On the Local Equivalence Between the Canonical and the Microcanonical Ensembles for Quantum Spin Systems

We study a quantum spin system on the d-dimensional hypercubic lattice \(\Lambda \) with \(N=L^d\) sites with periodic boundary conditions. We take an arbitrary translation invariant short-ranged Hamiltonian. For this system, we consider both the canonical ensemble with inverse temperature \(\beta _0\) and the microcanonical ensemble with the corresponding energy \(U_N(\beta _0)\). For an arbitrary self-adjoint operator \(\hat{A}\) whose support is contained in a hypercubic block B inside \(\Lambda \), we prove that the expectation values of \(\hat{A}\) with respect to these two ensembles are close to each other for large N provided that \(\beta _0\) is sufficiently small and the number of sites in B is \(o(N^{1/2})\). This establishes the equivalence of ensembles on the level of local states in a large but finite system. The result is essentially that of Brandao and Cramer (here restricted to the case of the canonical and the microcanonical ensembles), but we prove improved estimates in an elementary manner. We also review and prove standard results on the thermodynamic limits of thermodynamic functions and the equivalence of ensembles in terms of thermodynamic functions. The present paper assumes only elementary knowledge on quantum statistical mechanics and quantum spin systems.     

Higher Spin Dirac Operators Between Spaces of Simplicial Monogenics in Two Vector Variables

The higher spin Dirac operator \(\mathcal{Q}_{k,l}\) acting on functions taking values in an irreducible representation space for \(\mathfrak{so}(m)\) with highest weight \((k+\frac{1}{2},l+\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2})\), with k, l?∈?\(\mathbb{N}\) and \(k\geqslant l\), is constructed. The structure of the kernel space containing homogeneous polynomial solutions is then also studied.     

Holography of massive M2-brane theory: non-linear extension

We investigate the gauge/gravity duality between the \(\mathcal{N} = 6\) mass-deformed ABJM theory with \(\hbox {U}_k(N)\times \hbox {U}_{-k}(N)\) gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(2,1)\(\times \)SO(4)/\({\mathbb {Z}}_k\) \(\times \)SO(4)/\({\mathbb {Z}}_k\) isometry, in terms of a KK holography, which involves quadratic order field redefinitions. We establish the quadratic order KK mappings for various gauge invariant fields in order to obtain the canonical 4-dimensional gravity equations of motion and to reduce the LLM solutions to an asymptotically AdS\(_4\) gravity solutions. The non-linearity of the KK maps indicates that we can observe the true purpose of the non-linear KK holography of the LLM solutions. We read the vacuum expectation value of conformal dimension two operator from the asymptotically AdS\(_4\) gravity solutions. For the LLM solutions which are represented by