDCl3) : 163.73 (s), 163.33 
+(s), 162.30 (s), 156.52 (s), 155.74 (s), 155.25 (s), 145.34 (s), 134.47 (s), 131.82 (s), 125.25 
+(s), 125.02 (s), 122.66 (s), 122.60 (s), 104.65 (s), 103.83 (s), 95.43 (s), 68.67 (s), 40.13 (s), 
+28.92 (s), 28.17 (s), 22.42 (s), 14.00 (s). Anal. calcd. for C27H28N2O7: C 65.84, H 5.73, N 
+5.69; found C 65.59, H 5.78, N 5.65 %. 
+4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-(hexyloxy)benzoate (NF6) 
+The reaction of 4-(dimethylamino)-2-ethoxybenzoic acid (4-2, 2.10 g, 7.91 mmol) was 
+esterified with 4-nitrophenyl 4-hydroxybenzoate (6, 2.10 g, 8.10 mmol) using EDC (1.70 g, 
+8.69 mmol) and DMAP (0.96 g, 7.92 mmol) in dichloromethane (70 mL) yielded 1.63 g 
+(41 %). 1H NMR (CDCl3) : 8.32 (2 H, d, J=9.4 Hz), 8.23 (2 H, d, J=8.8 Hz), 7.99 (1 H, d, 
+J=9.4 Hz), 7.31 - 7.52 (4 H, m), 6.32 (1 H, dd, J=8.8, 2.3 Hz), 6.15 (1 H, d, J=1.8 Hz), 4.07 
+(2 H, t, J=6.7 Hz), 1.77 - 1.93 (2 H, m), 1.41 - 1.57 (2 H, m), 1.18 - 1.39 (4 H, m), 0.76 - 
+0.94 (3 H, m). 13C{H} NMR (CDCl3) : 163.70 (s), 163.32 (s), 162.27 (s), 156.54 (s), 155.74 
+(s), 155.25 (s), 145.35 (s), 134.46 (s), 131.77 (s), 125.23 (s), 125.01 (s), 122.64 (s), 122.56 
+(s), 104.69 (s), 103.85 (s), 95.48 (s), 68.70 (s), 40.07 (s), 31.52 (s), 29.20 (s), 25.69 (s), 22.53 
+(s), 13.99 (s). Anal. calcd. for C28H30N2O7: C 66.39, H 5.97, N 5.53; found C 66.21, H 5.90, 
+N 5.49 %. 
+
+8 
+ 
+1.3. 
+Equipment and apparatus 
+The compounds were studied by differential scanning calorimetry (DSC). Perkin-
+Elmer 7 Pyris calorimeter (Perkin Elmer, Shelton, CT, USA) was utilised and the 
+measurements were conducted on cooling/heating runs at a rate of 10 K/min. The 
+calorimeter was calibrated to the extrapolated onsets for the melting points of water, indium 
+and zinc. A small amount of the studied compound (2-5 mg) was sealed into an aluminium 
+pan and put into the calorimeter chamber. A nitrogen medium was utilised during the 
+calorimetric measurements. The phase transition temperatures and the corresponding 
+enthalpies were established from the second heating and the subsequent cooling runs.  
+Textures were observed under the polarising microscope Eclipse E600Pol (Nikon, 
+Tokyo, Japan). We analysed the samples in various geometries. Two kinds of commercial 
+cells were purchased with the thickness of 5 m: HG cells with homogeneous anchoring 
+(orienting molecules parallel to the cell surface) and HT cells with surfactant adjusting 
+homeotropic arrangement of molecules (perpendicular to the surface). These cells consist of 
+glasses with ITO transparent electrodes and materials were filled in the isotropic phase by 
+capillary action. The Linkam E350 heating/cooling stage with TMS 93 temperature 
+programmer (Linkam, Tadworth, UK) was utilised, with the temperature stabilisation  within 
+±0.1 K.  
+The switching current profile versus time was detected by a digital oscilloscope 
+Tektronix DPO4034 (Tektronix, Beaverton, OR, USA). Polarisation, P, was determined by 
+the integration of the current profile when the electric field of triangular modulation at a 
+frequency of 10 Hz was applied with the magnitude of 10 V/m. 
+We measured the dielectric spectroscopy by Schlumberger 1260 impedance analyser 
+(Schlumberger, Houston, TX, USA) and stabilised the temperature within ±0.1 K during the 
+frequency sweeps in a range of 1 Hz ÷ 1 MHz. The permittivity, (f) =−i which is 
+frequency dependent, was analysed with support of a modified version of the Cole-Cole 
+formula: 
+                    
+)
+2
+(
+)
+(
+1
+0
+)
+1
+(
+*
+m
+n
+r
+Af
+f
+i
+f
+if
++
+−
++
+
+=
+−
+−
+
+
+
+
+
+
+
+            (1), 
+where fr is the relaxation frequency,  is the dielectric strength,  is the distribution 
+parameter of relaxation,   is the permittivity of vacuum,  is the high frequency 
+permittivity, n, m, and A are the parameters of fitting. In formula (1) an ionic conductivity 
+and ITO electrode effects were taken into consideration. The measured values of the real 
+part of the permittivity,  and the imaginary part,  were simultaneously fitted to obtain 
+the parameters fr and .  
+The polarisation current profile of electric field was detected by Tektronix DPO4034 
+digital oscilloscope (Tektronix, Oregon, US). The driving voltage from a generator (Agilent, 
+California, US) was amplified by a linear amplifier providing the amplitude up to ±120 V.  
+The temperature-dependent second harmonic generation (SHG) measurements were 
+conducted using an optical setup based on Ti:sapphire femtosecond laser (Spitfire ACE), 
+
+9 
+ 
+which was amplified to produce 40 fs long pulses with 5 kHz repetition rate and central 
+wavelength of 800 nm. For SHG we utilised HG cells and placed them into a Linkam stage, 
+the temperature was stabilised with an accuracy ±0.1 K. The samples were illuminated by a 
+collimated beam with pulses fluence of approximately 0.01 mJ/cm2. The SHG signal 
+generated in transmission configuration was appropriately filtered, then detected by an 
+avalanche photodiode and amplified using a lock-in amplifier. The scheme of SHG 
+measurements is shown in Figure S1. 
+For the x-ray studies, the Bruker D8 GADDS system was utilised: parallel CuK 
+beam formed by Goebel mirror monochromator, 0.5 mm collimator, modified Linkam 
+heating stage, Vantec 2000 area detector. The samples for the diffraction experiments were 
+prepared in a form of droplets on heated surface. 
+ 
+Figure S1. 
+SHG measurement scheme. 
+ 
+2. 
+Mesomorphic properties   
+ 
+Clanek o kapalnych krystalech – Vladka Novotna
+Chopper
+Mirror
+800 nm
+Filters
+ND, High pass
+Mirror
+400 nm
+Lens
+Avalanche 
+photodiode
+Sample in HG cell
+Linkam stage
+Boxcar/Lock-in
+Lock-in 
+reference
+Temperature controller
+
+10 
+ 
+Figure S2. 
+Vitrification process and creation of a fibre after melting of NF5.  
+ 
+ 
+Figure S3. 
+The microphotograph of the texture for homologue NF2 detected in a 5 m 
+HG cell. The width of the photo corresponds to about 200m. 
+ 
+Figure S4. 
+The microphotograph of NF4 homologue in 5 m HG-A cell. The Polariser 
+orientation (white) and the rubbing direction (red) are marked. 
+
+R
+50μm11 
+ 
+ 
+Figure S5 
+The texture of NF5 in 5 m HG-A cell under a microscope with (a) crossed 
+polarisers, (b) and (c) with uncrossed position of polarisers (analyser rotated by an angle 
+about 20 degrees). All figures show the same part of the sample; red arrow represents the 
+rubbing direction and white arrows indicate the orientation of polarisers. 
+  
+Figure S6. 
+The microphotograph of NF6 homologue in 5 m HG-P cell. The rubbing 
+direction, R, is marked with a red line.  
+ 
+
+(a)
+(b)
+C
+50μm
+R50um12 
+ 
+ 
+Figure S7. 
+The photo of NF6 homologue in 5 m HG-A cell after the application of the 
+external electric field of about 2 V/m. The rubbing direction, R, is marked with a red line.  
+ 
+Figure S8. 
+The texture of NF6 in a special home-made gap-cell with a thickness of about 
+35 m, defined by two copper electrodes. One electrode is located at the right upper corner 
+out of the figure; the orientation of the applied electric field, E, is marked with the black 
+arrow. For (a) no electric field was applied and for (b) the electric field of about 0.2 V/m 
+was applied.  
+
+R
+P
+50μm(a)
+(b)
+E13 
+ 
+ 
+Figure S9. 
+3D-plot of (a) real, ’, and (b) imaginary, ’’, parts of the permittivity versus 
+frequency and temperature, T, for compound NF2. Dielectric measurements were performed 
+in 12 m cell with gold electrodes and no surfactant layer. 
+ 
+
+(a)
+25000
+20000
+15000
+3
+10000
+5000
+102
+40
+frequency (Hz)
+103
+60
+104
+80
+100
+105
+120
+(°C)
+T
+140
+106
+160
+(b)
+8000
+6000
+3
+4000
+2000
+Q
+102×
+40
+frequency (Hz)
+103×
+60
+80
+104
+100
+105
+120
+140
+T (C)
+106
+16014 
+ 
+ 
+Figure S10. 
+3D-plot of (a) real, ’, and (b) imaginary, ’’, parts of the permittivity versus 
+frequency and temperature, T, for compound NF3. Dielectric measurements were performed 
+in 12 m cell with gold electrodes and no surfactant layer. 
+ 
+ 
+
+(a)
+20000-
+15000
+000013
+5000
+60
+80
+101
+100
+102
+120
+(0d
+103
+frequency (Hz)
+104
+105
+140
+106
+(b)
+6000
+5000
+4000
+3
+3000
+2000
+1000
+60
+80
+101
+102
+100
+103
+frequency (Hz)
+104
+120
+105
+140
+10615 
+ 
+ 
+Figure S11. 
+3D-plot of (a) real, ’, and (b) imaginary, ’’, parts of the permittivity versus 
+frequency and temperature, T, for compound NF6. Dielectric measurements were performed 
+in 12 m cell with gold electrodes and no surfactant layer. 
+ 
+ 
+
+(a)
+18000
+16000
+14000
+12000
+10000
+32
+8000
+6000
+4000
+2000
+30
+0
+40
+101
+50
+102
+°℃)
+103
+60
+frequency (Hz)
+104
+70
+105
+80
+106
+(b)
+8000
+6000
+4000
+2000
+30
+40
+101
+50
+102
+103
+60
+frequency (Hz)
+104
+(0。
+70
+105
+106
+*8016 
+ 
+ 
+Figure S12. 
+Temperature dependences of the dielectric strength, , and the relaxation 
+frequency, fr, for NF5 in 12 m cell without surfactant layer. In the inset fr is presented in 
+the logarithmic scale versus reciprocal temperature, 1/T, in Kelvins and the activation energy 
+EA was established from the slope. 
+
+400
+14
+350
+12
+6
+300
+10
+5
+250
+.0)
+4
+8
+m
+3
+200
+(zH)
+6
+2
+=94 kJ/mol
+150
+0.0030
+0.0032
+4
+100
+10" RT (Jmol')
+2
+50
+0
+0
+30
+40
+50
+60
+70
+80
+T(C)17 
+ 
+ 
+Figure S13. 
+For NF5 at the temperature T=30° C (a) the x-ray intensity versus the 
+scattering angle, . (b) 2D pattern of the intensity at the same temperature. Scattering angles 
+are in the logarithmic scale. 
+ 
+Figure S14. 
+A model of NF1 molecule with the orientation of the dipole moment (blue 
+arrow). 
+
+(a)
+Intensity(arb.units)
+4.4 A
+10.4 A
+22.5 A
+0.1
+5
+10
+15
+20
+25
+30
+35
+(b)
+20(deg.)
+30
+28
+26
+24
+22
+20
+18
+16

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+arXiv:2301.00787v1  [gr-qc]  2 Jan 2023
+The Hamilton-Jacobi analysis for higher-order modified gravity
+Alberto Escalante∗ and Aldair Pantoja†
+Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla.
+Apartado Postal J-48 72570, Puebla Pue., M´exico,
+(Dated: January 3, 2023)
+The Hamilton-Jacobi [HJ] study for the Chern-Simons [CS] modification of general relativity
+[GR] is performed. The complete structure of the Hamiltonians and the generalized brackets are
+reported, from these results the HJ fundamental differential is constructed and the symmetries of
+the theory are found. By using the Hamiltonians we remove an apparent Ostrogradsky’s instability
+and the new structure of the hamiltonian is reported. In addition, the counting of physical degrees
+of freedom is developed and some remarks are discussed.
+PACS numbers: 98.80.-k,98.80.Qc
+I.
+INTRODUCTION
+It is well-known that GR is a successful framework for describing the classical behavior of the grav-
+itational field and its relation with the geometry of space-time [1–6]. From the canonical point of
+view, GR is a background independent gauge theory with diffeomorphisms invariance; the extended
+Hamiltonian is a linear combination of first class constraints and propagates two physical degrees
+of freedom [7]. From the quantum point of view, the quantization program of gravity is a difficult
+task to perform. In fact, from the nonperturbative scheme, the non-linearity of the gravitational
+field, manifested in the constraints, obscures the quantization making the complete description of
+a nonperturbative quantum theory of gravity still an open problem [8, 9]. On the other hand, the
+perturbative point of view of the path-integral method leads to the non-renormalizability problem
+[10, 11] with all the tools that have been developed in quantum field theory have not worked suc-
+cessfully. In this respect, it is common to study modified theories of gravity in order to obtain
+insights in the classical or quantum regime; with the expectation that these theories will provide
+new ideas or allow the development of new tools to carry out the quantization program, with an
+example of this being the so-called higher order theories [12–15]. In fact, higher-order theories are
+good candidates for fixing the infinities that appear in the renormalization problem of quantum
+gravity. It is claimed that adding higher order terms quadratic in the curvature to gravity could
+help avoid this problem; since these terms have a dimensionless coupling constant, which ensures
+∗Electronic address: [email protected]
+†Electronic address: [email protected]
+
+2
+that the final theory is divergence-free [16, 17]. The study of higher-order theories is a modern topic
+in physics, these theories are relevant in dark energy physics [18, 19], generalized electrodynamics
+[20–22] and string theories [23, 24]. Furthermore, an interesting model in four dimensions can be
+found in the literature, in which the Einstein-Hilbert [EH] action is extended by the addition of
+a Chern-Simons four-current coupled with an auxiliary field, thus, under a particular choice of the
+auxiliary field the resulting action will be a close model to GR [25]. In fact, at Lagrangian level
+the theory describes the propagation of two degrees of freedom corresponding to gravitational waves
+traveling with velocity c, but these propagate with different polarization intensities violating spatial
+reflection symmetry. Moreover, the Schwarzchild metric is a solution of the equations of motion,
+thus, the modified theory and the EH action share the same classical tests. On the other hand,
+at hamiltonian level the theory is a higher-order gauge theory [26] whose Hamiltonian analysis is
+known not to be easy to perform. In this respect, the analysis of constrained higher-order systems is
+usually developed by using the Ostrogradsky-Dirac [OD] [27–30] or the Gitman-Lyakhovich-Tyutin
+[GLT ] [31, 32] methods. OD scheme is based on the extension of the phase space by considering
+to the fields and their velocities as canonical coordinates and then introducing an extensi´on to their
+canonical momenta. However, the identification of the constraints is not easy to develop; in some
+cases, the constraints are fixed by hand in order to obtain a consistent algebra [33] and this yields
+the opportunity to work with alternative methods. On the other hand, the GLT framework is based
+on the introduction of extra variables which transforms a problem with higher time derivatives to
+one with only first-order ones then, by using the Dirac brackets the second class constraints and the
+extra variables can be removed [34].
+Nevertheless, there is an alternative scheme for analyzing higher-order theories:
+the so-called
+Hamilton-Jacobi method. The HJ scheme for regular field theories was developed by G¨uler [35, 36]
+and later extended for singular systems in [37, 38]. It is based on the identification of the constraints,
+called Hamiltonians. These Hamiltonians can be either involutive or non-involutive and they are used
+for constructing a generalized differential, where the characteristic equations, the gauge symmetries,
+and the generalized HJ brackets of the theory can be identified. It is important to remark that the
+identification of the Hamiltonians is performed by means of the null vectors, thus, the Hamiltonians
+will have the correct structure without fix them by hand as is done in other approaches, then the
+identification of the symmetries will be, in general, more economical than other schemes [39–43].
+With all of above the aims of this paper is to develop a detailed HJ analysis of the theory reported
+in [25].
+In fact, we shall analyze this model beyond the Lagrangian approach reported in [25];
+we shall see that the Jackiw-Yi [JY ] model is a higher-order theory and it is mandatory to study
+this theory due to its closeness with GR. However, it is well-known that in higher-order theories
+could be present ghost degrees of freedom associated to Ostrogradsky’s instabilities [44], namely,
+the hamiltonian function is unbounded and this is reflected with the presence of linear terms of the
+canonical momenta in the hamiltonian. In this respect, it is important to comment that if there are
+constraints, then it is possible to heal those instabilities [45, 46]; in our case the JY model will show
+an apparent Ostrogradsky’s instability since linear terms in the momenta will appear, however, we
+
+3
+will heal the theory by using the complete set of Hamiltonians, thereby exorcising the associated
+ghosts.
+The paper is organized as follows. In Sect. II, we start with the CS modification of GR, we will work
+in the perturbative context, say, we will expand the metric around the Minkowski background. We
+shall observe that the modified theory is of higher-order in the temporal derivatives, then we shall
+introduce a change of variables in order to express the action in terms of only first-order temporal
+derivatives. The change of variable will allows us to develop the HJ analysis in an easy way; the
+identification of the Hamiltonians, the construction of the generalized differential and the symmetries
+will be identified directly. In Sect. III we present the conclusions and some remarks.
+II.
+THE HAMILTON-JACOBI ANALYSIS
+The modified EH action is given by [25]
+S[gµν] =
+�
+M
+�
+R√−g + 1
+4θ∗Rσ
+τ
+µνRτ
+σµν
+�
+d4x,
+(1)
+where M is the space-time manifold, gµν the metric tensor, R the scalar curvature, g the determinant
+of the metric, Rαβµν the Riemman tensor and θ is a coupling field. In general, θ can be viewed as
+an external quantity or as a local dynamical variable, however, in order to obtain an action close to
+GR we are going to choose θ = t
+Ω. Along the paper we will use grek letters for labeling space-time
+indices µ = 0, 1, 2, 3 and latin letters for space indices i = 1, 2, 3. In addition, we will work within
+the perturbative context expanding the metric around the Minkowski background
+gµν = ηµν + hµν,
+(2)
+where hµν is the perturbation. By substituting the expression for θ and by taking into account eq.
+(2) in (1) we obtain the following linearized action
+S[hµν] = −1
+2
+�
+M
+hµν �
+Glin
+µν + Clin
+µν
+�
+d4x,
+(3)
+where Glin
+µν is the linearized version of the Einstein tensor and Clin
+µν is a linearized Cotton-type tensor
+Clin
+µν = − 1
+4Ω[ǫ0µλγ∂λ(□hγν − ∂ν∂αhαγ)+ ǫ0νλγ∂λ(□hγµ − ∂µ∂αhαγ)] [25] defined in four-dimensions.
+Now we shall suppose that the space-time has a topology M ∼= R × Σ, where R is an evolution
+parameter and Σ is a Cauchy hypersurface. Hence, by performing the 3 + 1 decomposition of the
+action (3) we write down the corresponding Lagrangian density
+L =
+� �1
+2
+˙hij ˙hij − ∂jh0i∂jh0i − 1
+2∂khij∂khij − 1
+2
+˙hii ˙hjj + ∂jh00∂jhii + 1
+2∂khii∂khjj − 2∂ih0i ˙hjj
+−∂ih00∂jhij − ∂ihij∂jhkk + 2∂jh0i ˙hij + ∂ihi0∂jh0j + ∂khki∂jhij + 1
+µǫijk(−¨hli∂jhlk
++2˙hli∂j∂lh0k + ∂lhmi∂m∂jhlk + ∇2h0i∂jh0k + ∇2hmi∂jhmk)
+�
+d3x,
+(4)
+where we have defined µ ≡ 2Ω and ǫijk ≡ ǫ0ijk. As it was commented above, we will reduce the
+order of the time derivatives of the Lagrangian (4) by extending the configuration space, this is done
+
+4
+by introducing the following change of variable
+Kij = 1
+2(˙hij − ∂ih0j − ∂jh0i),
+(5)
+here Kij is related with the so-called extrinsic curvature [47, 48]. Thus, by substituting (5) into (4)
+we rewrite the Lagrangian in the following new fashion
+L =
+� �
+2KijKij − 2KiiKjj − h00Rijij − hijRij + 1
+2hiiRijij + 1
+µǫijk(4Kil∂jKkl + ∂mhim∂j∂lhkl
++∇2him∂jhkm) + ψij(˙hij − ∂ih0j − ∂jh0i − 2Kij)
+�
+d3x,
+(6)
+where we have added the Lagrange multipliers ψij enforcing the the relation (5), and the expressions
+Rijij and Rij are defined in the following way
+Rijij ≡ ∂i∂jhij − ∇2hii,
+(7)
+Rij ≡ 1
+2(∂i∂khjk + ∂j∂khik − ∂i∂jhkk − ∇2hij).
+(8)
+Now, we calculate the canonical momenta associated with the dynamical variables
+π00 =
+∂L
+∂ ˙h00
+= 0,
+(9)
+π0i =
+∂L
+∂ ˙h0i
+= 0,
+(10)
+πij =
+∂L
+∂ ˙hij
+= ψij,
+(11)
+P ij =
+∂L
+∂ ˙Kij
+= 0,
+(12)
+Λij =
+∂L
+∂ ˙ψij
+= 0.
+(13)
+Thus, from the equations (9)-(13) we identify the following HJ Hamiltonians of the theory
+H′ ≡ H0 + Π = 0,
+(14)
+H00
+1
+≡ π00 = 0,
+(15)
+H0i
+2
+≡ π0i = 0,
+(16)
+Hij
+3
+≡ πij − ψij = 0,
+(17)
+Hij
+4
+≡ P ij = 0,
+(18)
+Hij
+5
+≡ Λij = 0,
+(19)
+where H0 is the canonical hamiltonian defined as usual H0 = ˙hµνπµν + ˙KijP ij + ˙ψijΛij − L and
+Π = ∂0S [39–43]. Moreover, the fundamental Poisson brackets [PB] between the canonical variables
+
+5
+are given by
+{hµν, παβ} = 1
+2(δα
+µδβ
+ν + δα
+ν δβ
+µ)δ3(x − y),
+(20)
+{Kij, πkl} = 1
+2(δk
+i δl
+j + δk
+j δl
+i)δ3(x − y),
+(21)
+{ψij, Λkl} = 1
+2(δi
+kδj
+l + δj
+kδi
+l)δ3(x − y).
+(22)
+Furthermore, in the HJ scheme, the dynamics of the system is governed by the fundamental differ-
+ential defined as
+dF = {F, HI}dωI,
+(23)
+where F is any function defined on the phase space, HI is the set of all Hamiltonians (14)-(19)
+and ωI are the parameters related to them. It is important to remark, that in the HJ method the
+Hamiltonians are classified as involutive and non-involutive. Involutive ones are those whose PB
+with all Hamiltonians, including themselves, vanish; otherwise, they are called non-involutive. Be-
+cause of integrability conditions, the non-involutive Hamiltonians are removed from the fundamental
+differential (23) by introducing the so-called generalized brackets, these new brackets are given by
+{f, g}∗ = {f, g} − {f, Ha′}C−1
+a′b′{Hb′, g},
+(24)
+where Ca′b′ is the matrix formed with the PB between all non-involutive Hamiltonians.
+From
+(14)-(19) the non-involutive Hamiltonians are Hij
+3 and Hij
+5 , whose PB is
+{Hij
+3 , Hij
+5 } = −1
+2(ηikηjl + ηilηkj)δ3(x − y),
+(25)
+therefore, the matrix Ca′b′ given by
+Ca′b′ =
+
+
+0
+− 1
+2(ηikηjl + ηilηkj)
+1
+2(ηikηjl + ηilηkj)
+0
+
+δ3(x − y),
+(26)
+and its inverse C−1
+a′b′ takes the form
+C−1
+a′b′ =
+
+
+0
+1
+2(ηikηjl + ηilηkj)
+− 1
+2(ηikηjl + ηilηkj)
+0
+
+ δ3(x − y).
+(27)
+In this manner, the following non-vanishing generalized brackets between the fields arise
+{hµν, παβ}∗ = 1
+2(δα
+µδβ
+ν + δβ
+µδα
+ν )δ3(x − y),
+(28)
+{Kij, P kl}∗ = 1
+2(δk
+i δl
+j + δl
+iδk
+j )δ3(x − y),
+(29)
+{hµν, ψαβ}∗ = 1
+2(δα
+µδβ
+ν + δβ
+µδα
+ν )δ3(x − y),
+(30)
+{ψij, Λkl}∗ = 0,
+(31)
+we observe from (31) that the canonical variables (ψij, Λkl) can be removed which implies that we
+can perform the substitution of πij = ψij and Λij = 0, hence, the canonical hamiltonian takes the
+
+6
+form
+H0 =
+�
+[2KiiKjj − 2KijKij + h00Rijij + hijRij − 1
+2hiiRijij − 1
+µǫijk(4Kil∂jKkl + ∂mhim∂j∂lhkl
++∇2him∂jhkm) − 2h0j∂iπij + 2Kijπij]d3x.
+(32)
+It is worth to comment, that the canonical hamiltonian has linear terms in the momenta πij and
+this fact could be related to Ostrogradsky’s instabilities. Nevertheless, it is well-known that those
+instabilities could be healed by means the correct identification of the constraints [45, 46]. In this
+respect, an advantage of the HJ scheme is that the constraints are identified directly and it is
+not necessary to fix them by hand, then with the generalized brackets and the identification of the
+Hamiltonians we can remove the linear canonical momenta terms. In fact, by using the Hamiltonians
+(14)-(19) the canonical hamiltonian takes the following form
+H′
+0 =
+�
+[1
+2πijπij − 1
+4πiiπjj + hijRij − 1
+µǫijk(4Kil∂jKkl + ∂mhim∂j∂lhkl + ∇2hil∂jhkl)
+− 4
+µ2 (2∂iKij∂jKkk + 2∂iKjk∂iKjk − 2∂jKik∂iKjk − ∂jKik∂kKij − ∂kKii∂kKjj]d3x.
+hence, the Ostrogradsky instability has been healed and the associated ghost was exorcised.
+On the other hand, with all these results we rewrite the fundamental differential in terms of either
+involutive Hamiltonians or generalized brackets, this is
+dF =
+�
+[{F, H′}∗dt + {F, H00
+1 }∗dω1
+00 + {F, H0i
+2 }∗dω2
+0i + {F, Hij
+4 }∗dω4
+ij]d3y.
+(33)
+thus, we will search if there are more Hamiltonians in the theory. For this aim, we shall take into
+account either the generalized differential (33) or the Frobenius integrability conditions which, ensure
+that system is integrable, this is
+dHa = 0,
+(34)
+where Ha ≡ (H00
+1 , H0i
+2 , Hij
+4 ) are all involutive Hamiltonians. From integrability conditions (34) the
+following 10 new Hamiltonians arise
+H00
+6
+≡ ∇2hii − ∂i∂jhij = 0,
+(35)
+H0i
+7
+≡ ∂jπij = 0,
+(36)
+Hij
+8
+≡ πij − 2Kij + 2ηijKkk − 2
+µ(ǫiklηjm + ǫjklηim)∂kKlm = 0,
+(37)
+Now, we observe that the Hamiltonians Hij
+4 , H00
+6
+and H8 are non-involutive, therefore they will be
+removed by introducing a new set of generalized brackets. In this respect, if we calculate the matrix
+whose entries will be all generalized brackets, say (28)-(31), between the non-involutive Hamiltonians,
+we will find null vectors, say vi = ( 1
+2∂i∂jζ, δikζ, 0), where ζ is an arbitrary function. Hence, from the
+contraction of the null vectors with the Hamiltonians [42, 43], we will find the following involutive
+Hamiltonian
+H9 = ∇2hii − ∂i∂jhij + 1
+2∂i∂jP ij,
+(38)
+
+7
+thus, there are only 12 non-involutive Hamiltonians (Hij
+4 , Hij
+8 ) whose generalized brackets are given
+by
+{Hij
+4 , Hij
+8 }∗ = 2[ 1
+2µ(ǫikmηjl + ǫjkmηil + ǫilmηjk + ǫilmηik)∂m + 1
+2(ηikηjl
++ηjkηil) − ηijηkl]δ3(x − y).
+(39)
+In this manner, we proceed to construct the new set of HJ generalized brackets, namely { , }∗∗, in
+the same way as we did before with the brackets (28)-(31). The non-trivial new generalized brackets
+are given by
+{hij, πkl}∗∗ = 1
+2(δk
+i δl
+j + δl
+iδk
+j )δ3(x − y),
+(40)
+{Kij, P kl}∗∗ = 0,
+(41)
+{hij, Kkl}∗∗ = 1
+4(ηikηjl + ηilηjk − ηijηkl)δ3(x − y) + µ2
+4Ξ[[(ηikηjl + ηilηjk − ηijηkl)∇2 + (ηij∂k∂l
++ηkl∂i∂j)](∇2 + µ2) − 3∂i∂j∂k∂l − 3µ2
+4 (ηik∂j∂l + ηil∂j∂k + ηjk∂i∂l + ηjl∂i∂k)
++µ
+4 [(ǫikmηjl + ǫjkmηil + ǫilmηjk + ǫjlmηik)(∇2 + µ2) + 3(ǫikm∂j∂l + ǫjkm∂i∂l
++ǫilm∂j∂k + ǫjlm∂i∂k)]∂m]δ3(x − y),
+(42)
+where Ξ ≡ −µ2(∇2 + µ2)(∇2 + µ2
+4 ). It is worth commenting, that some brackets were reported in
+[26], however, there are some differences. In fact, in this paper we have used an alternative analy-
+sis and new variables were introduced; the introduction of the variables allowed us to identify the
+brackets (42) directly and they have a more compact form than those reported in [26]. Moreover,
+the tedious classification of the constrains into first class and second class as usually is done, in the
+HJ scheme it is not necessary. Thus, we can observe that the HJ is more economical.
+With the new set of either involutives Hamiltonians or generalized brackets, the fundamental differ-
+ential takes the following new form
+dF =
+�
+[{F, H′(y)}∗∗dt + {F, H00
+1 (y)}∗∗dω1
+00 + {F, H0i
+2 (y)}∗∗dω2
+0i + {F, H0i
+7 (y)}∗∗dω7
+0i
++ {F, H9(y)}∗∗dω9]d3y,
+(43)
+where
+H00
+1
+= π00,
+(44)
+H0i
+2
+= π0i,
+(45)
+H0i
+7
+= ∂jπij,
+(46)
+H9 = ∇2hii − ∂i∂jhij.
+(47)
+From integrability conditions of H0i
+7 and H9 we find
+dH0i
+7
+= 0,
+(48)
+dH9 = −∂i∂jπij = −∂iH0i
+7 = 0,
+(49)
+
+8
+therefore, there are not further Hamiltonians. It is worth to comment, that the Hamiltonians given
+in (47) are related to those reported in [49] where only linearized gravity was studied. However, there
+are differences: from on side, the PB reported in [49] and the generalized brackets found in (40)-(42)
+are different. On the other hand, the contribution of the modification is present in the generalized
+brackets, and this fact will be relevant in the study of quantization because the generalized brackets
+will be changed to commutators and the contribution could provide differences with respect standard
+linearized gravity.
+Now, we will calculate the HJ characteristic equations, they are given by
+dh00 = dθ1
+00,
+(50)
+dh0i = 1
+2dθ2
+0i,
+(51)
+dhij = [2Kij + ∂ih0j + ∂jh0i]dt − 1
+2(δk
+i ∂j + δk
+j ∂i)dθ7
+0k,
+(52)
+dπ00 = −Rij
+ijdt,
+(53)
+dπ0i = 1
+2∂jπijdt,
+(54)
+dπij = [ηij∇2h00 − ∂i∂jh00 − ηijRkl
+kl − 2Rij − 1
+µ[(ǫikl∂j + ǫjkl∂i)∂k∂mhlm
+−(ǫiklηjm + ǫjklηim)∂k∇2hlm]]dt + (∂i∂j − ηij∇2)dθ9,
+(55)
+dKij = [−1
+2∂i∂jh00 − Rij + 1
+4ηijRkl
+kl]dt + 1
+2∂i∂jdθ9,
+(56)
+dP ij = [0]dt,
+(57)
+from the characteristic equations we can identify the following facts: from equations (50)-(51) we
+observe that the variables h00 and h0i are identified as Lagrange multipliers. Moreover, from (41)
+and (57) we discard to P ij as degree of freedom because its time evolution vanishes. Furthermore,
+we identify the equations of motion for hij and its momentum πij. In fact, by taking dθ7
+0k = 0 and
+dθ9 = 0, we obtain
+˙hij = 2Kij + ∂ih0j + ∂jh0i,
+(58)
+˙πij = ηij∇2h00 − ∂i∂jh00 − ηijRkl
+kl − 2Rij − 1
+µ[(ǫikl∂j + ǫjkl∂i)∂k∂mhlm
+−(ǫiklηjm + ǫjklηim)∂k∇2hlm],
+(59)
+˙Kij = −1
+2∂i∂jh00 − Rij + 1
+4ηijRklkl.
+(60)
+We observe that (58) corresponds to the definition of Kij, thus, if we use (58) and
+˙Kij we will
+obtain a second order time equation for hij as expected, then there are six degrees of freedom
+associated with the perturbation. In this manner, we calculate the number of physical degrees of
+freedom as follows: there are 12 canonical variables (hij, πij) and eight involutive Hamiltonians
+(H00
+1 , H0i
+2 , H0i
+7 , H9), thus
+DOF = 1
+2[12 − 8] = 2,
+and thus, the theory has two physical degrees of freedom just like GR [25, 26].
+On the other hand, if in the characteristics equations we take dt = 0, then we identify the following
+
+9
+canonical transformations
+δh00 = δω1
+00,
+(61)
+δh0i = 1
+2δω2
+0i,
+(62)
+δhij = −1
+2(δk
+i ∂j + δk
+j ∂i)δω7
+0k,
+(63)
+moreover, we can then identify the corresponding gauge transformations of the theory by considering
+that the Lagrangian (6) will be invariant under (61)-(63) if the variation δS = 0 [50], this is
+δS =
+� ∂S
+∂hµν
+δhµν +
+∂S
+∂(∂αhµν)δ(∂αhµν) +
+∂S
+∂(∂α∂βhµν)δ(∂α∂βhµν)
+�
+(64)
+=
+� ��
+−□hµν + □hλληµν − ∂α∂λhαληµν − ∂µ∂νhλλ + 2∂µ∂λhνλ + 1
+µǫ0µλγ(∂ν∂α∂λhαγ
+−∂λ□hν
+γ)) δhµν] d4x = 0,
+(65)
+thus, by taking account (61)-(63) into the variation, we obtain the following
+δS =
+�
+[Rij
+ijδω1
+00 + 1
+2[2∇2h0
+i + 2∂i ˙hj
+j − 2∂i∂jh0j − 2∂j ˙hij + 1
+µǫ0ijk(∂j∇2h0k − ∂j∂l ˙hkl)]δω2
+0i
+−1
+2[¨hij − ¨hk
+kηij + 2∂k ˙h0kηij − 2∂i ˙h0
+j + ∂i∂jh00 − ∇2h00ηij + 2Rij − Rkl
+klηij
++ 1
+µǫ0ikl(∂k¨hjl − ∂j∂k ˙h0l + ∂j∂k∂mhlm − ∂k∇2hjl)]δ(∂iω7
+0j + ∂jω7
+0i)]d4x = 0.
+(66)
+Now, we define ∂0ξ ≡ δω1
+00, so after long algebraic work we find that the variation takes the form
+δS =
+�
+[−∂j ˙hij + ∂ihjj + ∇2h0i − ∂i∂jh0j + 1
+2µǫ0ijk(∂j∇2h0k − ∂j∂l ˙hkl)](−∂iξ + δω2
+0i + ∂0δω7
+0i)d4x,
+= 0,
+(67)
+hence, the action will be invariant under (61)-(63) if the the parameters ω′s obey
+δω2
+0i = −∂0δω7
+0i + ∂iξ.
+(68)
+Now, we will write (68) in a new fashion.
+In fact, we introduce the following 4-vector ξµ ≡
+( 1
+2ξ, − 1
+2δω7
+0i) ≡ (ξ0, ξi); then ξ = 2ξ0 and δω7
+0i = −2ξi. Hence, the relation (68) takes the form
+1
+2δω2
+0i = ∂0ξi + ∂iξ0,
+(69)
+finally, from the equations (61)-(63) and (69) the following gauge transformations are identified
+δhµν = ∂µξν + ∂νξµ.
+(70)
+all these results are in agreement with those reported in [26], thus, our study complete and extends
+those reported in the literature.
+III.
+CONCLUSIONS AND REMARKS
+In this paper a detailed HJ analysis for the higher-order modified gravity has been performed.
+We introduced a new set of variables in a different way than other approaches and reported in
+
+10
+the literature, then the full set of involutive and non-involutive Hamiltonians were identified.
+The correct identification of the Hamiltonians allow us to avoid the Ostrogradsky instability by
+removing the terms with linear momenta, healing the canonical Hamiltonian. Furthermore, the HJ
+generalized brackets and the fundamental differential were obtained from which the characteristic
+equations and the gauge symmetries were identified. The complete identification of the Hamiltoni-
+ans allowed us to carry out the counting of the physical degrees of freedom, concluding that the
+modified theory and GR shares the same number of physical degrees of freedom. In this manner, we
+have all elements to analize the theory in the quantum context. In fact, with our perturbative HJ
+study either constraints or the generalized brackets are under control, thus, we could use the tools
+developed in the canonical quantization of field theories in order to make progress in this program
+[51]. Furthermore, our analysis will be relevant for the study of the theory in the non-perturbative
+scenario. In fact, now the modified theory will be full background independent then we will compare
+the differences between the canonical structure of GR reported in the literature [8, 9] and that for
+the modified theory. However, all those ideas are still in progress and will be reported soon [52].
+Data Availability Statement: No Data associated in the manuscript
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+

CtAyT4oBgHgl3EQf4fpz/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf,len=396
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='00787v1  [gr-qc]  2 Jan 2023  The Hamilton-Jacobi analysis for higher-order modified gravity  Alberto Escalante∗ and Aldair Pantoja†  Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  Apartado Postal J-48 72570, Puebla Pue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=', M´exico,  (Dated: January 3, 2023)  The Hamilton-Jacobi [HJ] study for the Chern-Simons [CS] modification of general relativity  [GR] is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' The complete structure of the Hamiltonians and the generalized brackets are  reported, from these results the HJ fundamental differential is constructed and the symmetries of  the theory are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' By using the Hamiltonians we remove an apparent Ostrogradsky’s instability  and the new structure of the hamiltonian is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In addition, the counting of physical degrees  of freedom is developed and some remarks are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  PACS numbers: 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='-k,98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='Qc  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  INTRODUCTION  It is well-known that GR is a successful framework for describing the classical behavior of the grav-  itational field and its relation with the geometry of space-time [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' From the canonical point of  view, GR is a background independent gauge theory with diffeomorphisms invariance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' the extended  Hamiltonian is a linear combination of first class constraints and propagates two physical degrees  of freedom [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' From the quantum point of view, the quantization program of gravity is a difficult  task to perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In fact, from the nonperturbative scheme, the non-linearity of the gravitational  field, manifested in the constraints, obscures the quantization making the complete description of  a nonperturbative quantum theory of gravity still an open problem [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' On the other hand, the  perturbative point of view of the path-integral method leads to the non-renormalizability problem  [10, 11] with all the tools that have been developed in quantum field theory have not worked suc-  cessfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In this respect, it is common to study modified theories of gravity in order to obtain  insights in the classical or quantum regime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' with the expectation that these theories will provide  new ideas or allow the development of new tools to carry out the quantization program, with an  example of this being the so-called higher order theories [12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In fact, higher-order theories are  good candidates for fixing the infinities that appear in the renormalization problem of quantum  gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' It is claimed that adding higher order terms quadratic in the curvature to gravity could  help avoid this problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' since these terms have a dimensionless coupling constant, which ensures  ∗Electronic address: aescalan@ifuap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='buap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='mx  †Electronic address: jpantoja@ifuap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='buap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='mx  2  that the final theory is divergence-free [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' The study of higher-order theories is a modern topic  in physics, these theories are relevant in dark energy physics [18, 19], generalized electrodynamics  [20–22] and string theories [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Furthermore, an interesting model in four dimensions can be  found in the literature, in which the Einstein-Hilbert [EH] action is extended by the addition of  a Chern-Simons four-current coupled with an auxiliary field, thus, under a particular choice of the  auxiliary field the resulting action will be a close model to GR [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In fact, at Lagrangian level  the theory describes the propagation of two degrees of freedom corresponding to gravitational waves  traveling with velocity c, but these propagate with different polarization intensities violating spatial  reflection symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Moreover, the Schwarzchild metric is a solution of the equations of motion,  thus, the modified theory and the EH action share the same classical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' On the other hand,  at hamiltonian level the theory is a higher-order gauge theory [26] whose Hamiltonian analysis is  known not to be easy to perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In this respect, the analysis of constrained higher-order systems is  usually developed by using the Ostrogradsky-Dirac [OD] [27–30] or the Gitman-Lyakhovich-Tyutin  [GLT ] [31, 32] methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' OD scheme is based on the extension of the phase space by considering  to the fields and their velocities as canonical coordinates and then introducing an extensi´on to their  canonical momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' However, the identification of the constraints is not easy to develop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' in some  cases, the constraints are fixed by hand in order to obtain a consistent algebra [33] and this yields  the opportunity to work with alternative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' On the other hand, the GLT framework is based  on the introduction of extra variables which transforms a problem with higher time derivatives to  one with only first-order ones then, by using the Dirac brackets the second class constraints and the  extra variables can be removed [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  Nevertheless, there is an alternative scheme for analyzing higher-order theories:  the so-called  Hamilton-Jacobi method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' The HJ scheme for regular field theories was developed by G¨uler [35, 36]  and later extended for singular systems in [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' It is based on the identification of the constraints,  called Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' These Hamiltonians can be either involutive or non-involutive and they are used  for constructing a generalized differential, where the characteristic equations, the gauge symmetries,  and the generalized HJ brackets of the theory can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' It is important to remark that the  identification of the Hamiltonians is performed by means of the null vectors, thus, the Hamiltonians  will have the correct structure without fix them by hand as is done in other approaches, then the  identification of the symmetries will be, in general, more economical than other schemes [39–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  With all of above the aims of this paper is to develop a detailed HJ analysis of the theory reported  in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  In fact, we shall analyze this model beyond the Lagrangian approach reported in [25];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  we shall see that the Jackiw-Yi [JY ] model is a higher-order theory and it is mandatory to study  this theory due to its closeness with GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' However, it is well-known that in higher-order theories  could be present ghost degrees of freedom associated to Ostrogradsky’s instabilities [44], namely,  the hamiltonian function is unbounded and this is reflected with the presence of linear terms of the  canonical momenta in the hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In this respect, it is important to comment that if there are  constraints, then it is possible to heal those instabilities [45, 46];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' in our case the JY model will show  an apparent Ostrogradsky’s instability since linear terms in the momenta will appear, however, we  3  will heal the theory by using the complete set of Hamiltonians, thereby exorcising the associated  ghosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' II, we start with the CS modification of GR, we will work  in the perturbative context, say, we will expand the metric around the Minkowski background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' We  shall observe that the modified theory is of higher-order in the temporal derivatives, then we shall  introduce a change of variables in order to express the action in terms of only first-order temporal  derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' The change of variable will allows us to develop the HJ analysis in an easy way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' the  identification of the Hamiltonians, the construction of the generalized differential and the symmetries  will be identified directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' III we present the conclusions and some remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  THE HAMILTON-JACOBI ANALYSIS  The modified EH action is given by [25]  S[gµν] =  �  M  �  R√−g + 1  4θ∗Rσ  τ  µνRτ  σµν  �  d4x,  (1)  where M is the space-time manifold, gµν the metric tensor, R the scalar curvature, g the determinant  of the metric, Rαβµν the Riemman tensor and θ is a coupling field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In general, θ can be viewed as  an external quantity or as a local dynamical variable, however, in order to obtain an action close to  GR we are going to choose θ = t  Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Along the paper we will use grek letters for labeling space-time  indices µ = 0, 1, 2, 3 and latin letters for space indices i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In addition, we will work within  the perturbative context expanding the metric around the Minkowski background  gµν = ηµν + hµν,  (2)  where hµν is the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' By substituting the expression for θ and by taking into account eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (2) in (1) we obtain the following linearized action  S[hµν] = −1  2  �  M  hµν �  Glin  µν + Clin  µν  �  d4x,  (3)  where Glin  µν is the linearized version of the Einstein tensor and Clin  µν is a linearized Cotton-type tensor  Clin  µν = − 1  4Ω[ǫ0µλγ∂λ(□hγν − ∂ν∂αhαγ)+ ǫ0νλγ∂λ(□hγµ − ∂µ∂αhαγ)] [25] defined in four-dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  Now we shall suppose that the space-time has a topology M ∼= R × Σ, where R is an evolution  parameter and Σ is a Cauchy hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Hence, by performing the 3 + 1 decomposition of the  action (3) we write down the corresponding Lagrangian density  L =  � �1  2  ˙hij ˙hij − ∂jh0i∂jh0i − 1  2∂khij∂khij − 1  2  ˙hii ˙hjj + ∂jh00∂jhii + 1  2∂khii∂khjj − 2∂ih0i ˙hjj  −∂ih00∂jhij − ∂ihij∂jhkk + 2∂jh0i ˙hij + ∂ihi0∂jh0j + ∂khki∂jhij + 1  µǫijk(−¨hli∂jhlk  +2˙hli∂j∂lh0k + ∂lhmi∂m∂jhlk + ∇2h0i∂jh0k + ∇2hmi∂jhmk)  �  d3x,  (4)  where we have defined µ ≡ 2Ω and ǫijk ≡ ǫ0ijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' As it was commented above, we will reduce the  order of the time derivatives of the Lagrangian (4) by extending the configuration space, this is done  4  by introducing the following change of variable  Kij = 1  2(˙hij − ∂ih0j − ∂jh0i),  (5)  here Kij is related with the so-called extrinsic curvature [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Thus, by substituting (5) into (4)  we rewrite the Lagrangian in the following new fashion  L =  � �  2KijKij − 2KiiKjj − h00Rijij − hijRij + 1  2hiiRijij + 1  µǫijk(4Kil∂jKkl + ∂mhim∂j∂lhkl  +∇2him∂jhkm) + ψij(˙hij − ∂ih0j − ∂jh0i − 2Kij)  �  d3x,  (6)  where we have added the Lagrange multipliers ψij enforcing the the relation (5), and the expressions  Rijij and Rij are defined in the following way  Rijij ≡ ∂i∂jhij − ∇2hii,  (7)  Rij ≡ 1  2(∂i∂khjk + ∂j∂khik − ∂i∂jhkk − ∇2hij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (8)  Now, we calculate the canonical momenta associated with the dynamical variables  π00 =  ∂L  ∂ ˙h00  = 0,  (9)  π0i =  ∂L  ∂ ˙h0i  = 0,  (10)  πij =  ∂L  ∂ ˙hij  = ψij,  (11)  P ij =  ∂L  ∂ ˙Kij  = 0,  (12)  Λij =  ∂L  ∂ ˙ψij  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (13)  Thus, from the equations (9)-(13) we identify the following HJ Hamiltonians of the theory  H′ ≡ H0 + Π = 0,  (14)  H00  1  ≡ π00 = 0,  (15)  H0i  2  ≡ π0i = 0,  (16)  Hij  3  ≡ πij − ψij = 0,  (17)  Hij  4  ≡ P ij = 0,  (18)  Hij  5  ≡ Λij = 0,  (19)  where H0 is the canonical hamiltonian defined as usual H0 = ˙hµνπµν + ˙KijP ij + ˙ψijΛij − L and  Π = ∂0S [39–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Moreover, the fundamental Poisson brackets [PB] between the canonical variables  5  are given by  {hµν, παβ} = 1  2(δα  µδβ  ν + δα  ν δβ  µ)δ3(x − y),  (20)  {Kij, πkl} = 1  2(δk  i δl  j + δk  j δl  i)δ3(x − y),  (21)  {ψij, Λkl} = 1  2(δi  kδj  l + δj  kδi  l)δ3(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (22)  Furthermore, in the HJ scheme, the dynamics of the system is governed by the fundamental differ-  ential defined as  dF = {F, HI}dωI,  (23)  where F is any function defined on the phase space, HI is the set of all Hamiltonians (14)-(19)  and ωI are the parameters related to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' It is important to remark, that in the HJ method the  Hamiltonians are classified as involutive and non-involutive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Involutive ones are those whose PB  with all Hamiltonians, including themselves, vanish;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' otherwise, they are called non-involutive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Be-  cause of integrability conditions, the non-involutive Hamiltonians are removed from the fundamental  differential (23) by introducing the so-called generalized brackets, these new brackets are given by  {f, g}∗ = {f, g} − {f, Ha′}C−1  a′b′{Hb′, g},  (24)  where Ca′b′ is the matrix formed with the PB between all non-involutive Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  From  (14)-(19) the non-involutive Hamiltonians are Hij  3 and Hij  5 , whose PB is  {Hij  3 , Hij  5 } = −1  2(ηikηjl + ηilηkj)δ3(x − y),  (25)  therefore, the matrix Ca′b′ given by  Ca′b′ =  \uf8eb  \uf8ed  0  − 1  2(ηikηjl + ηilηkj)  1  2(ηikηjl + ηilηkj)  0  \uf8f6  \uf8f8δ3(x − y),  (26)  and its inverse C−1  a′b′ takes the form  C−1  a′b′ =  \uf8eb  \uf8ed  0  1  2(ηikηjl + ηilηkj)  − 1  2(ηikηjl + ηilηkj)  0  \uf8f6  \uf8f8 δ3(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (27)  In this manner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' the following non-vanishing generalized brackets between the fields arise  {hµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' παβ}∗ = 1  2(δα  µδβ  ν + δβ  µδα  ν )δ3(x − y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (28)  {Kij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' P kl}∗ = 1  2(δk  i δl  j + δl  iδk  j )δ3(x − y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (29)  {hµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' ψαβ}∗ = 1  2(δα  µδβ  ν + δβ  µδα  ν )δ3(x − y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (30)  {ψij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Λkl}∗ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (31)  we observe from (31) that the canonical variables (ψij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Λkl) can be removed which implies that we  can perform the substitution of πij = ψij and Λij = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' the canonical hamiltonian takes the  6  form  H0 =  �  [2KiiKjj − 2KijKij + h00Rijij + hijRij − 1  2hiiRijij − 1  µǫijk(4Kil∂jKkl + ∂mhim∂j∂lhkl  +∇2him∂jhkm) − 2h0j∂iπij + 2Kijπij]d3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (32)  It is worth to comment, that the canonical hamiltonian has linear terms in the momenta πij and  this fact could be related to Ostrogradsky’s instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Nevertheless, it is well-known that those  instabilities could be healed by means the correct identification of the constraints [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In this  respect, an advantage of the HJ scheme is that the constraints are identified directly and it is  not necessary to fix them by hand, then with the generalized brackets and the identification of the  Hamiltonians we can remove the linear canonical momenta terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In fact, by using the Hamiltonians  (14)-(19) the canonical hamiltonian takes the following form  H′  0 =  �  [1  2πijπij − 1  4πiiπjj + hijRij − 1  µǫijk(4Kil∂jKkl + ∂mhim∂j∂lhkl + ∇2hil∂jhkl)  − 4  µ2 (2∂iKij∂jKkk + 2∂iKjk∂iKjk − 2∂jKik∂iKjk − ∂jKik∂kKij − ∂kKii∂kKjj]d3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  hence, the Ostrogradsky instability has been healed and the associated ghost was exorcised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  On the other hand, with all these results we rewrite the fundamental differential in terms of either  involutive Hamiltonians or generalized brackets, this is  dF =  �  [{F, H′}∗dt + {F, H00  1 }∗dω1  00 + {F, H0i  2 }∗dω2  0i + {F, Hij  4 }∗dω4  ij]d3y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (33)  thus, we will search if there are more Hamiltonians in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' For this aim, we shall take into  account either the generalized differential (33) or the Frobenius integrability conditions which, ensure  that system is integrable, this is  dHa = 0,  (34)  where Ha ≡ (H00  1 , H0i  2 , Hij  4 ) are all involutive Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' From integrability conditions (34) the  following 10 new Hamiltonians arise  H00  6  ≡ ∇2hii − ∂i∂jhij = 0,  (35)  H0i  7  ≡ ∂jπij = 0,  (36)  Hij  8  ≡ πij − 2Kij + 2ηijKkk − 2  µ(ǫiklηjm + ǫjklηim)∂kKlm = 0,  (37)  Now, we observe that the Hamiltonians Hij  4 , H00  6  and H8 are non-involutive, therefore they will be  removed by introducing a new set of generalized brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In this respect, if we calculate the matrix  whose entries will be all generalized brackets, say (28)-(31), between the non-involutive Hamiltonians,  we will find null vectors, say vi = ( 1  2∂i∂jζ, δikζ, 0), where ζ is an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Hence, from the  contraction of the null vectors with the Hamiltonians [42, 43], we will find the following involutive  Hamiltonian  H9 = ∇2hii − ∂i∂jhij + 1  2∂i∂jP ij,  (38)  7  thus, there are only 12 non-involutive Hamiltonians (Hij  4 , Hij  8 ) whose generalized brackets are given  by  {Hij  4 , Hij  8 }∗ = 2[ 1  2µ(ǫikmηjl + ǫjkmηil + ǫilmηjk + ǫilmηik)∂m + 1  2(ηikηjl  +ηjkηil) − ηijηkl]δ3(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (39)  In this manner, we proceed to construct the new set of HJ generalized brackets, namely { , }∗∗, in  the same way as we did before with the brackets (28)-(31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' The non-trivial new generalized brackets  are given by  {hij, πkl}∗∗ = 1  2(δk  i δl  j + δl  iδk  j )δ3(x − y),  (40)  {Kij, P kl}∗∗ = 0,  (41)  {hij, Kkl}∗∗ = 1  4(ηikηjl + ηilηjk − ηijηkl)δ3(x − y) + µ2  4Ξ[[(ηikηjl + ηilηjk − ηijηkl)∇2 + (ηij∂k∂l  +ηkl∂i∂j)](∇2 + µ2) − 3∂i∂j∂k∂l − 3µ2  4 (ηik∂j∂l + ηil∂j∂k + ηjk∂i∂l + ηjl∂i∂k)  +µ  4 [(ǫikmηjl + ǫjkmηil + ǫilmηjk + ǫjlmηik)(∇2 + µ2) + 3(ǫikm∂j∂l + ǫjkm∂i∂l  +ǫilm∂j∂k + ǫjlm∂i∂k)]∂m]δ3(x − y),  (42)  where Ξ ≡ −µ2(∇2 + µ2)(∇2 + µ2  4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' It is worth commenting, that some brackets were reported in  [26], however, there are some differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In fact, in this paper we have used an alternative analy-  sis and new variables were introduced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' the introduction of the variables allowed us to identify the  brackets (42) directly and they have a more compact form than those reported in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Moreover,  the tedious classification of the constrains into first class and second class as usually is done, in the  HJ scheme it is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Thus, we can observe that the HJ is more economical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  With the new set of either involutives Hamiltonians or generalized brackets, the fundamental differ-  ential takes the following new form  dF =  �  [{F, H′(y)}∗∗dt + {F, H00  1 (y)}∗∗dω1  00 + {F, H0i  2 (y)}∗∗dω2  0i + {F, H0i  7 (y)}∗∗dω7  0i  + {F, H9(y)}∗∗dω9]d3y,  (43)  where  H00  1  = π00,  (44)  H0i  2  = π0i,  (45)  H0i  7  = ∂jπij,  (46)  H9 = ∇2hii − ∂i∂jhij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (47)  From integrability conditions of H0i  7 and H9 we find  dH0i  7  = 0,  (48)  dH9 = −∂i∂jπij = −∂iH0i  7 = 0,  (49)  8  therefore, there are not further Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' It is worth to comment, that the Hamiltonians given  in (47) are related to those reported in [49] where only linearized gravity was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' However, there  are differences: from on side, the PB reported in [49] and the generalized brackets found in (40)-(42)  are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' On the other hand, the contribution of the modification is present in the generalized  brackets, and this fact will be relevant in the study of quantization because the generalized brackets  will be changed to commutators and the contribution could provide differences with respect standard  linearized gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' we will calculate the HJ characteristic equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' they are given by  dh00 = dθ1  00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (50)  dh0i = 1  2dθ2  0i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (51)  dhij = [2Kij + ∂ih0j + ∂jh0i]dt − 1  2(δk  i ∂j + δk  j ∂i)dθ7  0k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (52)  dπ00 = −Rij  ijdt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (53)  dπ0i = 1  2∂jπijdt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (54)  dπij = [ηij∇2h00 − ∂i∂jh00 − ηijRkl  kl − 2Rij − 1  µ[(ǫikl∂j + ǫjkl∂i)∂k∂mhlm  −(ǫiklηjm + ǫjklηim)∂k∇2hlm]]dt + (∂i∂j − ηij∇2)dθ9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (55)  dKij = [−1  2∂i∂jh00 − Rij + 1  4ηijRkl  kl]dt + 1  2∂i∂jdθ9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (56)  dP ij = [0]dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (57)  from the characteristic equations we can identify the following facts: from equations (50)-(51) we  observe that the variables h00 and h0i are identified as Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Moreover, from (41)  and (57) we discard to P ij as degree of freedom because its time evolution vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Furthermore,  we identify the equations of motion for hij and its momentum πij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In fact, by taking dθ7  0k = 0 and  dθ9 = 0, we obtain  ˙hij = 2Kij + ∂ih0j + ∂jh0i,  (58)  ˙πij = ηij∇2h00 − ∂i∂jh00 − ηijRkl  kl − 2Rij − 1  µ[(ǫikl∂j + ǫjkl∂i)∂k∂mhlm  −(ǫiklηjm + ǫjklηim)∂k∇2hlm],  (59)  ˙Kij = −1  2∂i∂jh00 − Rij + 1  4ηijRklkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (60)  We observe that (58) corresponds to the definition of Kij, thus, if we use (58) and  ˙Kij we will  obtain a second order time equation for hij as expected, then there are six degrees of freedom  associated with the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In this manner, we calculate the number of physical degrees of  freedom as follows: there are 12 canonical variables (hij, πij) and eight involutive Hamiltonians  (H00  1 , H0i  2 , H0i  7 , H9), thus  DOF = 1  2[12 − 8] = 2,  and thus, the theory has two physical degrees of freedom just like GR [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' if in the characteristics equations we take dt = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' then we identify the following  9  canonical transformations  δh00 = δω1  00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (61)  δh0i = 1  2δω2  0i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (62)  δhij = −1  2(δk  i ∂j + δk  j ∂i)δω7  0k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (63)  moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' we can then identify the corresponding gauge transformations of the theory by considering  that the Lagrangian (6) will be invariant under (61)-(63) if the variation δS = 0 [50],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' this is  δS =  � ∂S  ∂hµν  δhµν +  ∂S  ∂(∂αhµν)δ(∂αhµν) +  ∂S  ∂(∂α∂βhµν)δ(∂α∂βhµν)  �  (64)  =  � ��  −□hµν + □hλληµν − ∂α∂λhαληµν − ∂µ∂νhλλ + 2∂µ∂λhνλ + 1  µǫ0µλγ(∂ν∂α∂λhαγ  −∂λ□hν  γ)) δhµν] d4x = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (65)  thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' by taking account (61)-(63) into the variation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' we obtain the following  δS =  �  [Rij  ijδω1  00 + 1  2[2∇2h0  i + 2∂i ˙hj  j − 2∂i∂jh0j − 2∂j ˙hij + 1  µǫ0ijk(∂j∇2h0k − ∂j∂l ˙hkl)]δω2  0i  −1  2[¨hij − ¨hk  kηij + 2∂k ˙h0kηij − 2∂i ˙h0  j + ∂i∂jh00 − ∇2h00ηij + 2Rij − Rkl  klηij  + 1  µǫ0ikl(∂k¨hjl − ∂j∂k ˙h0l + ∂j∂k∂mhlm − ∂k∇2hjl)]δ(∂iω7  0j + ∂jω7  0i)]d4x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (66)  Now, we define ∂0ξ ≡ δω1  00, so after long algebraic work we find that the variation takes the form  δS =  �  [−∂j ˙hij + ∂ihjj + ∇2h0i − ∂i∂jh0j + 1  2µǫ0ijk(∂j∇2h0k − ∂j∂l ˙hkl)](−∂iξ + δω2  0i + ∂0δω7  0i)d4x,  = 0,  (67)  hence, the action will be invariant under (61)-(63) if the the parameters ω′s obey  δω2  0i = −∂0δω7  0i + ∂iξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (68)  Now, we will write (68) in a new fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  In fact, we introduce the following 4-vector ξµ ≡  ( 1  2ξ, − 1  2δω7  0i) ≡ (ξ0, ξi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' then ξ = 2ξ0 and δω7  0i = −2ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Hence, the relation (68) takes the form  1  2δω2  0i = ∂0ξi + ∂iξ0,  (69)  finally, from the equations (61)-(63) and (69) the following gauge transformations are identified  δhµν = ∂µξν + ∂νξµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  (70)  all these results are in agreement with those reported in [26], thus, our study complete and extends  those reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  CONCLUSIONS AND REMARKS  In this paper a detailed HJ analysis for the higher-order modified gravity has been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  We introduced a new set of variables in a different way than other approaches and reported in  10  the literature, then the full set of involutive and non-involutive Hamiltonians were identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  The correct identification of the Hamiltonians allow us to avoid the Ostrogradsky instability by  removing the terms with linear momenta, healing the canonical Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Furthermore, the HJ  generalized brackets and the fundamental differential were obtained from which the characteristic  equations and the gauge symmetries were identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' The complete identification of the Hamiltoni-  ans allowed us to carry out the counting of the physical degrees of freedom, concluding that the  modified theory and GR shares the same number of physical degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In this manner, we  have all elements to analize the theory in the quantum context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In fact, with our perturbative HJ  study either constraints or the generalized brackets are under control, thus, we could use the tools  developed in the canonical quantization of field theories in order to make progress in this program  [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Furthermore, our analysis will be relevant for the study of the theory in the non-perturbative  scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' In fact, now the modified theory will be full background independent then we will compare  the differences between the canonical structure of GR reported in the literature [8, 9] and that for  the modified theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' However, all those ideas are still in progress and will be reported soon [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  Data Availability Statement: No Data associated in the manuscript  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Einstein, The Foundation of the General Theory of Relativity, Annalen Phys 49, 769-822 (1916).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Einstein, The Field Equations of Gravitation, Sitzungsberichte, Royal Pruss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=', Berlin, 844-847  (1915).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content='  [3] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Dyson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Eddington and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Davison, A Determination of the Deflection of Light by the Sun’s  Gravitational Field, from Observations Made at the Total Eclipse of May 29 1919, Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'}
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DNA0T4oBgHgl3EQfAv8K/content/tmp_files/2301.01965v1.pdf.txt

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+Inference on the intraday spot volatility from high-frequency order
+prices with irregular microstructure noise
+Markus Bibinger∗a
+aFaculty of Mathematics and Computer Science, Julius-Maximilians-Universität Würzburg,
+[email protected]
+Abstract
+We consider estimation of the spot volatility in a stochastic boundary model with one-sided mi-
+crostructure noise for high-frequency limit order prices. Based on discrete, noisy observations of an
+Itô semimartingale with jumps and general stochastic volatility, we present a simple and explicit
+estimator using local order statistics. We establish consistency and stable central limit theorems
+as asymptotic properties. The asymptotic analysis builds upon an expansion of tail probabilities
+for the order statistics based on a generalized arcsine law. In order to use the involved distribution
+of local order statistics for a bias correction, an efficient numerical algorithm is developed. We
+demonstrate the finite-sample performance of the estimation in a Monte Carlo simulation.
+Keywords:
+arcsine law, limit order book, market microstructure, nonparametric boundary
+model, volatility estimation
+MSC Classification: 62M09, 60J65, 60F05
+1. Introduction
+Time series of intraday prices are typically described as a discretized path of a continuous-time
+stochastic process. To have arbitrage-free markets the log-price process should be a semimartin-
+gale. Risk estimation based on high-frequency data at highest available observation frequencies
+has to take microstructure frictions into account. Disentangling these market microstructure ef-
+fects from dynamics of the long-run price evolution has led to observation models with additive
+noise, see, for instance, [9], [2] and [14]. The market microstructure noise, modelling for instance
+the oscillation of traded prices between bid and ask order levels in an electronic market, is clas-
+sically a centred (white) noise process with expectation equal to zero. These models can explain
+many stylized facts of high-frequency data. Having available full limit order books including data
+of submissions, cancellations and executions of bid and ask limit orders, however, it is not clear
+which time series to consider at all. While challenging the concept of one price process it raises
+the question if the information can be exploited more efficiently, in particular to improve risk
+quantification. The considered stochastic boundary model for limit order prices of an order book
+has been discussed by [5], [15] and Chapter 1.8 of [4]. It preserves the concept of an underlying
+efficient, semimartingale log-price which determines long-run price dynamics and an additive, ex-
+ogenous noise which models market-specific microstructure frictions. Its key idea is that ask order
+prices should (in most cases) lie above the unobservable efficient price and bid prices below the
+∗Financial support from the Deutsche Forschungsgemeinschaft (DFG) under grant 403176476 is gratefully ac-
+knowledged.
+1
+arXiv:2301.01965v1  [math.ST]  5 Jan 2023
+
+efficient price. This leads to observation errors which are irregular in the sense of having non-zero
+expectation and a distribution with a lower- or upper-bounded support. Considering without loss
+of generality a model for (best) ask order prices, we obtain lower-bounded observation errors and
+use local minima for the estimation. Modelling (best) bid prices instead would yield a model with
+upper-bounded observation errors and local maxima could be used for an analogous estimation.
+Both can be combined in practice. Inference on the spot volatility is one of the most important
+topics in the financial literature, see, for instance, [17] and the references therein. In this work,
+we address spot volatility estimation for the model from [5].
+It is known that the statistical and probabilistic properties of models with irregular noise are
+very different than for regular noise and require other methods, see, for instance, [18], [13] and
+[19]. Therefore, our estimation methods and asymptotic theory are quite different compared to
+the market microstructure literature, while we can still profit from some of the techniques used
+there. In [5] an estimator for the quadratic variation of a continuous semimartingale, that is, the
+integrated volatility, was proposed with convergence rate n−1/3, based on n discrete observations
+with one-sided noise. Optimality of the rate was proved in the standard asymptotic minimax
+sense.
+A main insight was that this convergence rate is better than the optimal rate, n−1/4,
+under regular market microstructure noise. Using local minima over blocks of shrinking lengths
+hn ∝ n−2/3 ∝ (nhn)−2, the resulting distribution of local minima is involved and infeasible, such
+that in [5] a central limit theorem for the estimator could not be obtained. Our estimator is
+related to a localized version of the one from [5], combined with truncation methods to eliminate
+jumps of the semimartingale. For the asymptotic theory, however, we follow a different approach
+choosing blocks of lengths hn, where hnn2/3 → ∞ slowly. This allows to establish stable central
+limit theorems with the best achievable rate, arbitrarily close to n−1/6, in the important special
+case of a semimartingale volatility. We exploit this to construct pointwise asymptotic confidence
+intervals.
+Although the asymptotic theory relies on block lengths that are slightly unbalanced by smooth-
+ing out the impact of the noise distribution on the distribution of local minima asymptotically, our
+numerical study demonstrates that the confidence intervals work well in realistic scenarios with
+block lengths which optimize the estimator’s performance. Robustness to different noise specifi-
+cations is an advantage that is naturally implied by our approach. Our estimator is surprisingly
+simple, it is a local average of squared differences of block-wise minima times a constant factor
+which comes from moments of the half-normal distribution of the minimum of a Brownian motion
+over the unit interval. This estimator is consistent. However, the stable central limit theorem at
+fast convergence rate requires a subtle bias correction which incorporates a more precise approxi-
+mation of the asymptotic distribution of local minima. For that purpose, our analysis is based on
+a generalization of the arcsine law which gives the distribution of the proportion of time over some
+interval that a Brownian motion is positive. For a numerical computation of the bias-correction
+function, we introduce an efficient algorithm. Reducing local minima over many random variables
+to iterated minima of two random variables in each step combined with a convolution step, it can
+be interpreted as a kind of dynamic programming approach. It turns out to be much more efficient
+compared to the natural approximation by a Monte Carlo simulation and is a crucial ingredient
+of our numerical application. Our convergence rate is much faster than the optimal rate, n1/8, for
+spot volatility estimation under regular noise, see [10]. The main contribution of this work is to
+2
+
+develop the probabilistic foundation for the asymptotic analysis of the estimator and to establish
+the stable central limit theorems, asymptotic confidence and a numerically practicable method.
+The methods and proof techniques to deal with jumps are inspired by the truncation methods
+pioneered in [16] and summarized in Chapter 13 of [11]. Overall, the strategy and restrictions on
+jump processes are to some extent similar, while several details under irregular noise using order
+statistics are rather different compared to settings without noise or with regular centred noise as
+in [7].
+We introduce and further discuss our model in Section 2. Section 3 presents estimation methods
+and Section 4 asymptotic results. The numerical application is considered in Section 5 and a
+Monte Carlo simulation study illustrates an appealing finite-sample performance of the method.
+All proofs are given in Section 6.
+2. Model with lower-bounded, one-sided noise and assumptions
+Consider an Itô semimartingale
+Xt
+=
+X0 +
+� t
+0
+as ds +
+� t
+0
+σs dWs +
+� t
+0
+�
+R
+δ(s, z)1{|δ(s,z)|≤1}(µ − ν)(ds, dz)
++
+� t
+0
+�
+R
+δ(s, z)1{|δ(s,z)|>1}µ(ds, dz) , t ≥ 0 ,
+(1)
+with a one-dimensional standard Brownian motion (Wt), defined on some filtered probability
+space (ΩX, FX, (FX
+t ), PX). For the drift process (at), and the volatility process (σt), we impose
+the following quite general assumptions.
+Assumption 1. The processes (at)t≥0 and (σt)t≥0 are locally bounded. The volatility process is
+strictly positive, inft∈[0,1] σt > 0, PX-almost surely. For all 0 ≤ t + s ≤ 1, t ≥ 0, s ≥ 0, with some
+constants Cσ > 0, and α > 0, it holds that
+E
+�
+(σ(t+s) − σt)2�
+≤ Cσs2α .
+(2)
+Condition (2) introduces a regularity parameter α, governing the smoothness of the volatility
+process. The parameter α is crucial, since it will naturally influence convergence rates of spot
+volatility estimation. Inequality (2) is less restrictive than α-Hölder continuity, since it does not
+rule out volatility jumps. This is important as empirical evidence for volatility jumps, in particular
+simultaneous price and volatility jumps, has been reported for intraday high-frequency financial
+data, see, for instance, [22] and [6].
+The presented theory is moreover for general stochastic
+volatilities, allowing as well for rough volatility.
+Rough fractional stochastic volatility models
+recently became popular and are used, for instance, in the macroscopic model of [8] and [20].
+The jump component of (1) is illustrated as in [11] and related literature, where the predictable
+function δ is defined on Ω × R+ × R, and the Poisson random measure µ is compensated by
+ν(ds, dz) = λ(dz) ⊗ ds, with a σ-finite measure λ. We impose the following standard condition
+with a generalized Blumenthal-Getoor or jump activity index r, 0 ≤ r ≤ 2.
+3
+
+Assumption 2. Assume that supω,x |δ(t, x)|/γ(x) is locally bounded with a non-negative, deter-
+ministic function γ which satisfies
+�
+R
+(γr(x) ∧ 1)λ(dx) < ∞ .
+(3)
+We use the notation a ∧ b = min(a, b), and a ∨ b = max(a, b), throughout this paper. The
+assumption is most restrictive in the case r = 0, when jumps are of finite activity. The larger r,
+the more general jump components are allowed. We will develop results under mild restrictions
+on r.
+The process (Xt), which can be decomposed
+Xt = Ct + Jt ,
+(4)
+with the continuous component (Ct), and the càdlàg jump component (Jt), provides a model for
+the latent efficient log-price process in continuous time.
+High-frequency (best) ask order prices from a limit order book at times tn
+i , 0 ≤ i ≤ n, on the
+fix time interval [0, 1], cannot be adequately modelled by discrete recordings of (Xt). Instead, we
+propose the additive model with lower-bounded, one-sided microstructure noise:
+Yi = Xtn
+i + ϵi , i = 0, . . . , n,
+ϵi
+iid
+∼ Fη, ϵi ≥ 0 .
+(5)
+The crucial property of the model is that the support of the noise is lower bounded. It is not that
+important, that this boundary is zero, it could be as well a different constant, or even a regularly
+varying function over time. We set the bound equal to zero which appears to be the most natural
+choice for limit orders.
+Assumption 3. The i.i.d. noise (ϵi)0≤i≤n, has a cumulative distribution function (cdf) Fη satis-
+fying
+Fη(x) = ηx
+�
+1 + O(1)
+�
+, as x ↓ 0 .
+(6)
+This is a nonparametric model in that the extreme value index is −1 for the minimum domain
+of attraction close to the boundary. This standard assumption on one-sided noise has been used
+by [13] and [19] within different frameworks, as well. We do not require assumptions about the
+maximum domain of attraction, moments and the tails of the noise distribution. Parametric exam-
+ples which satisfy (6) are, for instance, the uniform distribution on some interval, the exponential
+distribution and the standard Pareto distribution with heavy tails.
+The i.i.d. assumption on the noise is crucial and generalizations to weakly dependent noise
+will require considerable work and new proof concepts. Heterogeneity instead, that is, a time-
+dependent noise level η(t), could be included in our asymptotic analysis under mild assumptions.
+4
+
+3. Construction of spot volatility estimators
+We partition the observation interval [0, 1] in h−1
+n
+equispaced blocks, h−1
+n
+∈ N, and take local
+minima on each block. We hence obtain for k = 0, . . . , h−1
+n
+− 1, the local, block-wise minima
+mk,n = min
+i∈In
+k
+Yi , In
+k = {i ∈ {0, . . . , n} : tn
+i ∈ (khn, (k + 1)hn)} .
+(7)
+While h−1
+n
+is an integer, nhn is in general not integer-valued. For a simple interpretation, however,
+one can think of nhn as an integer-valued sequence which gives the number of noisy observations
+per block. A spot volatility estimator could be obtained as a localized version of the estimator from
+Eq. (2.9) in [5] for the integrated volatility in the analogous model. The idea is that differences
+mk,n − mk−1,n of local minima estimate differences of efficient prices and a sum of squared differ-
+ences can be used to estimate the volatility. However, things are not that simple. To determine
+the expectation of squared differences of local minima we introduce the function
+Ψn(σ2) =
+π
+2(π − 2)h−1
+n E
+��
+min
+i∈{0,...,nhn−1}
+�
+σB i
+n + ϵi
+�
+−
+min
+i∈{1,...,nhn}
+�
+σ ˜B i
+n + ϵi
+��2�
+,
+(8)
+where (Bt) and ( ˜Bt) denote two independent standard Brownian motions. For hnn2/3 → ∞, we
+have that
+Ψn(σ2) = σ2 + O(1) ,
+(9)
+such that we do not require Ψ−1
+n
+for a consistent estimator in this case. Note that we defined Ψn
+different compared to [5] with the constant factor π/(π − 2).
+When there are no price jumps, a simple consistent estimator for the spot squared volatility
+σ2
+τ is given by
+ˆσ2
+τ− =
+π
+2(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n
+�
+mk,n − mk−1,n)2 ,
+(10)
+for suitable sequences hn → 0 and Kn → ∞. Using only observations before time τ, the estimator
+is available on-line at time τ ∈ (0, 1] during a trading day. Working with ex-post data over the
+whole interval, instead of using only observations before time τ, one may use as well
+ˆσ2
+τ+ =
+π
+2(π − 2)Kn
+(⌊h−1
+n τ⌋+Kn)∨(h−1
+n −1)
+�
+k=⌊h−1
+n τ⌋+1
+h−1
+n
+�
+mk,n − mk−1,n)2 ,
+(11)
+or an estimator with an average centred around time τ ∈ (0, 1). The difference of the two estimators
+(11) and (10) can be used to infer a possible jump in the volatility process at time τ ∈ (0, 1), as
+well.
+To construct confidence intervals for the spot volatility, it is useful to establish also a spot
+5
+
+quarticity estimator:
+�
+σ4τ − =
+π
+4(3π − 8)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−2
+n
+�
+mk,n − mk−1,n)4 .
+(12)
+A spot volatility estimator which is robust with respect to jumps in (Xt) is obtained with
+threshold versions of these estimators. We truncate differences of local minima whose absolute
+values exceed a threshold un = hκ
+n, κ ∈ (0, 1/2), which leads to
+ˆσ2,(tr)
+τ−
+=
+π
+2(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n
+�
+mk,n − mk−1,n)21{|mk,n−mk−1,n|≤un} ,
+(13)
+and analogous versions of the estimators (11) and (12).
+4. Asymptotic results
+We establish asymptotic results for equidistant observations, tn
+i = i/n. We begin with the
+asymptotic theory in a setup without jumps in (Xt).
+Theorem 1 (Stable central limit theorem for continuous (Xt)). Set hn, such that hnn2/3 → ∞,
+and Kn = CKhδ−2α/(1+2α)
+n
+for arbitrary δ, 0 < δ < 2α/(1 + 2α), and with some constant CK > 0.
+If (Xt) is continuous, i.e. Jt = 0 in (4), under Assumptions 1 and 3, the spot volatility estimator
+(10) is consistent, ˆσ2
+τ−
+P→ σ2
+τ−, and satisfies the stable central limit theorem
+K1/2
+n
+�
+ˆσ2
+τ− − Ψn
+�
+σ2
+τ−
+��
+st
+−→ N
+�
+0, 7π2/4 − 2π/3 − 12
+(π − 2)2
+σ4
+τ−
+�
+.
+(14)
+There is only a difference between σ2
+τ and its left limit σ2
+τ− in case of a volatility jump at
+time τ.
+In particular, the estimator is as well consistent for σ2
+τ, for any fix τ ∈ (0, 1).
+The
+convergence rate K−1/2
+n
+gets arbitrarily close to n−2α/(3+6α), which is optimal in our model. In
+the important special case when α = 1/2, for a semimartingale volatility, the rate is arbitrarily
+close to n−1/6. This is much faster than the optimal rate of convergence in the model with additive
+centred microstructure noise, which is known to be n−1/8, see [10]. The constant in the asymptotic
+variance is obtained from several variance and covariance terms including (squared) local minima
+and is approximately 2.44. The function Ψn was shown to be monotone and invertible in [5] and
+Ψn and its inverse Ψ−1
+n
+can be approximated using Monte Carlo simulations, see Section 5.1. The
+asymptotic distribution of the estimator does not hinge on the noise level η, different to methods
+for centred noise. Hence, we do not require any pre-estimation of noise parameters and the theory
+directly extends to a time-varying noise level ηt in (6) under the mild assumption that 0 < ηt < ∞,
+for all t. The stable convergence in (14) is stronger than weak convergence and is important, since
+the limit distribution is mixed normal depending on the stochastic volatility. We refer to [11],
+Section 2.2.1, for an introduction to stable convergence. For a normalized central limit theorem,
+we can use the spot quarticity estimator (12).
+Proposition 2 (Feasible central limit theorem). Under the conditions of Theorem 1, the spot
+quarticity estimator (12) is consistent, such that we get for the spot volatility estimation the
+6
+
+normalized central limit theorem
+K1/2
+n
+π − 2
+�
+�
+σ4τ −(7π2/4 − 2π/3 − 12)
+�
+ˆσ2
+τ− − Ψn
+�
+σ2
+τ−
+��
+d
+−→ N(0, 1) .
+(15)
+Proposition 2 yields asymptotic confidence intervals for spot volatility estimation. For q ∈
+(0, 1), it holds true that
+P
+�
+σ2
+τ− ∈
+�
+Ψ−1
+n
+�
+ˆσ2
+τ− −
+π − 2
+�
+�
+σ4τ −(7π2/4 − 2π/3 − 12)
+K−1/2
+n
+Φ−1(1 − q)
+�
+,
+Ψ−1
+n
+�
+ˆσ2
+τ− +
+π − 2
+�
+�
+σ4τ −(7π2/4 − 2π/3 − 12)
+K−1/2
+n
+Φ−1(1 − q)
+���
+→ 1 − q ,
+by monotonicity of Ψ−1
+n
+with Φ the cdf of the standard normal distribution. Since Ψ−1
+n
+is differ-
+entiable and the derivative is
+�
+Ψ−1
+n
+�′ = 1 + O(1), the delta method (for stable convergence) yields
+as well asymptotic confidence intervals and the central limit theorem
+K1/2
+n
+�
+Ψ−1
+n
+�
+ˆσ2
+τ−
+�
+− σ2
+τ−
+�
+st
+−→ N
+�
+0, 7π2/4 − 2π/3 − 12
+(π − 2)2
+σ4
+τ−
+�
+.
+(16)
+We may not simply replace Ψn
+�
+σ2
+τ−
+�
+by its first-order approximation σ2
+τ− in (14), since the bias
+multiplied with K1/2
+n
+does in general not converge to zero. If the condition hnn2/3 → ∞ is violated,
+this central limit theorem does not apply.
+Theorem 3 (Stable central limit theorem with jumps in (Xt)). Set hn, such that hnn2/3 → ∞,
+and Kn = CKhδ−2α/(1+2α)
+n
+for arbitrary δ, 0 < δ < 2α/(1 + 2α), and with some constant CK > 0.
+Under Assumptions 1, 2 and 3, with
+r < 2 + 2α
+1 + 2α ,
+(17)
+the truncated spot volatility estimator (13) with
+κ ∈
+�
+1
+2 − r
+α
+2α + 1, 1
+2
+�
+,
+(18)
+is consistent, ˆσ2,(tr)
+τ−
+P→ σ2
+τ−, and satisfies the stable central limit theorem
+K1/2
+n
+�
+ˆσ2,(tr)
+τ−
+− Ψn
+�
+σ2
+τ−
+��
+st
+−→ N
+�
+0, 7π2/4 − 2π/3 − 12
+(π − 2)2
+σ4
+τ−
+�
+.
+(19)
+In order to obtain a central limit theorem at (almost) optimal rate, we thus have to impose
+mild restrictions on the jump activity. For the standard model with a semimartingale volatility,
+i.e. α = 1/2, we need that r < 3/2, and for α = 1 we have the strongest condition that r < 4/3.
+These conditions are equivalent to the ones of Theorem 1 in [7], which gives a central limit
+theorem for spot volatility estimation under similar assumptions on (Xt), but with slower rate
+of convergence for centred microstructure noise. Using a truncated quarticity estimator with the
+same thresholding yields again a feasible central limit theorem and asymptotic confidence intervals.
+7
+
+Remark 1. From a theoretical point of view one might ponder why we do not work out an asymp-
+totic theory for hn ∝ n−2/3, when noise and efficient price both influence the asymptotic distribu-
+tion of the local minima. However, in this balanced case, the asymptotic distribution is infeasible.
+For this reason, [5] could not establish a central limit theorem for their integrated volatility esti-
+mator. Moreover, their estimator was only implicitly defined depending on the unknown function
+Ψ−1
+n . Even imposing a parametric assumption on the noise as an exponential distribution would
+not render a feasible limit theory for hn ∝ n−2/3, see the discussion in [5]. Choosing hn, such
+that hnn2/3 → ∞ slowly, yields instead a simple, explicit and consistent estimator and a fea-
+sible central limit theorem for spot volatility estimation. In particular, we use Ψn only for the
+bias-correction of the simple estimator, while the estimator itself and the (estimated) asymptotic
+variance do not hinge on Ψn. Central limit theorems for spot volatility estimators are in general
+only available at almost optimal rates, when the variance dominates the squared bias in the mean
+squared error, see, for instance, Theorem 13.3.3 and the remarks below in [11]. Therefore, (14)
+is the best achievable central limit theorem. Our choice of hn avoids moreover strong assumptions
+on the noise that would be inevitable for smaller blocks. Our numerical work will demonstrate that
+the presented asymptotic results are useful in practice and can be applied without loosing (much)
+efficiency compared to a different selection of blocks.
+5. Implementation and simulations
+5.1. Monte Carlo approximation of Ψn
+Although the function Ψn from (8) tends to the identity asymptotically, it has a crucial role
+for a bias correction of our estimator in (14). We can compute the function numerically based
+on a Monte Carlo simulation. Hence, we have to compute Ψn(σ2) as a Monte Carlo mean over
+many iterations and over a fine grid of values for the squared volatility.
+Then, we can also
+numerically invert the function and use Ψ−1
+n ( · ). To make this procedure feasible without too
+high computational expense we require an algorithm to efficiently sample from the law of the local
+minima for some given n and block length hn.
+Consider for nhn ∈ N, with Zi
+iid
+∼ N(0, 1), and the observation errors (ϵk)k≥0, the minimum
+M nhn
+1
+:=
+min
+k=1,...,nhn
+� σ
+√n
+k
+�
+i=1
+Zi + ϵk
+�
+,
+for some fix σ > 0, and for l ∈ {0, . . . , nhn}:
+M nhn
+l
+:=
+min
+k=l,...,nhn
+� σ
+√n
+k
+�
+i=0
+Zi + ϵk
+�
+,
+where we set Z0 := 0. Since
+Ψn(σ2) = 1
+2
+π
+π − 2h−1
+n E
+��
+M nhn−1
+0
+− M nhn
+1
+�2�
+,
+with M nhn−1
+0
+generated independently from M nhn
+1
+, we want to simulate samples distributed as
+M nhn−1
+0
+and M nhn
+1
+, respectively. Note that the moments of M nhn−1
+0
+and M nhn
+1
+slightly differ
+8
+
+what can be relevant for moderate values of nhn. As in the simulation of Section 5.2, we im-
+plement exponentially distributed observation errors (ϵk), with some given noise level η. In data
+applications, we can do the same with an estimated noise level
+ˆη =
+� 1
+2n
+n
+�
+i=1
+�
+Yi − Yi−1
+�2
+�−1/2
+= η + OP
+�
+n−1/2�
+.
+This estimator works for all noise distributions with finite fourth moments. To simulate the local
+minima for given n, hn, η, and squared volatility σ2, in an efficient way we use a specific dynamic
+programming principle. Observe that
+M nhn
+1
+=
+σ
+√nZ1 + min
+�
+ϵ1, M nhn
+2
+�
+=
+σ
+√nZ1 + min
+�
+ϵ1, σ
+√nZ2 + min
+�
+ϵ2, M nhn
+3
+��
+=
+σ
+√nZ1 + min
+�
+. . . min
+�
+ϵnhn−2, σ
+√nZnhn−1 + min
+�
+ϵnhn−1, σ
+√nZnhn + ϵnhn
+��
+. . .
+�
+.
+In the baseline noise model, ϵk
+iid
+∼ Exp(η), the random variable
+σ
+√nZnhn+ϵnhn has an exponentially
+modified Gaussian (EMG) distribution. With any fixed noise distribution, we can easily generate
+realizations from this convolution. A pseudo random variable which is distributed as M nhn
+1
+is now
+generated following the last transformation in the reverse direction. In pseudo code, this reads
+1. Generate U_{nh_n}~ EMG(sigma^2/n,eta)~ Exp(eta)+sigma/sqrt(n)*Norm(1)
+2. U_{nh_n-1}=min(U_{nh_n},Exp(eta))+sigma/sqrt(n)*Norm(1)
+3. iterate until U_1
+where the end point U1 has the target distribution of M nhn
+1
+. In each iteration step, we thus take
+the minimum of the current state of the process with one independent exponentially distributed
+random variable and the convolution with one independent normally distributed random variable.
+To sample from the distribution of M nhn−1
+0
+instead, we use the same algorithm and just drop the
+convolution with the normal distribution in the last step.
+It turns out that this algorithm facilitates a many times faster simulation compared to a
+classical simulation starting with a discretized path of (Xt).
+Figure 1 plots the result of the Monte Carlo approximation of Ψn(σ2) for n = 23,400 and
+n · hn = 15, on a grid of 1500 values of σ2. In this case, hn is quite small, but this configuration
+turns out to be useful below in Section 5.2. We know that Ψn(σ2) is monotone, such that the
+oscillation of the blue line in Figure 1 is due to the inaccuracy of the Monte Carlo means although
+we use N = 100,000 iterations for each grid point. Nevertheless, we can see that the function is
+rather close to a linear function with slope 1.046 based on a least squares estimate. The left plot of
+Figure 1 draws a comparison to the identity function which is illustrated by the dotted line, while
+the plot right-hand side draws a comparison to the linear function with slope 1.046. We see that it
+is crucial to correct for the bias in (14) when using such small values of hn. Although the function
+Ψn(σ2) is not exactly linear, a simple bias correction dividing estimates by 1.046 is almost as good
+as using the more precise numerical inversion based on the Monte Carlo approximation. Since the
+Monte Carlo approximations of Ψn(σ2) look close to linear functions in all considered cases, we
+9
+
+Figure 1: Monte Carlo means to estimate Ψn(σ2) over a fine grid (blue line) for n = 23,400 and n · hn = 15. Left,
+the dotted line shows the identity function, right the dotted line is a linear function with slope 1.046.
+Table 1: Regression slopes to measure the bias of estimator (10) and deviation Ψn(σ2) − σ2.
+n · hn
+10
+15
+25
+39
+78
+234
+h−1
+n
+2340
+1560
+936
+600
+300
+100
+hn · n2/3
+0.350
+0.524
+0.874
+1.36
+2.73
+8.18
+slope
+1.077
+1.046
+1.025
+1.016
+1.008
+1.003
+approx. bias
+7.7%
+4.6%
+2.5%
+1.6%
+0.8%
+0.3%
+report the estimated slopes based on least squares and N = 100,000 Monte Carlo iterations for
+different values of hn in Table 1 to summarize concisely how far the distance between the function
+Ψn(σ2) and the identity is. Simulating all iterations for all grid points with our algorithm takes
+only a few hours with a standard computer.
+5.2. Simulation study of estimators
+We simulate n = 23,400 observations corresponding to one observation per second over a
+(NASDAQ) trading day of 6.5 hours. The efficient price process is simulated from the model
+dXt = νtσt dWt ,
+dσ2
+t = 0.0162 ·
+�
+0.8465 − σ2
+t
+�
+dt + 0.117 · σt dBt ,
+νt =
+�
+6 − sin(3πt/4)
+�
+· 0.002 , t ∈ [0, 1] .
+The factor (νt) generates a typical U-shaped intraday volatility pattern.
+(Wt, Bt) is a two-
+dimensional Brownian motion with leverage d[W, B]t = 0.2 dt. The stochastic volatility component
+has several realistic features and the simulated model is in line with recent literature, see [6] and
+references therein. Observations with lower-bounded, one-sided microstructure noise are generated
+by
+Yi = X i
+n + ϵi , 0 ≤ i ≤ n ,
+10
+
+0.00012
+80000'0
+f(c3)
+0.00004
+00000'
+0.00000
+0.00004
+0.00008
+0.000120.00012
+80000'0
+4(c3)
+0.00004
+00000'
+0.00000
+0.00004
+0.00008
+0.00012
+3Figure 2: True and estimated spot volatility with pointwise confidence sets.
+with exponentially distributed noise, ϵi
+iid
+∼ Exp(η), with η = 10,000. The noise variance is then
+rather small, but this is in line with stylized facts of real NASDAQ data as, for instance, those
+analysed in [6].1
+The black line in Figure 2 shows a fixed path of the squared volatility. We fix this path for the
+following Monte Carlo simulation and generate new observations of (Xt) and (Yi) in each iteration
+according to our model. The blue line in Figure 2 gives the estimated volatility by the Monte
+Carlo means over N = 50,000 iterations based on n·hn = 15 observations per block using the non-
+adjusted estimator (10) with windows which are centred around the block on that we estimate
+the spot volatility and with Kn = 180. We plot estimates on each block, where the estimates
+close to the boundaries rely on less observations. The red line gives the bias-corrected volatility
+estimates using the numerically evaluated function Ψn, based on the algorithm from Section 5.1
+with n · hn = 15 and n = 23,400. We determined the values n · hn = 15 and Kn = 180 as suitable
+values to obtain a small mean squared error. In fact, the choice of Kn = 180 is rather large in
+favour of a smaller variance what yields a rather smooth estimated spot volatility in Figure 2.
+The estimated volatility hence appears smoother compared to the true semimartingale volatility,
+but the intraday pattern is well captured by our estimation. We expect that this is typically
+an appealing implementation in practice as smaller Kn results in a larger variance. Choosing
+Kn = 180 rather large, we have to use quite small block sizes hn, to control the overall bias of
+the estimation. Since hn · n2/3 ≈ 0.52 is small, the bias correction becomes crucial here. Still,
+1Note that the noise level estimate is analogous to the one used for regular market microstructure noise. Typical
+noise levels obtained for trades of e.g. Apple are approx. 15,000 and approx. 4,000 for 3M. For mid quotes or best
+ask/bid prices the levels are only slightly larger (variance smaller).
+11
+
+0.00018
+0.00014
+0.00010
+90000'0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+timeTable 2: Summary statistics of estimation for different values of hn and Kn, MSD = mean standard deviation,
+MAB = mean absolute bias, MABC = MAB of bias-corrected estimator.
+Kn
+120
+180
+240
+nhn
+MSD
+MAB
+MABC
+MSD
+MAB
+MABC
+MSD
+MAB
+MABC
+10
+14.6
+7.59
+0.73
+12.0
+7.51
+0.90
+10.5
+7.60
+1.13
+15
+14.4
+4.59
+0.88
+11.8
+4.57
+1.17
+10.3
+4.46
+1.43
+25
+14.3
+2.56
+1.24
+11.8
+2.63
+1.66
+10.3
+2.86
+1.91
+78
+14.7
+2.44
+2.52
+12.3
+3.53
+3.42
+11.0
+4.33
+4.16
+All values multiplied with factor 106.
+our asymptotic results work well for this implementation. This can be seen by the comparison of
+pointwise empirical 10% and 90% quantiles from the Monte Carlo iterations illustrated by the grey
+area and the 10% and 90% quantiles of the limit normal distribution with the asymptotic variance
+from (14). The latter are drawn as dotted lines for the blocks with larger distance than Kn/2
+from the boundaries where the variances are of order K−1
+n . Close to the boundaries the empirical
+variances increase due to the smaller number of blocks used for the estimates. Moreover, the bias
+correction which is almost identical to dividing each estimate by 1.046, correctly scales the simple
+estimates which have a significant positive bias for the chosen tuning parameters. Overall, our
+asymptotic results provide a good finite-sample fit even though we have hn · n2/3 < 1 here. Note,
+however, that σt · η ≈ 100, and our asymptotic expansion requires in fact that hn · n2/3σt · η is
+large when taking constants into account.
+Table 2 summarizes the performance of the estimation along different choices of nhn and Kn.
+We give the following quantities:
+1. MSD: the mean standard deviation of N iterations averaged over all grid points;
+2. MAB: the mean absolute bias of N iterations averaged over all grid points and for estimator
+(10) without any bias correction;
+3. MABC: the mean absolute bias of N iterations averaged over all grid points and for estimator
+(10) with a simple bias correction dividing estimates by the factors given in Table 1.
+All results are based on N = 50,000 Monte Carlo iterations. First of all, the values used for Figure
+2 are not unique minimizers of the mean squared error. Several other combinations given in Table
+2 render equally well results. Overall, the performance is comparable within a broad range of
+block lengths and window sizes. The variances decrease for larger Kn, while the bias increases
+with larger Kn for fixed hn. Important for the bias is the total window size, Kn · hn, over that
+the volatility is approximated constant for the estimation. The variance only depends on Kn,
+changing the block length for fix Kn does not significantly affect the variance. While the MSD is
+hence almost constant within the columns of Table 2, the bias after correction, MABC, increases
+from top down due to the increasing window size. Without the bias correction two effects interfere
+for MAB. Larger blocks reduce the systematic bias due to Ψn(σ2
+t ) − σ2
+t , but the increasing bias
+due to the increasing window size prevails for n · hn = 78, and the two larger values of Kn.
+12
+
+6. Proofs
+6.1. Law of the integrated negative part of a Brownian motion
+A crucial lemma for our theory is on an upper bound for the cdf of the integrated negative
+part of a Brownian motion. We prove a lemma based on a generalization of Lévy’s arc-sine law
+by [21]. The result is in line with the conjecture in Eq. (261) of [12] where one finds an expansion
+of the density with a precise constant of the leading term. Denote by f+ the positive part and by
+f− the negative part of some real-valued function f.
+Lemma 4. For a standard Brownian motion (Wt)t≥0, it holds that
+P
+� � 1
+0
+(Wt)− dt ≤ x
+�
+= O(x1/3), x → 0 .
+Proof. Observe the equality in distribution
+� 1
+0 (Wt)− dt
+d=
+� 1
+0 (Wt)+ dt, such that
+P
+� � 1
+0
+(Wt)− dt ≤ x
+�
+= P
+� � 1
+0
+(Wt)+ dt ≤ x
+�
+, x > 0 .
+For any ϵ > 0, the inequality
+� 1
+0
+(Wt)+ dt ≥
+� 1
+0
+Wt · 1(Wt > ϵ) dt ≥ ϵ
+� 1
+0
+1(Wt > ϵ) dt
+leads us to
+P
+� � 1
+0
+(Wt)+ dt ≤ x
+�
+≤ P
+�
+ϵ
+� 1
+0
+1(Wt > ϵ) dt ≤ x
+�
+= P
+�
+1 −
+� 1
+0
+1(Wt ≤ ϵ) dt ≤ x/ϵ
+�
+= P
+� � 1
+0
+1(Wt ≤ ϵ) dt ≥ 1 − x/ϵ
+�
+.
+Using (15) and (16) from [21], we obtain that
+P
+� � 1
+0
+1(Wt ≤ ϵ) dt ≥ 1 − x/ϵ
+�
+= 1
+π
+� 1
+1−x/ϵ
+exp(−ϵ2/(2u))
+�
+u(1 − u)
+du + 2Φ(ϵ) − 1 ,
+with Φ the cdf of the standard normal distribution. Thereby, we obtain that
+P
+� � 1
+0
+(Wt)+ dt ≤ x
+�
+≤ 1
+π
+� 1
+1−x/ϵ
+exp(−ϵ2/(2u))
+�
+u(1 − u)
+du + 2
+� ϵ
+0
+exp(−u2/2)
+√
+2π
+du ,
+and elementary bounds give the upper bound
+P
+� � 1
+0
+(Wt)+ dt ≤ x
+�
+≤ 2
+π
+�x
+ϵ
+1
+�
+1 − x/ϵ
++
+2ϵ
+√
+2π .
+13
+
+Choosing ϵ = x1/3, we obtain the upper bound
+P
+� � 1
+0
+(Wt)+ dt ≤ x
+�
+≤ 2
+π x1/3
+1
+√
+1 − x2/3 + 2x1/3
+√
+2π .
+6.2. Asymptotics of the spot volatility estimation in the continuous case
+6.2.1. Proof of Theorem 1
+In the sequel, we write An ≲ Bn for two real sequences, if there exists some n0 ∈ N and a
+constant K, such that An ≤ KBn, for all n ≥ n0.
+Step 1
+In the first step, we prove the approximation
+ˆσ2
+τ− =
+π
+2(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n
+�
+mk,n − mk−1,n)2
+=
+π
+2(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n
+�
+˜mk,n − ˜m∗
+k−1,n)2 + OP
+�
+hα∧1/2
+n
+�
+with
+˜mk,n = min
+i∈In
+k
+�
+ϵi + σ(k−1)hn(Wtn
+i − Wkhn)
+�
+, and
+˜m∗
+k−1,n = min
+i∈In
+k−1
+�
+ϵi − σ(k−1)hn(Wkhn − Wtn
+i )
+�
+.
+We show that for k ∈ {1, . . . , h−1
+n
+− 1}, it holds that
+mk,n − mk−1,n = ˜mk,n − ˜m∗
+k−1,n + OP
+�
+h1/2
+n
+�
+.
+(20)
+We subtract Xkhn from mk,n and mk−1,n, and use that it holds for all i that
+�
+Yi − Xkhn
+�
+−
+�
+Xtn
+i −
+�
+Xkhn + σ(k−1)hn(Wtn
+i − Wkhn)
+��
+=
+�
+σ(k−1)hn(Wtn
+i − Wkhn) + ϵi
+�
+.
+This implies that
+min
+i∈In
+k
+�
+Yi−Xkhn
+�
+−max
+i∈In
+k
+�
+Xtn
+i −
+�
+Xkhn+σ(k−1)hn(Wtn
+i −Wkhn)
+��
+≤ min
+i∈In
+k
+�
+σ(k−1)hn(Wtn
+i −Wkhn)+ϵi
+�
+.
+Changing the roles of
+�
+Yi − Xkhn
+�
+and
+�
+σ(k−1)hn(Wtn
+i − Wkhn) + ϵi
+�
+, we obtain by the analogous
+inequalities and the triangle inequality, with Mt := Xkhn +
+� t
+khn σ(k−1)hn dWs, that
+���mk,n − Xkhn − ˜mk,n
+��� ≤ max
+i∈In
+k
+��Xtn
+i − Mtn
+i
+�� ≤
+sup
+t∈[khn,(k+1)hn]
+��Xt − Mt
+��
+≤
+sup
+t∈[khn,(k+1)hn]
+���Ct − Ckhn −
+t
+∫
+khn
+σ(k−1)hn dWs
+��� .
+14
+
+We write (Ct) for (Xt) to emphasize continuity, see (4). (20) follows from
+sup
+t∈[khn,(k+1)hn]
+���Ct − Ckhn −
+t
+∫
+khn
+σ(k−1)hn dWs
+��� = OP(h1/2
+n ) ,
+(21)
+and the analogous estimate for mk−1,n and ˜m∗
+k−1,n. We decompose
+sup
+t∈[khn,(k+1)hn]
+���Ct − Ckhn −
+t
+∫
+khn
+σ(k−1)hn dWs
+��� ≤
+sup
+t∈[khn,(k+1)hn]
+���
+t
+∫
+khn
+(σs − σ(k−1)hn) dWs
+���
++
+sup
+t∈[khn,(k+1)hn]
+� t
+khn
+|as|ds .
+Under Assumption 1, we can assume that (σt) and (at) are bounded on [0, 1] by the localization
+from Section 4.4.1 in [11]. Using Itô’s isometry and Assumption 1, we obtain that
+E
+�� � t
+khn
+(σs − σ(k−1)hn) dWs
+�2�
+=
+� t
+khn
+E
+�
+(σs − σ(k−1)hn)2�
+ds
+= O
+� � t
+khn
+(s − (k − 1)hn)2α ds
+�
+= O
+�
+(t − (k − 1)hn)2α+1�
+.
+By Doob’s martingale maximal inequality and since supt∈[khn,(k+1)hn]
+� t
+khn |as|ds = OP(hn), it
+holds that
+sup
+t∈[khn,(k+1)hn]
+���Ct − Ckhn −
+t
+∫
+khn
+σ(k−1)hn dWs
+��� = OP
+�
+h(1/2+α)∧1
+n
+�
+.
+We conclude that (21) holds, since α > 0. Since
+h−1
+n
+�
+mk,n − mk−1,n
+��
+mk,n − ˜mk,n
+�
+= OP
+�
+hα∧1/2
+n
+�
+,
+and analogously for (mk−1,n − ˜m∗
+k−1,n), we conclude Step 1.
+Step 2
+We bound the bias of the spot volatility estimation using Step 1. For ⌊h−1
+n τ⌋ > Kn, we obtain
+with the function Ψn from (8) that
+E
+�
+ˆσ2
+τ− − Ψn
+�
+σ2
+τ−
+��
+=
+=
+1
+Kn
+π
+2(π − 2)
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n E
+��
+˜mk,n − ˜m∗
+k−1,n)2�
+− E
+�
+Ψn
+�
+σ2
+τ−
+��
++ O
+�
+hα∧1/2
+n
+�
+=
+1
+Kn
+π
+2(π − 2)
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+2(π − 2)
+π
+E
+�
+Ψn
+�
+σ2
+(k−1)hn
+��
+− E
+�
+Ψn
+�
+σ2
+τ−
+��
++ O
+�
+hα∧1/2
+n
+�
+≲
+1
+Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+E
+�
+σ2
+(k−1)hn − σ2
+τ−
+�
++ O
+�
+hα∧1/2
+n
+�
+≲
+1
+Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+E
+�
+σ(k−1)hn − στ−
+�
++ O
+�
+hα∧1/2
+n
+�
+15
+
+≲
+1
+Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+�
+E
+��
+σ(k−1)hn − στ−
+�2��1/2
++ O
+�
+hα∧1/2
+n
+�
+= O
+�
+(Kn hn)α�
+= O
+�
+hα/(1+2α)
+n
+�
+= O
+�
+K−1/2
+n
+�
+.
+We used that (α ∧ 1/2) > α/(2α + 1) for all α. For the asymptotic upper bounds we used the
+binomial formula and Hölder’s inequality to conclude with (2) from Assumption 1.
+Step 3
+For (9) and the consistency of ˆσ2
+τ−, we prove that
+E
+�
+ˆσ2
+τ−
+�
+= σ2
+τ− + O(1) .
+(22)
+Denote by Pσ(k−1)hn the regular conditional probabilities conditioned on σ(k−1)hn, and Eσ(k−1)hn
+the expectations with respect to the conditional measures. We obtain by the tower rule that
+E
+�
+h−1
+n
+�
+˜mk,n − ˜m∗
+k−1,n)2�
+= E
+�
+h−1
+n Eσ(k−1)hn
+��
+˜mk,n − ˜m∗
+k−1,n)2��
+= E
+�
+Eσ(k−1)hn
+��
+h−1/2
+n
+˜mk,n)2�
++ Eσ(k−1)hn
+��
+h−1/2
+n
+˜m∗
+k−1,n)2�
+− 2 Eσ(k−1)hn
+�
+h−1/2
+n
+˜mk,n
+�
+Eσ(k−1)hn
+�
+h−1/2
+n
+˜m∗
+k−1,n
+��
+,
+by the conditional independence of ˜mk,n and ˜m∗
+k−1,n.
+We establish and use an approximation of the tail probabilities of ( ˜mk,n) and ( ˜m∗
+k−1,n), re-
+spectively. For x ∈ R, we have that
+Pσ(k−1)hn
+�
+h−1/2
+n
+min
+i∈In
+k
+�
+ϵi + σ(k−1)hn(Wtn
+i − Wkhn)
+�
+> xσ(k−1)hn
+�
+= Pσ(k−1)hn
+�
+min
+i∈In
+k
+�
+h−1/2
+n
+�
+Wtn
+i − Wkhn
+�
++ h−1/2
+n
+σ−1
+(k−1)hnϵi
+�
+> x
+�
+= Eσ(k−1)hn
+� ⌊(k+1)nhn⌋
+�
+i=⌊knhn⌋+1
+P
+�
+ϵi > h1/2
+n σ(k−1)hn
+�
+x − h−1/2
+n
+(Wtn
+i − Wkhn)
+�
+|FX��
+= Eσ(k−1)hn
+�
+exp
+� ⌊(k+1)nhn⌋
+�
+i=⌊knhn⌋+1
+log
+�
+1 − Fη
+�
+h1/2
+n σ(k−1)hn
+�
+x − h−1/2
+n
+(Wtn
+i − Wkhn)
+�����
+by the tower rule for conditional expectations, and since ϵi
+iid
+∼ Fη. It holds that
+Wtn
+i − Wkhn =
+i−⌊knhn⌋
+�
+j=1
+˜Uj, ˜Uj
+iid
+∼ N(0, n−1), j ≥ 2, ˜U1 ∼ N
+�
+0, tn
+⌊knhn⌋+1 − khn
+�
+,
+Uj = h−1/2
+n
+˜Uj
+iid
+∼ N
+�
+0, (nhn)−1�
+, j ≥ 2, U1 ∼ N
+�
+0, h−1
+n
+�
+tn
+⌊knhn⌋+1 − khn
+��
+.
+We apply a Riemann sum approximation with a standard Brownian motion (Bt)t≥0. With (6),
+and a first-order Taylor expansion of z �→ log(1 − z), we obtain that
+Pσ(k−1)hn
+�
+h−1/2
+n
+min
+i∈In
+k
+�
+ϵi + σ(k−1)hn(Wtn
+i − Wkhn)
+�
+> xσ(k−1)hn
+�
+=
+16
+
+= Eσ(k−1)hn
+�
+exp
+�
+− h1/2
+n σ(k−1)hnη
+⌊(k+1)nhn⌋
+�
+i=⌊knhn⌋+1
+�
+x −
+i−⌊knhn⌋
+�
+j=1
+Uj
+�
++(1 + O(1))
+��
+= Eσ(k−1)hn
+�
+exp
+�
+− h1/2
+n nhnσ(k−1)hnη
+� 1
+0
+(Bt − x)− dt (1 + O(1))
+��
+.
+If nh3/2
+n
+→ ∞, we deduce that
+Pσ(k−1)hn
+�
+h−1/2
+n
+min
+i∈In
+k
+�
+ϵi + σ(k−1)hn(Wtn
+i − Wkhn)
+�
+> xσ(k−1)hn
+�
+= P
+�
+inf
+0≤t≤1 Bt ≥ x
+�
++ Eσ(k−1)hn
+�
+1
+�
+inf
+0≤t≤1 Bt < x
+�
+exp
+�
+− h3/2
+n nσ(k−1)hnη
+� 1
+0
+(Bt − x)− dt (1 + O(1))
+��
+= P
+�
+inf
+0≤t≤1 Bt ≥ x
+�
++ P
+�
+inf
+0≤t≤1 Bt < x
+�
+· O(1) .
+(23)
+We do not have a lower bound for
+� 1
+0 (Bt − x)− dt. However, using that the first entry time Tx of
+(Bt) in x, conditional on {inf0≤t≤1 Bt < x}, has a continuous conditional density f(t|Tx < 1), by
+Lemma 4 and properties of the Brownian motion we obtain for any δ > 0 that
+Eσ(k−1)hn
+�
+1
+�
+inf
+0≤t≤1 Bt < x
+�
+exp
+�
+− h3/2
+n nσ(k−1)hnη
+� 1
+0
+(Bt − x)− dt
+��
+≤ exp
+�
+−
+�
+h3/2
+n n
+�δσ(k−1)hnη
+�
+P( inf
+0≤t≤1 Bt < x) + P
+�
+inf
+0≤t≤1 Bt < x,
+� 1
+0
+(Bt − x)− dt ≤
+�
+h3/2
+n n
+�−1+δ�
+≤
+�
+exp
+�
+−
+�
+h3/2
+n n
+�δσ(k−1)hnη
+�
++
+� 1
+0
+P
+� � 1
+s
+(Bt)− dt ≤
+�
+h3/2
+n n
+�−1+δ�
+f(s|Tx < 1) ds
+�
+P( inf
+0≤t≤1 Bt < x)
+≤
+�
+exp
+�
+−
+�
+h3/2
+n n
+�δσ(k−1)hnη
+�
++
+� 1
+0
+P
+�
+(1 − s)
+� 1
+0
+(Bt)− dt ≤
+�
+h3/2
+n n
+�−1+δ�
+f(s|Tx < 1) ds
+�
+× P( inf
+0≤t≤1 Bt < x)
+= P( inf
+0≤t≤1 Bt < x) · Rn ,
+with a remainder
+Rn = O
+��
+h3/2
+n n
+�− 1+δ
+3 �
+.
+We applied Lemma 4 in the last step. From the unconditional Lévy distribution of Tx, f(s|Tx < 1)
+is explicit, but its precise form does not influence the asymptotic order. Under the condition
+nh3/2
+n
+→ ∞, the minimum of the Brownian motion over the interval hence dominates the noise
+in the distribution of local minima, different than for a choice hn ∝ n−2/3. By the reflection
+principle, it holds that
+P
+�
+− inf
+0≤t≤1 Bt ≥ x
+�
+= P
+�
+sup
+0≤t≤1
+Bt ≥ x
+�
+= 2P
+�
+B1 ≥ x
+�
+= P
+�
+|B1| ≥ x
+�
+,
+(24)
+for x ≥ 0.
+Using the illustration of moments by integrals over tail probabilities we exploit this, and a
+completely analogous estimate for ˜m∗
+k−1,n, to approximate conditional expectations. This yields
+17
+
+that
+Eσ(k−1)hn
+�
+h−1/2
+n
+˜mk,n
+�
+=
+=
+� ∞
+0
+Pσ(k−1)hn
+�
+h−1/2
+n
+˜mk,n > x
+�
+dx −
+� ∞
+0
+Pσ(k−1)hn
+�
+− h−1/2
+n
+˜mk,n > x
+�
+dx
+= −
+� ∞
+0
+Pσ(k−1)hn
+�
+σ(k−1)hn sup
+0≤t≤1
+Bt > x
+�
+dx + OP(1)
+= −
+� ∞
+0
+Pσ(k−1)hn
+�
+σ(k−1)hn|B1| > x
+�
+dx + OP(1)
+= −Eσ(k−1)hn
+�
+σ(k−1)hn|B1|
+�
++ OP(1)
+= −
+�
+2
+π σ(k−1)hn + OP(1) .
+We used (24). An analogous computation yields the same result for ˜m∗
+k−1,n:
+Eσ(k−1)hn
+�
+h−1/2
+n
+˜m∗
+k−1,n
+�
+= −
+�
+2
+π σ(k−1)hn + OP(1) .
+For the second conditional moments, we obtain that
+Eσ(k−1)hn
+�
+h−1
+n
+�
+˜mk,n
+�2�
+= 2
+� ∞
+0
+x Pσ(k−1)hn
+�
+|h−1/2
+n
+˜mk,n| > x
+�
+dx
+= 2
+� ∞
+0
+x Pσ(k−1)hn
+�
+σ(k−1)hn sup
+0≤t≤1
+Bt > x
+�
+dx + OP(1)
+= 2
+� ∞
+0
+x Pσ(k−1)hn
+�
+σ(k−1)hn|B1| > x
+�
+dx + OP(1)
+= σ2
+(k−1)hn + OP(1) .
+The last identity uses the illustration of the second moment of the normal distribution as an
+integral over tail probabilities. An analogous computation yields that
+Eσ(k−1)hn
+�
+h−1
+n
+�
+˜m∗
+k−1,n
+�2�
+= σ2
+(k−1)hn + OP(1) .
+This proves (22).
+Step 4
+We determine the asymptotic variance of the estimator. Illustrating moments as integrals over
+tail probabilities, with the analogous approximation as above, we obtain that
+Varσ(k−1)hn
+�
+˜m2
+k,n
+�
+= Eσ(k−1)hn
+�
+˜m4
+k,n
+�
+−
+�
+Eσ(k−1)hn
+�
+˜m2
+k,n
+��2
+= 2σ4
+(k−1)hnh2
+n + OP(h2
+n),
+Covσ(k−1)hn
+�
+˜m2
+k,n, ˜mk,n ˜m∗
+k−1,n
+�
+= Eσ(k−1)hn
+�
+˜m3
+k,n
+�
+Eσ(k−1)hn
+�
+˜m∗
+k−1,n
+�
+− Eσ(k−1)hn
+�
+˜m2
+k,n
+�
+Eσ(k−1)hn
+�
+˜mk,n
+�
+Eσ(k−1)hn
+�
+˜m∗
+k−1,n
+�
+= 2
+π σ4
+(k−1)hnh2
+n + OP(h2
+n),
+Varσ(k−1)hn
+�
+˜mk,n ˜m∗
+k−1,n
+�
+= Eσ(k−1)hn
+�
+˜m2
+k,n
+�
+Eσ(k−1)hn
+��
+˜m∗
+k−1,n
+�2�
+18
+
+−
+�
+Eσ(k−1)hn
+�
+˜mk,n
+�
+Eσ(k−1)hn
+�
+˜m∗
+k−1,n
+��2
+= σ4
+(k−1)hn
+�
+1 − 4
+π2
+�
+h2
+n + OP(h2
+n).
+We have used the first four moments of the half-normal distribution and their illustration via
+integrals over tail probabilities. The dependence structure between ˜mk,n and ˜m∗
+k,n also affects the
+variance of ˆσ2
+τ−. We perform approximation steps for covariances similar as for the moments of
+local minima above, using that
+h−1
+n Covσ(k−1)hn
+�
+˜mk,n, ˜m∗
+k,n
+�
+=
+� ∞
+−∞
+� ∞
+−∞
+�
+Pσ(k−1)hn
+�
+h−1/2
+n
+˜mk,n > x, h−1/2
+n
+˜m∗
+k,n > y
+�
+− Pσ(k−1)hn
+�
+h−1/2
+n
+˜mk,n > x
+�
+Pσ(k−1)hn
+�
+h−1/2
+n
+˜m∗
+k,n > y
+��
+dx dy
+=
+� ∞
+0
+� ∞
+0
+�
+Pσ(k−1)hn
+�
+σ(k−1)hn sup
+0≤t≤1
+Bt > x, σ(k−1)hn
+�
+sup
+0≤t≤1
+Bt − B1
+�
+> y
+�
+− Pσ(k−1)hn
+�
+σ(k−1)hn sup
+0≤t≤1
+Bt > x
+�
+Pσ(k−1)hn
+�
+σ(k−1)hn
+�
+sup
+0≤t≤1
+Bt − B1
+�
+> y
+��
+dx dy + OP(1).
+This shows that the joint distribution of ( ˜mk,n, ˜m∗
+k,n) relates to the distribution of the minimum
+and the difference between minimum and endpoint of Brownian motion over an interval, or equiv-
+alently the distribution of the maximum and the difference between maximum and endpoint. The
+latter is readily obtained from the joint density of maximum and endpoint which is a well-known
+result on stochastic processes. Utilizing this, we obtain that
+Covσ(k−1)hn
+�
+˜mk,n , ˜m∗
+k,n
+�
+=
+�1
+2 − 2
+π
+�
+hn σ2
+(k−1)hn(1 + OP(hα
+n)) + OP
+�
+hn
+�
+.
+The additional remainder of order hα
+n in probability is due to the different approximations of (σt)
+in ˜mk,n and ˜m∗
+k,n. This implies that
+Covσ(k−1)hn
+�
+˜mk,n ˜m∗
+k−1,n, ˜mk+1,n ˜m∗
+k,n
+�
+=
+�
+Eσ(k−1)hn
+�
+˜mk,n ˜m∗
+k,n
+�
+− Eσ(k−1)hn
+�
+˜mk,n
+�
+Eσ(k−1)hn
+�
+˜m∗
+k,n
+��
+Eσ(k−1)hn
+�
+˜m∗
+k−1,n
+�
+E
+�
+˜mk+1,n
+�
+= σ4
+(k−1)hn
+� 1
+π − 4
+π2
+�
+h2
+n + OP
+�
+h2
+n
+�
+.
+With analogous steps, we deduce two more covariances which contribute to the asymptotic vari-
+ance:
+Covσ(k−1)hn
+�
+˜m2
+k,n,
+�
+˜m∗
+k,n
+�2�
+= −h2
+n
+σ4
+(k−1)hn
+2
++ OP
+�
+h2
+n
+�
+,
+Covσ(k−1)hn
+��
+˜m∗
+k,n
+�2, mk ˜m∗
+k−1,n
+�
+= −h2
+n
+2
+3π σ4
+(k−1)hn + OP
+�
+h2
+n
+�
+.
+All covariance terms which enter the asymptotic variance are of one of these forms.
+For the
+conditional variance given σ2
+τ−, we obtain that
+Varσ2
+τ−
+�
+ˆσ2
+τ−
+�
+=
+1
+K2n
+π2
+4(π − 2)2
+�
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−2
+n Varσ2
+τ−
+�
+˜m2
+k,n + ( ˜m∗
+k,n)2 − 2 ˜mk,n ˜m∗
+k−1,n
+�
+19
+
+−
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧2
+4h−2
+n Covσ2
+τ−
+�
+˜mk,n ˜m∗
+k−1,n , ˜m2
+k−1,n + ( ˜m∗
+k−1,n)2 − 2 ˜mk−1,n ˜m∗
+k−2,n
+��
++ OP
+�
+K−1
+n
+�
+=
+1
+K2n
+π2
+4(π − 2)2
+�
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−2
+n
+�
+2Varσ2
+τ−
+�
+˜m2
+k,n
+�
++ 4Varσ2
+τ−
+�
+˜mk,n ˜m∗
+k−1,n
+�
++ 2 Covσ2
+τ−
+�
+˜m2
+k,n, ( ˜m∗
+k,n)2�
+− 4 Covσ2
+τ−
+�
+˜m2
+k,n, ˜mk,n ˜m∗
+k−1,n
+�
+− 4 Covσ2
+τ−
+�
+( ˜m∗
+k,n)2, ˜mk,n ˜m∗
+k−1,n
+��
++
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧2
+4h−2
+n
+�
+2 Covσ2
+τ−
+�
+˜mk,n ˜m∗
+k−1,n, ˜mk−1,n ˜m∗
+k−2,n
+�
+− Covσ2
+τ−
+�
+˜mk,n ˜m∗
+k−1,n, ˜m2
+k−1,n
+�
+− Covσ2
+τ−
+�
+˜mk,n ˜m∗
+k−1,n, ( ˜m∗
+k−1,n)2���
++ OP
+�
+K−1
+n
+�
+=
+1
+Kn
+π2
+4(π − 2)2 σ4
+τ−
+�
+8 − 16
+π2 − 1 − 8
+π + 8
+3π + 2
+� 4
+3π − 16
+π2
+��
++ OP
+�
+K−1
+n
+�
+=
+1
+Kn
+1
+(π − 2)2
+�7π2
+4
+− 2π
+3 − 12
+�
+σ4
+τ− + OP
+�
+K−1
+n
+�
+.
+Step 5
+For a central limit theorem, the squared bias needs to be asymptotically negligible compared to
+the variance, which is satisfied for Kn = O(h−2α/(1+2α)
+n
+). By the existence of higher moments of
+˜mk,n and ˜m∗
+k−1,n, a Lyapunov-type condition is straightforward, such that asymptotic normality
+conditional on σ2
+τ− is implied by a classical central limit theorem for m-dependent triangular
+arrays as the one by [3]. A feasible central limit theorem is implied by this conditional asymptotic
+normality in combination with FX-stable convergence.
+For the stability, we show that αn =
+K1/2
+n
+�
+ˆσ2
+τ− − σ2
+τ−
+�
+satisfy
+E [Zg(αn)] → E [Zg(α)] = E[Z]E [g(α)] ,
+(25)
+for any FX-measurable bounded random variable Z and continuous bounded function g, where
+α = σ2
+τ−
+1
+(π − 2)
+�
+7π2
+4
+− 2π
+3 − 12 U ,
+(26)
+with U a standard normally distributed random variable which is independent of FX. By the
+above approximations it suffices to prove this for the statistics based on ˜mk,n and ˜m∗
+k−1,n from
+(20), and Z measurable w.r.t. σ(
+� t
+0 σs dWs, 0 ≤ t ≤ 1). Set
+An = [τ − (Kn + 1)hn, τ] , ˜X(n)t =
+� t
+0
+1An(s)σ⌊sh−1
+n ⌋hn dWs , ¯X(n)t = Xt − ˜X(n)t .
+Denote with Hn the σ-field generated by ¯X(n)t and FX
+0 . The sequence
+�
+Hn
+�
+n∈N is isotonic with
+limit �
+n Hn = σ(
+� t
+0 σs dWs, 0 ≤ t ≤ 1). Since E[Z|Hn] → Z in L1(P) as n → ∞, it is enough to
+show that E[Zg(αn)] → E[Z]E[g(α)], for Z being Hn0-measurable for some n0 ∈ N. Observe that
+αn includes only increments of local minima based on ˜X(n)t, which are uncorrelated from those
+of ¯X(n)t. For all n ≥ n0, we hence obtain that E[Zg(αn)] = E[Z]E[g(αn)] → E[Z]E[g(α)] by a
+standard central limit theorem. This shows (25) and completes the proof of (14).
+20
+
+6.2.2. Proof of Proposition 2
+For the quarticity estimator (12), when ⌊h−1
+n τ⌋ > Kn, we have that
+E
+��
+σ4τ − − σ4
+τ−
+�
+=
+π
+4(3π − 8)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−2
+n E
+�
+˜m4
+k,n + ( ˜m∗
+k−1,n)4 − 4 ˜m3
+k,n ˜m∗
+k−1,n
+− 4 ˜mk,n( ˜m∗
+k−1,n)3 + 6 ˜m2
+k,n( ˜m∗
+k−1,n)2�
+− E[σ4
+τ−] + O
+�
+hα∧1/2
+n
+�
+=
+�
+π
+4(3π − 8)
+�
+6 − 16/π − 16/π + 6
+�
+− 1
+�
+E[σ4
+τ−] + O(1)
+= O(1) ,
+by using the same moments as in the computation of the asymptotic variance. We can bound its
+variance by
+Var
+��
+σ4τ −
+�
+≤
+π2
+16(3π − 8)2K2n
+2Knh−4
+n Var
+��
+˜mk,n − ˜m∗
+k−1,n
+�4�
++ O
+�
+K−1
+n
+�
+≤
+1
+Kn
+π2
+8(3π − 8)2 h−4
+n E
+��
+˜mk,n − ˜m∗
+k−1,n
+�8�
++ O
+�
+K−1
+n
+�
+≤
+1
+Kn
+π2
+8(3π − 8)2 h−4
+n 256 E
+�
+˜m8
+k,n
+�
++ O
+�
+K−1
+n
+�
+= O(K−1
+n ) ,
+what readily implies Proposition 2.
+6.3. Asymptotics of the truncated spot volatility estimation with jumps
+Denote by
+DX
+k := mk,n − mk−1,n, k = 1, . . . , h−1
+n
+− 1 ,
+the differences of local minima based on the observations (5), with the general semimartingale (4)
+with jumps. Denote by
+DC
+k := ˜mk,n − ˜m∗
+k−1,n, k = 1, . . . , h−1
+n
+− 1 ,
+the differences of the unobservable local minima considered in Section 6.2.
+In particular, the
+statistics DC
+k are based only on the continuous part (Ct) in (4) such that the jumps are eliminated.
+Theorem 3 is implied by Proposition 2, if we can show that
+π
+2(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n
+��
+DX
+k
+�21{|DX
+k |≤un} −
+�
+DC
+k
+�2�
+= OP
+�
+h
+α
+2α+1
+n
+�
+= OP
+�
+K−1/2
+n
+�
+.
+We decompose this difference of the truncated estimator, which is based on the available observa-
+tions with jumps, and the non-truncated estimator, which uses non-available observations without
+21
+
+jumps, in the following way:
+π
+2(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n
+��
+DX
+k
+�21{|DX
+k |≤un} −
+�
+DC
+k
+�2�
+=
+π
+2(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n
+�
+1{|DC
+k |>cun}
+��
+DX
+k
+�21{|DX
+k |≤un} −
+�
+DC
+k
+�2�
++ 1{|DC
+k |≤cun}1{|DX
+k |≤un}
+��
+DX
+k
+�2 −
+�
+DC
+k
+�2�
+− 1{|DC
+k |≤cun}1{|DX
+k |>un}
+�
+DC
+k
+�2
+�
+,
+with some arbitrary constant c ∈ (0, 1). We consider the three addends which are different error
+terms by
+1. large absolute statistics based on the continuous part (Ct);
+2. non-truncated statistics which contain (small) jumps;
+3. the truncation of also the continuous parts in statistics (DX
+k ) which exceed the threshold;
+separately. The probability P(|DC
+k | > cun) can be bounded using the estimate from (23) and
+Gaussian tail bounds. Observe that the remainder in (23) is non-negative. This yields that for
+some y > 0, we have that
+P
+�
+h−1/2
+n
+�� ˜mk,n
+�� > y
+�
+≤ P
+�
+sup
+0≤t≤1
+Bt > y
+�
+,
+what is intuitive, since the errors (ϵi) are non-negative. We apply the triangular inequality and
+then Hölder’s inequality to the expectation of the absolute first error term and obtain for any
+p ∈ N that
+π
+2(π − 2)Kn
+E
+�����
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n 1{|DC
+k |>cun}
+��
+DX
+k
+�21{|DX
+k |≤un} −
+�
+DC
+k
+�2�����
+�
+≤
+π
+2(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n E
+�
+1{|DC
+k |>cun}
+���
+�
+DX
+k
+�21{|DX
+k |≤un} −
+�
+DC
+k
+�2���
+�
+≤
+π
+2(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n
+�
+P
+�
+|DC
+k | > cun
+�
+2
+�
+u4
+n + E
+��
+DC
+k
+�4���1/2
+≤
+π
+2(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n
+�
+P
+�
+h−1/2
+n
+|DC
+k | > chκ−1/2
+n
+��1/2√
+2 u2
+n
+≤
+π
+2(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n
+�
+2 P
+�
+|B1| > c
+2hκ−1/2
+n
+��1/2√
+2 u2
+n
+≤
+√
+2π
+(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h2κ−1
+n
+exp
+�
+− c2
+4 h2κ−1
+n
+�
+22
+
+= O
+�
+h(−p+1)(2κ−1)
+n
+�
+= O
+�
+h
+α
+2α+1
+n
+�
+.
+Since 2κ − 1 < 0 and p arbitrarily large, we conclude that the first error term is asymptotically
+negligible. We will use the elementary inequalities
+DX
+k = min
+i∈In
+k
+�
+C i
+n + J i
+n + ϵi
+�
+− min
+i∈In
+k−1
+�
+C i
+n + J i
+n + ϵi
+�
+≤ min
+i∈In
+k
+�
+C i
+n + ϵi
+�
++ max
+i∈In
+k
+J i
+n − min
+i∈In
+k−1
+�
+C i
+n + ϵi
+�
+− min
+i∈In
+k−1
+J i
+n
+= DC
+k + max
+i∈In
+k
+J i
+n − min
+i∈In
+k−1
+J i
+n + OP
+�
+hα∧1/2
+n
+�
+,
+and
+DX
+k = min
+i∈In
+k
+�
+C i
+n + J i
+n + ϵi
+�
+− min
+i∈In
+k−1
+�
+C i
+n + J i
+n + ϵi
+�
+≥ min
+i∈In
+k
+�
+C i
+n + ϵi
+�
++ min
+i∈In
+k
+J i
+n − min
+i∈In
+k−1
+�
+C i
+n + ϵi
+�
+− max
+i∈In
+k−1
+J i
+n
+= DC
+k + min
+i∈In
+k
+J i
+n − max
+i∈In
+k−1
+J i
+n + OP
+�
+hα∧1/2
+n
+�
+.
+Therefore, we can bound |DX
+k − DC
+k | by
+sup
+i∈In
+k ,j∈In
+k−1
+|J i
+n − J j
+n | ≤
+sup
+s∈[khn,(k+1)hn],t∈[(k−1)hn,khn]
+|Js − Jt|
+≤
+sup
+s∈[khn,(k+1)hn]
+|Js − Jkhn| +
+sup
+t∈[(k−1)hn,khn]
+|Jkhn − Jt| ,
+and the remainder term of the approximation for the continuous part which is OP
+�
+hα∧1/2
+n
+�
+. Since
+the compensated small jumps of a semimartingale admit a martingale structure, Doob’s inequality
+for càdlàg L2-martingales can be used to bound these suprema. Based on these preliminaries, we
+obtain for the expected absolute value of the second error term that
+π
+2(π − 2)Kn
+E
+�����
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n 1{|DC
+k |≤cun}1{|DX
+k |≤un}
+��
+DX
+k
+�2 −
+�
+DC
+k
+�2�����
+�
+≤
+π
+2(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n E
+�
+1{|DC
+k |≤cun}1{|DX
+k |≤un}
+���
+�
+DX
+k
+�2 −
+�
+DC
+k
+�2���
+�
+≲
+1
+Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n E
+�
+sup
+i∈In
+k ,j∈In
+k−1
+|J i
+n − J j
+n |2 ∧ (1 + c)2u2
+n
+�
+≲
+1
+Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n E
+�
+sup
+t∈[khn,(k+1)hn]
+|Jt − Jkhn|2 ∧ u2
+n
+�
+≲
+1
+Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n E
+�
+|J(k+1)hn − Jkhn|2 ∧ u2
+n
+�
+= O
+�
+u2−r
+n
+�
+.
+23
+
+Applying the elementary inequalities from above, a cross term in the upper bound for
+�
+DX
+k
+�2 −
+�
+DC
+k
+�2 is of smaller order and directly neglected. It can be handled using the Cauchy-Schwarz
+inequality. In the last step, we adopt a bound on the expected absolute thresholded jump incre-
+ments from Equation (54) in [1]. For the negligibility of the second error term, we thus get the
+condition that
+κ(2 − r) ≥
+α
+1 + 2α .
+(27)
+Doob’s inequality yields as well that
+P
+�
+sup
+t∈[khn,(k+1)hn]
+|Jt − Jkhn| ≥ (1 − c)un
+�
+≤ E
+���J(k+1)hn − Jkhn
+��r∧1�
+�
+(1 − c)un
+�r∧1
++ O(hn) = O
+�
+hnu−r
+n
+�
+.
+For this upper bound, we decomposed the jumps in the sum of large jumps and the martingale of
+compensated small jumps to which we apply Doob’s inequality. We derive the following estimate
+for the expectation of the third (absolute) error term
+π
+2(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n E
+�
+1{|DC
+k |≤cun}1{|DX
+k |>un}
+�
+DC
+k
+�2�
+≤
+π
+2(π − 2)Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n E
+�
+1{2 sups∈[(k−1)hn,(k+1)hn] |Js−Jkhn|≥(1−c)un}
+�
+DC
+k
+�2�
+≲
+1
+Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+h−1
+n P
+�
+sup
+t∈[khn,(k+1)hn]
+|Jt − Jkhn| ≥ (1 − c)un
+�
+E
+��
+DC
+k
+�2�
+≲
+1
+Kn
+⌊h−1
+n τ⌋−1
+�
+k=(⌊h−1
+n τ⌋−Kn)∧1
+�E
+���J(k+1)hn − Jkhn
+��r∧1�
+�
+(1 − c)un
+�r∧1
++ O(hn)
+�
+= O
+�
+hnu−r
+n
+�
+.
+For the negligibility of the third error term, we thus get the condition that
+1 − κr ≥
+α
+1 + 2α .
+(28)
+Since under the conditions of Theorem 3, (27) and (28) are satisfied, the proof is finished by the
+negligibility of all addends in the decomposition above.
+References
+[1] Aït-Sahalia, Y. and Jacod, J. (2010). Is Brownian motion necessary to model high-frequency data? Annals
+of Statistics, 38(5), 3093–3128.
+[2] Aït-Sahalia, Y. and Jacod, J. (2014). High-frequency financial econometrics. Princeton University Press.
+[3] Berk, K. N. (1973). A central limit theorem for m-dependent random variables with unbounded m. Annals
+of Probability, 1(2), 352–354.
+[4] Bishwal, J.P.N. (2022). Parameter Estimation in Stochastic Volatility Models. Springer, Cham.
+24
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+[5] Bibinger, M., Jirak, M. and Reiß, M. (2016). Volatility estimation under one-sided errors with applications
+to limit order books. Annals of Applied Probability, 26(5), 2754–2790.
+[6] Bibinger, M., Neely, C. and Winkelmann, L. (2019). Estimation of the discontinuous leverage effect:
+Evidence from the Nasdaq order book. Journal of Econometrics, 209(2), 158–184.
+[7] Bibinger, M. and Winkelmann, L. (2018). Common price and volatility jumps in noisy high-frequency data.
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+[8] El Euch, O., Fukasawa, M. and Rosenbaum, M. (2018). The microstructural foundations of leverage effect
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+one-sided errors. Annals of Statistics 42(5), 1970–2002.
+[14] Li, Z. M. and Linton, O. (2022). A ReMeDI for microstructure noise, Econometrica, 90(1), 367–389.
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+[16] Mancini, C. (2009). Non-parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient
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+356–371.
+25
+

E9AyT4oBgHgl3EQfSfeg/content/tmp_files/2301.00088v1.pdf.txt

@@ -0,0 +1,1972 @@
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+1
+A Transient Electrical-Thermal Co-Simulation
+Method with LTS for Multiscale Structures
+Kai Zhu, Graduate Student Member, IEEE, and Shunchuan Yang, Senior Member, IEEE
+Abstract—In this article, an efficient transient electrical-
+thermal co-simulation method based on the finite element method
+(FEM) and the discontinuous Galerkin time-domain (DGTD)
+method is developed for electrical-thermal coupling analysis of
+multiscale structures. Two Independent meshes are adopted by
+the steady electrical analysis and the transient thermal simulation
+to avoid redundant overhead. In order to enhance the feasibility
+and efficiency of solving multiscale and sophisticated structures, a
+local time stepping (LTS) technique coupled with an interpolation
+method is incorporated into the co-simulation method. Several
+numerical examples from simple structures to complex multi-
+scale PDN structures are carried out to demonstrate the accuracy
+and efficiency of the proposed method by comparing with
+the COMSOL. Finally, two practical numerical examples are
+considered to confirm the performance of the proposed method
+for complex and multiscale structures.
+Index Terms—Discontinuous Galerkin time-domain (DGTD)
+method, electrical-thermal co-simulation, finite element method
+(FEM), local time stepping (LTS), PDN.
+I. INTRODUCTION
+W
+ITH the development of semiconductor technique and
+packaging technology over the past few decades, the
+typical size of components in integrated circuits (ICs) has
+kept shrinking while the integration density sees an upward
+trend. It is well known that the Joule heating effect in
+ICs is a challenging issue which has attracted substantial
+attention from researchers. Since the electrical malfunction
+is frequently related to temperature increment and improper
+power distribution, a reasonable design both in circuit structure
+and thermal sinks becomes crucially important. Taking the
+through-silicon via (TSV) for instance, it plays a key role in
+2.5D/3D ICs design [1], [2], for enabling the high-speed signal
+processing by smoothing paths for continuing the interconnect
+scaling. However, currents flowing through TSVs leads to local
+temperature rise, then potentially influences the transistors
+switching states and induces circuit failures or performance
+deterioration [3], [4]. Generally, electromigration, estimated
+by the Black’s equation, works as the primary cause of circuit
+failure [5], [6]. From them we can obtain that the mean time of
+This work was supported in part by the National Natural Science Foundation
+of China under Grant 62141405, 62101020, 62071125, in part by Defense In-
+dustrial Technology Development Program under Grant JCKY2019601C005,
+in part by Pre-Research Project under Grant J2019-VIII-0009-0170 and
+Fundamental Research Funds for the Central Universities. (Corresponding
+author: Shunchuan Yang.)
+K. Zhu is with the School of Electronic and Information Engineering,
+Beihang University, Beijing, 100083, China (e-mail: [email protected]).
+S. Yang is with the Research Institute for Frontier Science and the School of
+Electronic and Information Engineering, Beihang University, Beijing, 100083,
+China (e-mail: [email protected]).
+Manuscript received xxx; revised xxx.
+failure abides by a negative exponential multiplier relationship
+[7]. Therefore, an efficient and accurate electrical-thermal co-
+simulation algorithm can be indispensable for ICs design.
+There have been a great deal of numerical algorithms
+developed for solving thermal or electrical problems with
+respective advantages and deficiencies either in accuracy or
+efficiency. The analytical method, such as the equivalent circuit
+model, can be adopted in some circumstances and efficiency
+improvements can be obtained. However, it suffers from the
+lack of generality [8]. The finite volume method (FVM) can
+be adopted to analyze the heat transfer problems [9], which
+introduces the numerical flux to represent the information
+exchange between adjacent subdivision elements [10]. The
+widely used finite difference method (FDM) benefits from
+simplicity and efficiency [11], [12]. However, it is subject
+to staircase errors caused by structured meshes [13]. The
+finite method time domain (FETD) is practically restrained for
+solving a large matrix equation at each time step, which can
+be computationally intensive [14], [15] and an ill-conditioned
+matrix may be obtained. The domain decomposition method
+(DDM) coupled by the finite element tearing and intercon-
+necting (FETI) can contribute to alleviating the computation
+burden [16], [17], which have been applied for coping with
+large scale problems [18].
+The discontinuous galerkin time-domain (DGTD) method
+[19], [20] has attracted much attention and developed rapidly
+for inheriting the advantages of the FVTD method and the
+FETD method. It can be regarded as an element-level DDM
+[21], which implies an innate parallel characteristic and pro-
+vides the possibility of utilizing adaptive orders or types of
+basis functions in different elements [22].
+In this article, we introduce an electrical-thermal co-
+simulation scheme, which integrates the electrical analysis and
+thermal simulation through an iteration procedure. The electri-
+cal analysis is based on the finite element method (FEM) for its
+capacity of modeling sophisticated structures [14]. The thermal
+simulation can be divided into two phases and is based on the
+DGTD method. It is noteworthy that the thermal conduction
+equation is a parabolic partial differential equation, which is
+difficult to be solved directly by the traditional DGTD method
+[20]. In order to address this issue, an auxiliary equation need
+to be introduced to degrade the parabolic partial differential
+equation to a hyperbolic partial differential equation, which
+can be solved directly by the DGTD method [23].
+The DGTD method generally leads to a series of compact
+linear systems, the dimension of which is equal to the degrees
+of freedom within the corresponding element, and those matrix
+equations are required to be solved at each time step. The merit
+arXiv:2301.00088v1  [math.NA]  31 Dec 2022
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+2
+of this method over the FETD method lies in that the matrices
+dimension is quite small, which indicates the inverses of ele-
+mental matrices can be calculated readily and stored before the
+time iteration begins. The computational complexity mainly
+depends upon the number of elements, the order of chosen
+local basis functions, and the simulation time steps. If explicit
+time discretization scheme is adopted, the time step is limited
+rigorously by the minimal size of the discretized elements
+according to the Courant–Friedrichs–Lewy (CFL) stability
+condition [24]. For multiscale structures, the disparities of
+the mesh size in different regions can be large. If the global
+time step (GTS) scheme is adopted, the time step is restricted
+to the global smallest mesh size to guarantee the stability,
+thereby leading to substantial computational overhead. Some
+implicit-explicit methods have been developed to alleviate
+this issue, where an implicit time-marching scheme is used
+in regions with fine mesh, while the explicit time-marching
+scheme is adopted in coarse mesh regions [25], [26]. However,
+these methods dramatically increase the computational time
+and memory consumption for large scale problems and the
+highly disparate mesh element sizes may cause ill-conditioned
+problem when the implicit time-marching scheme is applied.
+Therefore, this method cannot cope with the problems effi-
+ciently encountered in analyzing multiscale structures [27].
+To tackle the aforementioned challenge, a local time step-
+ping (LTS) scheme is integrated into the electrical-thermal co-
+simulation method, with which the computational efficiency
+can be significantly improved and the capability of simulating
+multiscale and locally refined structures can be facilitated [28].
+The structure can be divided into several groups according
+to different mesh sizes, then elements in each group advance
+in time with local time steps [29], [30], which can reduce
+simulation time while maintaining accurate solutions. It is
+worth noting that the time step in each group is required to
+be selected carefully in order to guarantee the global stability
+and accuracy requirements. Then, a rigorous analysis of the
+stability of the LTS scheme is introduced based on the Von-
+Neuman method [31], [32].
+The article is organized as follows. In section II, the detailed
+formulation for electrical and thermal problem is presented.
+Then, the co-simulation procedure as well as the concept and
+implementation of the LTS technique is developed. In section
+III, several simple examples are presented to demonstrate
+the accuracy and efficiency of the proposed scheme, as well
+as the efficiency enhanced by the LTS technique compared
+with the GTS scheme. In section IV, the proposed scheme is
+applied to some practical structures to verify the capability
+of co-simulation for some practical structures. Finally, some
+conclusions are drawn in Section V.
+II. THEORIES AND FORMULATIONS
+In this section, the detailed formulations for current conti-
+nuity equation and heat conduction equation are introduced.
+Then the coupling simulation flow algorithm is elaborated,
+including the application of independent meshes for electrical
+and thermal simulations and the LTS technique developed for
+coping with multiscale structures.
+A. Formulations of Electrical and Thermal Analysis
+The current continuity equation is considered in the elec-
+trostatic analysis, which can be written as
+∇ · (σ∇φ + ε∇∂φ
+∂t ) = 0,
+(1)
+where σ and ε are the electrical conductivity and the per-
+mittivity of the medium, respectively. The Dirichlet boundary
+condition subjected to the governing equation can be expressed
+as
+φ = φ0,
+(2)
+The impedance boundary condition adopted for modeling
+lossy conductors can be imposed on the surface of the con-
+ductor with the form
+ˆn · σ∇φ =
+φ
+RS .
+(3)
+where ˆn is the unit normal vector pointing outward from
+the boundary of the computational domain, R denotes the
+surface impedance of a conductor, S represents the area of
+the boundary surface. In our implementation, the potential
+distribution is considered constant during an interval, and the
+steady-state solution is analyzed through the FEM method.
+As for the thermal simulation, the temperature evolution of
+a spatial point is governed by the transient heat conduction
+equation, which can be written as
+ρc∂T
+∂t = ∇ · (k∇T) + Q,
+(4)
+where ρ represents the density of the medium, c denotes
+the heat capacity, k is the thermal conductivity, Q represents
+the heat source, respectively. To solve (4), the corresponding
+boundary conditions include the Dirichlet boundary condition
+T = T0,
+(5)
+and the convective boundary condition
+ˆn · (k∇T) = −h (T − Ta) ,
+(6)
+where h is the convective heat transfer coefficient, Ta denotes
+the ambient temperature. Since the traditional DGTD method
+is incapable to solve the parabolic differential equation di-
+rectly, an auxiliary vector variable q is introduced to transform
+(4) to a hyperbolic differential equation [20], which can be
+rewritten as
+q = −k∇T,
+(7)
+ρc∂T
+∂t = −∇ · q + Q.
+(8)
+By implementing the Galerkin’s spatial testing procedure
+to (7) and (8) in the ith subdomain, the formulation can be
+obtained as
+�
+Vi
+Nk
+�
+qx + k ∂T
+∂x
+�
+dV = 0,
+(9)
+�
+Vi
+Nk
+�
+ρc∂T
+∂t +∇ · q − Q
+�
+dV = 0,
+(10)
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+3
+where Nk denotes the kth test basis function, qx represents
+the component of q in the x-direction, and the components in
+other two directions can be obtained in a similar manner. With
+the application of spatial integration and Gauss’s theorem, (9)
+and (10) can be rewritten as
+�
+Vi
+NkqxdV = k
+�
+Vi
+T ∂Nk
+∂x dV
+− k
+4
+�
+f=1
+�
+∂Vi
+nxT ∗NkdS,
+(11)
+�
+Vi
+Nkρc∂T
+∂t dV =
+�
+Vi
+(q · ∇Nk + NkQ) dV
+−
+4
+�
+f=1
+�
+∂Vi
+Nk ˆn · q∗dS,
+(12)
+where T ∗ and q∗ are the numerical fluxes which represent the
+information exchange between adjacent elements, which can
+be expressed as the linear combination of variables in adjacent
+elements like (13).
+ˆn · q∗ = D0
+�ˆn · qi + ˆn · qj�
++ D1
+�ˆn · qi − ˆn · qj�
++ D2
+�
+T i − T j�
+,
+T ∗ = D3
+�
+T i + T j�
++ D4
+�
+T i − T j�
++ D5
+�ˆn · qi − ˆn · qj�
+.
+(13)
+where T i and T j denote the temperature in self element and
+external neighboring element, respectively. The same notations
+are adopted for q, and Di (i = 0, . . . 5) are constants, which
+depend on the chosen numerical flux form. In our imple-
+mentation, the upwind flux is adopted for better convergence
+properties [33]. Therefore, the coefficients can be defined as
+D0 = D3 = 0.5, D2 = −4 and D1 = D4 = D5 = 0. In addi-
+tion, for the numerical flux on convective boundary surfaces,
+the coefficients are revised as D3 = D4 = 0.5, D2 = −h and
+D0 = D1 = D5 = 0 correspondingly.
+To establish the semi-discrete matrix system, (13) is applied
+to (11) and (12), with T and q approximated by nodal basis
+functions, which leads to
+Miqi
+x = kSi
+xTi −
+4
+�
+f=1
+�
+k(D3 + D4)Gi
+xTi
++k(D3 − D4)Gj
+xTj�
+,
+(14)
+ρcMi ∂Ti
+∂t = Si
+xqi
+x + Si
+yqi
+y + Si
+zqi
+z + Qi
+−
+4�
+f=1
+�
+D0Gi
+xqi
+x + D0Gi
+yqi
+y + D0Gi
+zqi
+z + D0Gj
+xqj
+x
++D0Gj
+yqj
+y + D0Gj
+zqj
+z + D2CiTi − D2CjTj�
+.
+(15)
+where Mi denotes the local mass matrix and Si
+x, Si
+y and Si
+z are
+the local stiffness matrices, the detailed expression of matrices
+and vectors in (14) and (15) can be written as
+�
+Mi�
+kl =
+�
+Vi
+N i
+kN i
+l dV,
+(16)
+�
+Si
+x
+�
+kl =
+�
+Vi
+∂N i
+k
+∂x N i
+l dV,
+(17)
+�
+Qi�
+k =
+�
+Vi
+N i
+kQdV,
+(18)
+�
+Gi
+x
+�
+kl =
+�
+∂Vi
+nxN i
+kN i
+l dS,
+(19)
+�
+Gj
+x
+�
+kl =
+�
+∂Vi
+nxN i
+kN j
+l dS,
+(20)
+�
+Ci�
+kl =
+�
+∂Vi
+N i
+kN i
+l dS,
+(21)
+�
+Cj�
+kl =
+�
+∂Vi
+N i
+kN j
+l dS.
+(22)
+where nx denotes the component in the x-direction of the
+outward normal vector. The detailed forms of other matrices
+including Si
+y, Si
+z, Gi
+y, Gi
+z, Gj
+y and Gj
+z can be obtained simi-
+larly.
+Since the obtained matrix equation is still semi-discrete, the
+derivative in the temporal dimension is required to be dis-
+cretized. The backward difference is unconditionally stable but
+yields a global matrix operation thereby losing the advantage
+of the DGTD method. Therefore, the forward difference is
+adopted in our implementation, where the time derivative term
+can be approximated by
+∂T
+∂t = T (t + ∆t) − T (t)
+∆t
++ O (∆t) .
+(23)
+to obtain the finial matrix equation in the thermal simulation.
+The forward difference is conditionally stable, with the
+convergence dependent on selected time step and related
+system coef��cients. In order to guarantee the stability, the
+Courant–Friedrichs–Lewy (CFL) condition is required to be
+satisfied, hence finding a valid approach for estimating the
+time step bound is of vital importance. In this implementation,
+the Von-Neuman stability analysis is introduced to validate the
+stability of the chosen time step. Firstly, a column vector is
+constructed to include all the unknowns of the system equation
+at a specific time. For instance, unknowns related to heat flux
+can be written as Uq =
+��
+q1
+x, q1
+y, q1
+z
+�
+, . . . ,
+�
+qN
+x , qN
+y , qN
+z
+��T ,
+where qi
+x, qi
+y and qi
+z denote the components of the heat flux of
+the ith element in different directions, respectively. N denotes
+the number of split elements. If the number of basis func-
+tions is M, qi
+x can be represented as
+�
+qi,1
+x , qi,2
+x , . . . , qi,M
+x
+�
+.
+Similarly, the column vector for T can be written as UT =
+�
+T1, . . . , TN�T . According to (14), Uq at tn can be obtained
+by
+Uq (tn) = AqUT (tn) ,
+(24)
+By substituting (24) into (15), the time-marching relation-
+ship between U (tn+1) and U (tn) can be rewritten in a
+compact matrix form
+UT (tn+1) = AT UT (tn) .
+(25)
+where the dimension of Aq and AT are 3MN × 3MN and
+MN × MN, respectively, which assemble the information of
+all element matrices and corresponding numerical flux. The
+concrete form of A can vary with the discretizing parameters
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+4
+adopted in the DGTD method, including the chosen time step
+(∆t), the order and form of the basis function (M), the form
+of numerical flux (q∗, T ∗), and the properties of material and
+meshes.
+The stability of the system can be analyzed by computing
+the MN eigenvalues of AT (λi, i = 1, . . . , MN). If all
+the eigenvalues are located inside the unit circle, it can be
+concluded that it is stable.
+B. The Electrical-Thermal Co-Simulation Procedure
+The flowchart of the procedure of the electrical-thermal co-
+simulation is represented in Algorithm 1. The initialization
+includes inputting the material parameters required in the
+electrical and thermal simulation, such as the electrical con-
+ductivity for electrical analysis, material density, heat capacity
+and thermal conductivity for thermal simulation. In addition,
+the time step, the simulation duration, and the relationship
+coefficients between material parameters and temperature need
+to be taken into consideration. Over each iteration, the elec-
+trical problem is solved through the FEM solver with the
+voltage and current density distribution obtained. Then, the
+dissipated power calculated in each element is considered as
+the source for subsequent thermal simulation. The dissipated
+power during a time step period can be written as
+Q = σ| ⃗E|2 = σ|∇φ|2.
+(26)
+In this article, the influence of temperature on electrical
+conductivity is considered, and its value can be calculated by
+the fitted interpolation function, the fourth-order form can be
+written as
+σ(T) =
+4
+�
+n=0
+AnT n
+T0 ≤ T ≤ T1.
+(27)
+where An (n = 0, . . . 4) are the fitting coefficients of mate-
+rial properties, [T0, T1] denotes the interpolation interval, the
+related coefficients of the materials used are shown in Table
+I [34], [35]. It is worth noting that the co-simulation method
+can also be used when other parameters vary with temperature
+without sacrificing its generality.
+TABLE I
+TEMPERATURE INTERPOLATION COEFFICIENTS OF TWO MATERIALS
+Cu
+Poly-Si
+A0
+2.91 × 108
+7.45 × 104
+A1
+−1.56 × 106
+−1.08 × 102
+A2
+3.70 × 103
+1.01 × 10−1
+A3
+−3.93
+−5.17 × 10−5
+A4
+−1.56 × 103
+1.07 × 10−7
+After the temperature distribution at a time step is obtained,
+the material parameters are updated within each split element
+based on this distribution and the interpolation function con-
+necting material properties and temperature if the simulation
+is not finished. Otherwise, the time iteration phase completes.
+Although some structures are electrically insulated and can
+be ignored in electrical analysis, they are still required to be
+considered for the heat transfer effect to ensure that the tem-
+perature distribution analysis in the overall region is accurate.
+Moreover, fine meshes for the electrical analysis may be used
+to accurately model complex structures, while relatively coarse
+meshes can be adopted in the thermal analysis for the slow
+pace of change. There can also be large gaps of mesh densities
+in different parts of the structure. If the structure is meshed in a
+unified density for electrical analysis and thermal simulation,
+the time step can be restricted to extremely small values to
+guaranteed stability, thereby leading to a great amount of
+computational overhead, even the time consumption can be
+unacceptable. To address this issue, two independent meshes
+for the electrical and thermal simulation are used to improve
+the flexibility of the algorithm and to avoid the redundant
+degrees of freedom (DoFs). A mapping relationship between
+meshes is built and a interpolation method is applied for
+imposing heat source and updating the material coefficients
+in the time-marching process.
+With the growing of unknowns, the construction of mapping
+relationship bridging the discontinuities between electrical and
+thermal meshes can be increasingly time-consuming. In order
+to alleviate this problem, two tree structures can be constructed
+in advance and split elements in electrical and thermal analysis
+are stored in leaf nodes to accelerate the traversal process. In
+this implement, two full octrees are adopted for simplicity,
+where within each level every node has 8 children, and the
+computational domain is divided into 512 blocks if the depth
+is set to 4, as illustrated in Fig. 1. The jth node of the
+ith level is denoted as Ci,j, where a node element includes
+the location information (tetrahedron indexes, space boundary)
+and pointers to each of its 8 children.
+...
+...
+Child
+Child
+Child
+0,0
+C
+1,0
+C
+1,4
+C
+1,7
+C
+...
+...
+2,0
+C
+3,0
+C
+3,4
+C
+...
+3,7
+C
+...
+…...
+Key
+Key
+Key
+...
+(a)
+1,0
+C
+1,1
+C
+1,2
+C
+1,3
+C
+1,4
+C
+1,5
+C
+2,0
+C
+2,1
+C
+2,2
+C
+2,3
+C
+2,4
+C
+2,5
+C
+3,0
+C
+Level 1
+Level 2
+Level 3
+(b)
+Fig. 1.
+Illustration of the construction of octree (a) Topology of the tree, (b)
+Geometric space representation for different levels.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+5
+Algorithm 1 Transient electrical-thermal co-simulation
+Input: Parameters of material properties
+Control factors of time stepping
+Output: Distribution of temperature and electric potential
+1: Initiate t = 0 and σ = σ0
+2: repeat
+3:
+Analyze current density through electrical analysis;
+4:
+Calculate produced Joule heat;
+5:
+Simulate temperature through thermal analysis;
+6:
+Update σ, set t = t + ∆t;
+7: until t >= tmax
+8: Output T and φ.
+C. The Explicit Local Time Stepping
+For a multiscale structure, elements can be partitioned into
+groups and different time steps are allowed in separate groups
+based on the explicit local time stepping (LTS) scheme, with
+time step in each group constrained by the local mesh size.
+Therefore, the total number of equation calculation needed
+for advancing the numerical solution from tn to tn + ∆t
+will decrease in comparison with using the global smallest
+time step. For simplicity, assuming the split elements are
+divided into four groups, the average sizes in each group
+are h, h/2, h/4 and h/8, respectively. Since the time step
+in each group is required to satisfy the stability condition,
+which can be chosen as ∆t1 = 2∆t2 = 4∆t3 = 8∆t4, with
+∆ti (i = 1, . . . , 4) denote the time step for the ith group. The
+implementation of time stepping is shown schematically in
+Fig. 2. The recursive procedure can be given as
+1) Elements in each group advance one step with local time
+steps ∆ti (i = 1, . . . , 4), the sequence of the advance-
+ment is Group #1 (averagely coarsest mesh), followed
+by Group #2, Group #3, and finally Group #4, as shown
+in Fig. 2(a).
+2) Elements in Group#4 (averagely finest mesh) advance one
+step with local time step ∆t4, then elements in Group#4
+and Group#3 are at the same time level, as shown in Fig.
+2(b).
+3) Elements in Group#3 and Group#4 advance one step in
+sequence with local time steps ∆t3 and ∆t4, as illustrated
+in Fig. 2(c).
+4) Elements in Group#4 advance one step with local time
+step ∆t4, then the elements in Group#2, Group#3, and
+Group#4 are at the same time level, as outlined in Fig.
+2(d).
+5) Elements in Group#2, Group#3, Group#4 advance one
+steps with local time steps ∆ti (i = 2, 3, 4), as presented
+in Fig. 2(e).
+6) The processes in 2–4 repeat until all elements reach the
+final interval.
+While analyzing an element in Group#i with a neighboring
+element located in Group#j (j < i), the temperature of the
+latter may be unknown at the current time step, which leads
+to a difficulty in constructing the numerical flux. Taking Fig.
+2(g) for instance, if the element in Group#4 has an adjacent
+element located in group 1-3, the temperature of the adjacent
+Group 1
+t
+size
+Group 2
+t
+size
+Group 3
+Group 4
+t
+size
+t
+size
+△t1
+(a)
+(b)
+(c)
+(d)
+t
+size
+t
+size
+t
+size
+t
+size
+(e)
+(f)
+(g)
+(h)
+△t1/2
+△t1/4
+△t1/8
+△t1/2
+3△t1/8
+△t1
+△t1/2
+△t1
+3△t1/4
+△t1
+△t1/2
+△t1/4
+△t1
+5△t1/8
+3△t1/4
+△t1
+7△t1/8
+△t1
+△t1
+Fig. 2.
+Example of the LTS stepping process for four groups with time steps
+∆t1 = 2∆t2 = 4∆t3 = 8∆t4.
+element is unknown at 7∆t1/8. However, after the process (1)
+presented in Fig. 2(a) finished, the temperature of elements
+in Group#1 at ∆t1 has been obtained, the similar situation
+is also for elements in Group#2 and Group#3. Therefore, a
+linear interpolation strategy can be adopted to estimate the
+temperature of adjacent elements in different groups. While
+considering the element in Group#i at n∆ti, the temperature
+of the adjacent element in Group#j at this time can be
+interpolated by
+T j
+app =
+�
+T j (nratio) ,
+j ≥ i
+(1 − C)T j (nlow) + CT j (nlow + 1) ,
+j < i
+(28)
+where the parameters are given by
+nratio = n∆ti
+∆tj
+,
+(29)
+nlow =
+�n∆ti
+∆tj
+�
+,
+(30)
+C = [n% (∆tj/∆ti)] / (∆tj/∆ti) .
+(31)
+and ⌊a⌋ denotes the maximum integer less than a. In the
+thermal analysis, the interpolation strategy for adjacent ele-
+ments in different groups is also needed to handle the auxiliary
+variable q. There are two types of interpolation methods for
+q. The first type is similar to that for T in (28), and nearly
+no extra storage is required except for some assigned for
+symbol marks. The other type consists of two substeps, firstly
+obtaining the interpolation temperature for adjacent elements
+by (28), then (14) is solved to get the approximated qx, and the
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+6
+same process for qy and qz. The second interpolation strategy
+is applied in our implementation, with a higher accuracy
+obtained, notwithstanding extra computational resources at an
+acceptable level are required.
+For the stability analysis of the LTS technique, the similar
+Von-Neuman method is also introduced. To better fulfill the
+matrix-filling process, the column vector including the un-
+knowns are divided according to their groups owning different
+time steps. Taking four groups in Fig. 2 for example, interme-
+diate unknowns of elements in different group at a time are
+stored in four arrays noted as Ui
+q and Ui
+T (i = 1, . . . , 4), then
+the time march from tn to tn + ∆ti in the ith group can be
+written in a compact form as
+Uq (tn) = AqiUT (tn) ,
+(32)
+UT (tn + ∆ti) = AT iUT (t0) + UT (tn) ,
+(33)
+where t0 denotes the initial simulation time. If there are Ni
+elements in Group#i, the dimensions of Aqi and AT i are
+3MNi × 3MN and MNi × MN, respectively. After the
+substeps in each group according to recursive order shown
+in Fig. 2 are finished, the time-marching from tn to tn + ∆t1
+in each group can be filled into a compact form
+UT (tn + ∆t1) = AiUT (t0) , i = 1, . . . , 4.
+(34)
+Based on the mapping relationship of the local node indexes to
+the global node indexes, Ai can be sorted into the final matrix
+A. Given that all the eigenvalues of A are located inside the
+unit circle, the proposed LTS method can be regarded as stable.
+III. VERIFICATION AND DEMONSTRATION OF THE
+STABILITY, ACCURACY AND EFFICIENCY
+In this section, several basic numerical examples are pro-
+vided to verify the efficiency and accuracy of the proposed
+electrical-thermal co-simulation method by comparing with
+the COMSOL software. The accuracy and stability of the LTS
+method are also verified, then the speedup performance is
+investigated for different time step ratios and mesh densities.
+All computations in this section are performed on the computer
+with Intel i9-10900 2.8 GHz CPU and 32 GB memory.
+A. Accuracy and Stability Verification
+A copper block with the dimension of 1.2×6.6×0.6 mm3 is
+tested to demonstrate the proposed co-simulation scheme, and
+the thermal property of the copper is shown in Table II. The
+convection boundary condition is applied on all six surfaces
+of the structure with h = 1000 W/(m2K), and the ambient
+temperature is set as Ta = 300 K. Two Gaussian pulses are
+imposed on one face of the structure sequentially, with the
+form of
+VGauss(t) = V0e−(t−t0)2/τ 2
+(35)
+where V0 = 0.02 V, τ 2 = 0.1, t0 = 0.6 and 3.6, respectively. It
+is noteworthy that V0 represents the peak voltage of the pulse,
+t0 is the time when the pulse reaches its peak, τ controls the
+width of the pulse.
+The temporary temperature at P1 (0.6, 3.3, 0.3) (mm) is
+illustrated in Fig. 3(a), it can be found that results obtained
+from the proposed method show excellent agreement with
+those from the COMSOL. To have a better clarification, the
+relative error of the observation point at each time point is
+outlined in Fig. 3(b), which is defined as
+RE = (Tp − Tc) /Tc
+(36)
+where Tp and Tc denote the results obtained from the proposed
+method and the COMSOL, respectively.
+TABLE II
+THERMAL PROPERTIES OF DIFFERENT MATERIALS
+Thermal
+conductivity
+(W · m−1 · K−1)
+Heat
+capacity
+(J · kg−1 · K−1)
+Density
+(kg · m−3)
+Copper
+400
+385
+8.7 × 103
+Nickel
+91
+440
+8.9 × 103
+Al2O3
+10
+750
+3.9 × 103
+Silicon
+130
+700
+2.3 × 103
+0
+2
+4
+6
+Time (s)
+300
+320
+340
+360
+380
+400
+Temperature (K)
+0.01
+0.02
+0.03
+0.04
+0.05
+Voltage (V)
+COMSOL
+Proposed
+V
+(a)
+0
+2
+4
+6
+Time (s)
+-6
+-3
+0
+3
+6
+8
+Relative Error
+10-6
+Error
+(b)
+Fig. 3.
+Simulation of the temperature with voltage pulses imposed. (a)
+Temperature at P1 (0.6, 3.3, 0.3) (mm) obtained from the proposed scheme
+and the COMSOL, (b) The relative error.
+-1
+-0.5
+0
+0.5
+1
+Real(
+i)
+-1
+-0.5
+0
+0.5
+1
+Imag(
+i)
+0.9 1
+-0.2
+0
+0.2
+Fig. 4.
+The distribution of the eigenvalues of AT with ∆t = 2 × 10−5 s.
+To test the stability, eigenvalues of AT in this context are
+analyzed and presented in Fig. 4. Since all the eigenvalues are
+located inside the unit circle, the stability in this circumstance
+can be concluded.
+In Fig. 5, the temperature distribution of the plane z = 0.3
+(mm) of the structure at t = 4 (s) obtained from these two
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+7
+methods are presented. Then, the electric potential distribution
+of the same plane at t = 4 (s) is shown in Fig. 6. For a
+fair comparison, the structure is split into a similar number
+of tetrahedrons, and the same time step is adopted for the
+proposed scheme and the COMSOL. The CPU time and
+memory consumption are only about 265 s and 15 MB for
+the proposed scheme. By comparison, the figures are 2014 s
+and 3.1 GB for the COMSOL, which implies an efficiency
+improvement of the proposed scheme.
+Temperature (K)
+389.5
+389.0
+(a)
+(b)
+389.1
+389.2
+389.3
+389.4
+y
+(mm)
+x  (mm)
+0
+1
+2
+3
+4
+5
+6
+0.5
+1
+y
+(mm)
+x  (mm)
+0
+1
+2
+3
+4
+5
+6
+0.5
+1
+Fig. 5.
+Temperature distribution of the plane z = 0.03 mm at 4 s obtained
+from (a) the COMSOL, (b) the proposed scheme.
+ (mV)
+10
+0
+(a)
+(b)
+2
+4
+6
+8
+y
+(mm)
+x  (mm)
+0
+1
+2
+3
+4
+5
+6
+0.5
+1
+y
+(mm)
+x  (mm)
+0
+1
+2
+3
+4
+5
+6
+0.5
+1
+12
+14
+
+Fig. 6.
+Electric potential distribution of the plane z = 0.03 mm at 4 s
+obtained from (a) the COMSOL, (b) The proposed scheme.
+0
+2
+4
+6
+Time (s)
+300
+320
+340
+360
+380
+Temperature (K)
+0
+0.01
+0.02
+0.03
+Voltage (V)
+COMSOL
+Proposed
+V
+(a)
+0
+2
+4
+6
+Time (s)
+-8
+-6
+-4
+-2
+0
+2
+4
+Relative Error
+10-6
+Error
+(b)
+Fig. 7.
+Simulation of temperature with a step voltage imposed. (a)
+Temperature at P1 (0.6, 3.3, 0.3) (mm) obtained from the proposed scheme
+and the COMSOL, (b) The relative error.
+In addition, a ladder signal is imposed for testing, the results
+and relative errors are shown in Fig. 7. Similar to the previous
+situation, results obtained from the proposed scheme agree
+well with the COMSOL.
+B. Efficiency Improvement by the LTS Method
+To further demonstrate the stability and efficiency improve-
+ment of the LTS technique, the structure of two conductors
+composed of copper and nickel covered by a silicon box is
+considered, as illustrated in Fig. 8. The thermal properties of
+the media are listed in Table II. In this implementation, a volt-
+age pulse varying over time is imposed on one face of the cop-
+per, which can be written as V1 = 0.04 + VGauss sin (300πt)
+(V), where VGauss is described in (35) with V0 = 0.03,
+τ = 0.01, and t0 = 0.001. For the electrical analysis, only
+the copper is considered and discretized. For the thermal
+analysis, the whole structure is discretized into tetrahedrons
+and then divided into three groups according to material. The
+convection boundary condition is applied on six surfaces of
+the silicon block to represent the thermal transfer between the
+object and the environment, with h = 1000 W/(m2 · K) and
+the ambient temperature Ta = 300 K.
+Group II
+0.86
+0.58
+Group III
+0.18
+0.06
+0.66
+0.12
+Nickle
+Copper
+Silicon
+Unit: mm
+0.34
+Electrical 
+mesh
+Heat 
+mesh
+Mapping
+1t
+
+2t
+
+3t
+
+Group I
+Port
+Sink
+Fig. 8.
+Geometry of the conductors and silicon box, and illustration of the
+independent meshes adopted in electrical and thermal analysis in different
+groups.
+0
+0.005
+0.01
+0.015
+0.02
+Time (s)
+300
+320
+340
+360
+380
+400
+Temperature (K)
+2
+4
+6
+8
+10
+Voltage (mV)
+COMSOL-p1
+COMSOL-p2
+COMSOL-p3
+Proposed-p1
+Proposed-p2
+Proposed-p3
+V
+(a)
+0
+0.005
+0.01
+0.015
+0.02
+Time (s)
+-10
+-8
+-6
+-4
+-2
+0
+2
+Relative Error
+10-4
+Error-p1
+Error-p2
+Error-p3
+(b)
+0
+0.005
+0.01
+0.015
+0.02
+Time (s)
+-5
+0
+5
+10
+Relative Error
+10-4
+Error-p1
+Error-p2
+Error-p3
+(c)
+0
+1
+2
+3
+4
+5
+Number of tetrahedron
+105
+0
+1
+2
+3
+4
+Simulation Time (s)
+104
+5
+5.5
+6
+Ratio
+Proposed-GTS
+Proposed-LTS
+Ratio
+(d)
+Fig. 9.
+Simulation of the temperature with the voltage pulses imposed.
+(a) Temperature at P1 (-0.06, 0.33, 0.03) (mm), P2 (-0.52, 0.33, 0.03) (mm)
+and P3 (-0.29, 0.33, 0.03) (mm) obtained from the proposed-LTS scheme
+and COMSOL, (b) The relative error between the proposed-GTS scheme and
+the COMSOL, (c) The relative error between the proposed-LTS scheme and
+the COMSOL, (d) The comparison of the CPU time for the proposed-GTS
+scheme and the proposed-LTS scheme with the structures discretized into
+different number of tetrahedrons.
+
+0:06+00
+2.0
+00+90.5
+lewbetaineS.0 00+90.0
+0'4
+0.0
+8.0
+00+98.10'06+00
+2.0
+.
+00+00.50'06+000:5
+06+000.Q88
+380°2
+LGbGLLG6 (K)LGGLLG
+0.8
+380'
+380'5
+380'3
+380°4
+(K)
+38020.0
+o's
+04
+oe
+8.0
+b!0.Q88
+380°2
+LGbGLLG6 (K)JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+8
+Since the temperature changes more dramatically over time
+for the Joule heating caused by conducting current in the
+copper, it is discretized into finer elements and recognized
+as Group#3. The larger volume of the outer layer composed
+of silicon is desired to be discretized into coarser elements to
+save computational cost. Three observing probes are placed
+at P1 (-0.06, 0.33, 0.03) (mm), P2 (-0.52, 0.33, 0.03) (mm),
+and P3 (-0.29, 0.33, 0.03) (mm) to record the temperature
+variation. In Fig. 9(a), the temperature of probes obtained from
+the proposed-LTS scheme and the COMSOL are presented. An
+excellent agreement can be observed as expected.
+The CPU time and memory consumption of the COMSOL,
+and the proposed scheme with global time stepping, as well as
+the proposed scheme with local time stepping are compared
+in Table III, which indicates the efficiency improvement of
+the proposed scheme. For the proposed-LTS scheme, time
+steps adopted in different groups have the relationship ∆t3 =
+2∆t2 = 6∆t1, with the minimum time step ∆t1 = 5 × 10−8
+s. For the COMSOL and the proposed-GTS scheme, the time
+step is selected as ∆t = 5 × 10−8 s. The relative error of
+the proposed-GTS scheme and the proposed-LTS scheme in
+comparison with the COMSOL are shown in Fig. 9(b) and
+Fig. 9(c), respectively. It can be observed that about four times
+speed up is achieved with a reasonable accuracy loss.
+In this occasion, the eigenvalues of AT are illustrated in
+Fig. 11(a), since all the eigenvalues are located inside the unit
+circle, the stability can be guaranteed theoretically. When time
+steps are chosen as ∆t3 = 5∆t2 = 100∆t1 with ∆t1 =
+2.5 × 10−8 s, the eigenvalues distribution is shown in Fig.
+11(b), and the stability can still be verified.
+Temperature (K)
+370
+372
+374
+376
+378
+380
+382
+(a)
+y
+(mm)
+x  (mm)
+0
+0.2
+0.1
+0.4
+0.6
+0.8
+0.3
+0.2
+0.4
+0.5
+(b)
+y
+(mm)
+x  (mm)
+0
+0.2
+0.1
+0.4
+0.6
+0.8
+0.3
+0.2
+0.4
+0.5
+Fig. 10.
+Temperature distribution of the plane z = 0.03 mm at 0.01 s
+obtained from (a) the COMSOL, (b) the proposed-LTS scheme with ∆t3 =
+2∆t2 = 6∆t1.
+TABLE III
+COMPARISON OF THE COMPUTATIONAL COST BETWEEN THE
+PROPOSED-GTS SCHEME, THE PROPOSED-LTS SCHEME AND THE
+COMSOL
+Tetrahedrons
+Memory
+CPU Time (s)
+COMSOL
+1016
+3.7 GB
+9239
+Proposed-GTS
+1357
+25 MB
+698
+Proposed-LTS
+1357
+27 MB
+172
+When the structure is discretized into finer meshes, the sim-
+ulation time is compared in Fig. 9(d), it can be obtained that
+the speedup ratio, which is defined by the ratio of the runtime
+of the two schemes, is relatively stable with varied number of
+tetrahedrons if the time step relationship is constant. However,
+the saved time has seen an upward trend with the number
+of tetrahedrons growing. As for the memory consumption,
+when 418,546 tetrahedrons are generated in this context, the
+memory consumption is about 3 GB. Therefore, the capability
+of the proposed-LTS scheme applied for simulating multiscale
+problem can be demonstrated.
+-1
+-0.5
+0
+0.5
+1
+Real(
+i)
+-1
+-0.5
+0
+0.5
+1
+Imag(
+i)
+0.9 1
+-0.2
+0
+0.2
+(a)
+-1
+-0.5
+0
+0.5
+1
+Real(
+i)
+-1
+-0.5
+0
+0.5
+1
+Imag(
+i)
+0.9 1
+-0.2
+0
+0.2
+(b)
+Fig. 11.
+The eigenvalues distribution of AT for the proposed-LTS scheme
+with different time step relationship. (a) ∆t3 = 2∆t2 = 6∆t1, (b) ∆t3 =
+5∆t2 = 100∆t1.
+IV. CO-SIMULATION EXAMPLES
+In this section, two representative PDN structures are pre-
+sented to demonstrate the capability of the proposed method.
+By comparing the results of electrical–thermal coupling sim-
+ulation with the results of electrical simulation, the effect of
+temperature variation on potential distribution is analyzed.
+A. PDN structure with power planes
+A simplified PDN structure is considered in this example
+[36], as shown in Fig. 12, which can be recognized as a
+combination of units. The overhead and cross-sectional view
+of the grid unit are presented in Fig. 13 (a) and Fig. 13 (b),
+respectively. The dimension of the cross section of conductor
+grid is 10×1 mm2, the radius and height of micro bumps are
+10 and 20 mm, respectively, with the figures for connecting
+conductors between different grid layers are 7 and 30 mm. The
+dimension of vias connecting the neighboring two conductor
+layers is 5×5×0.6 mm3. For simplicity, the conductor layers,
+and the vias, as well as the micro bumps are all made of copper
+and the structure is covered by a silicon rectangular with the
+dimension of 920 × 1040 × 75.4 mm3.
+Similar to previous example, a periodic voltage pulse is
+imposed on the upper surfaces of micro bumps, as shown in
+Fig. 14 (a), while the connecting conductors at the bottom
+layer are recognized as the ground. The related parameters of
+the voltage pulses are shown in Table IV. For the electrical
+analysis, only the units through which current flows are
+considered, including micro bumps, power planes and con-
+necting conductors. For the thermal analysis, all the structures
+are discretized into tetrahedrons and then divided into three
+regions, including power region, ground region and silicon
+region. The time steps adopted in different regions have the
+
+X
+30:0
+3150
+340
+31e:0
+318°0
+380°0
+385°0
+LGLG3100
+315'0
+3140
+318'0
+380'0
+385'0JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+9
+relationship ∆t3 = 2∆t2 = 4∆t1, with the minimum time
+step ∆t1 = 2 × 10−11 s. The convection boundary condition
+is applied on the six surfaces of silicon block to represent the
+thermal transfer between the structure and the environment,
+with h = 20 W/(m2 · K) and the ambient temperature
+Ta = 300 K.
+Unit I
+Fig. 12.
+Illustration of the PDN with power planes.
+10
+10
+8
+ 
+ 
+(a)
+10
+10
+8
+Unit: mm
+Ground 
+pad
+Power 
+pad
+(b)
+1
+0.6
+5
+20
+20
+14
+30
+Unit: mm
+bump
+Via
+p3
+p6
+p7
+p4
+p5
+p1
+p2
+(c)
+Fig. 13.
+The overhead and side view of the PDN structure. (a) one type of
+unit cell, (b) another type of unit cell for the conductor grid, (c) the size of
+the structure and the location of seven observing probes.
+There are 25,1889 and 545,915 tetrahedrons generated in
+electrical and thermal simulation, respectively, which result in
+2,282,092 unknowns during an interval. In order to compare
+the temperature rise effect in different sections, seven observe
+probes are placed to record the temperature variation, with
+the coordinates of these probes listed in Table VI, which are
+also marked in Fig. 13(c): (1) P1, P3 and P5 are on the vias
+connecting conductor layers; (2) P2 and P4 are on the power
+grids; (3) P6 and P7 are on the connecting conductors between
+different grid layers.
+The temporary temperature at probes on three different units
+obtained from the proposed-LTS scheme is shown in Fig.
+14 (b)–(d). Then, the voltage difference distribution between
+considering thermal effect and without heat impact is con-
+sidered in Fig. 15. Due to the simulation time constraints, the
+temperature rises are not conspicuous, but it can be anticipated
+to keep rising at subsequent time. The total number of time
+steps for the finest group is 16,000, and the simulation costs
+41,960 s in total and 4.1 GB memory, with 1,365 s spent on
+pre-processing and 40,595 s on time stepping, respectively.
+TABLE IV
+THE SPATIAL COORDINATES OF SEVEN PROBES (UNITS: mm)
+P1
+P2
+P3
+P4
+P5
+P6
+P7
+x
+-9
+-9
+-9
+-9
+-9
+-9
+-9
+y
+79.5
+77
+77
+72
+79.5
+79.5
+77
+z
+-0.3
+0.5
+1.3
+2.1
+2.9
+-1.2
+-1.2
+0
+50
+100
+150
+200
+Time (ns)
+10
+20
+30
+40
+50
+Voltage (V)
+(a)
+(b)
+(c)
+(d)
+Fig. 14.
+The imposed voltage pulse and transient temperature at probes on
+different units. (a) the imposed pulse, (b) temperature on the first unit, (c) the
+second unit, (d) the third unit.
+
+T at Ps
+T
+T at P6
+T at P-
+302
+300
+0
+100
+200
+Time (ns)1
+300306
+T at Pi
+T at P2
+T at P3
+304
+T at n306
+P4
+()
+T at Ps
+304
+_T at P7
+302
+300
+0
+100
+200
+Time (ns)300310
+T at Pi
+T at P2
+308
+T at P3
+TT at Ps
+302
+_T at P6
+T
+T at P-
+301
+300
+0
+100
+200
+Time (ns)300304
+4rTatPi
+T at P2
+TatP3
+303
+T at nJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+10
+ (mV)
+2.9
+2.0
+1.0
+0.0
+-1.0
+-2.1
+Fig. 15.
+Electric potential difference between considering and ignoring
+thermal effects at t = 300 ns.
+B. PDN structure with chips
+To further demonstrate the capability of the proposed
+scheme, a PDN structure with 4 simplified chips is tested
+in this section, as shown in Fig. 16 (a). The PDN structure
+consists of three conductor layers, with the size of each layer
+identical to the former example. The size information of the
+chips are shown in Fig. 16 (b). In view of the scale difference
+of the structure, it can be divided vertically into three regions
+including chip region, connecting conductor region and PDN
+region. The structure is made of copper for simplicity.
+Similar to the former example, for the electrical analysis, a
+periodic voltage signal is imposed on the upper surface of the
+chips, which is represented in Fig. 17 (a), while the bottom
+of the conductor layer is recognized as the ground. For the
+thermal analysis, time steps adopted in different groups have
+the relationship ∆t3 = 2∆t2 = 6∆t1, with the minimum
+time step ∆t1 = 8 × 10−11 s. The convection boundary
+condition is applied on the outer surfaces of the structure
+to represent the thermal transfer with the environment, with
+h = 20 W/(m2 · K) and the ambient temperature Ta = 300
+K.
+(a)
+Unit: mm
+170
+150
+30
+1
+3
+Region I
+Region II
+Region III
+(b)
+Fig. 16.
+Illustration of the PDN structure with chips. (a) 3-D diagram, (b)
+overhead overview and size information.
+0
+2000
+4000
+6000
+Time (ns)
+4
+6
+8
+10
+12
+Voltage (V)
+(a)
+0
+2000
+4000
+6000
+Time (ns)
+300
+302
+304
+306
+308
+T (K)
+T at p1
+T at p2
+T at p3
+T at p4
+T at p5
+T at p6
+T at p7
+(b)
+0
+2000
+4000
+6000
+Time (ns)
+300
+301
+302
+303
+304
+305
+T (K)
+T at p1
+T at p2
+T at p3
+T at p4
+T at p5
+T at p6
+T at p7
+(c)
+0
+2000
+4000
+6000
+Time (ns)
+300
+300.2
+300.4
+300.6
+300.8
+301
+T (K)
+T at p1
+T at p2
+T at p3
+T at p4
+T at p5
+T at p6
+T at p7
+(d)
+Fig. 17.
+The imposed voltage pulse and transient temperature at probes on
+different layers. (a) the imposed pulse, (b) temperature on the third layer, (c)
+the second layer, (d) the first layer.
+There are 232,172 and 110,682 tetrahedrons generated in the
+electrical and thermal simulation, respectively, which results
+in 530,146 unknowns during an interval. The total number of
+time steps for the finest group is 80,000, which costs 59,597 s
+in total and 1.1 GB memory, with 72 s spent on pre-processing
+and 59,525 s on time stepping. In order to compare the
+temperature rises in different layers, seven observing probes
+are placed on each layer to record the temperature variation,
+with the coordinates of probes on the top layer identical to the
+former example, as listed in Table VI.
+(a)
+(b)
+(c)
+(d)
+Fig. 18.
+Temperature profiles of plane y = 85 mm of the PDN structure at
+four instances. (a) 2560 ns, (b) 5120 ns, (c) 6400 ns, (d) colormap.
+(a)
+(b)
+Fig. 19.
+Current density amplitude of plane y = 85 mm of the PDN structure
+at 6400 ns.
+The temporary temperature at the probes on different layers
+obtained from the proposed-LTS scheme is shown in Fig. 17
+(b)–(d). Fig. 18 shows the temperature profiles at the plane
+
+303
+(D)
+3:02
+301
+3001.8e+09
+1.5e+9
+Amplitude (A/m)
+1.0e+9
+5.0e+8
+8.4e-04I.S
+0.1-
+0.0
+J'O
+5'O
+e.sJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+11
+(a)
+(b)
+Fig. 20.
+Temperature profile of the exterior surface of the PDN structure at
+t = 6400 ns.
+y = 85 mm at four instances, 2560, 5120, and 6400 ns. The
+current density amplitude distribution of this plane at 6400 ns
+is also presented in Fig. 19. It can be obtained that temperature
+rise concentrates in small areas for greater current density and
+spread away from those locations.
+Then, the voltage difference distribution between consid-
+ering thermal effect and without heat impact is considered
+in Fig. 20. Despite the maximum number is quite small for
+the limited time in this simulation, it is foreseeable that the
+influence will keep accumulating over time, which indicates
+the indispensability of taking thermal effect into consideration.
+V. CONCLUSION
+In this article, a transient electrical–thermal co-simulation
+scheme has been developed based on the FEM and the DGTD
+method. In the thermal simulation, an auxiliary variable is
+introduced to degrade the parabolic equation to a hyperbolic
+one, which can be solved by DGTD method directly. By
+adopting different discretized volumes and independent grids
+for the electrical solver and the thermal solver, redundant
+computational overhead can be avoided. On the premise of
+guaranteeing the stability, a flexible explicit LTS technique
+based on interpolation method is incorporated into the solver
+to improve the capability of solving multi-scale problem. With
+the LTS technique, the sophisticated structure can be divided
+into groups and different time steps are allowed in separate
+groups. Two numerical examples are provided to demonstrate
+the validity, flexibility, as well as the efficiency improvement
+by the LTS technique in comparison with COMSOL. Further-
+more, the electrical–thermal behavior of two multiscale PDN
+systems is analyzed. Oriented to the increasingly miniaturized
+and multiscale electronic devices, the proposed co-simulation
+algorithm provides an accurate and effective alternative to
+analyze their potential distribution and thermal effects in real
+time.
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+

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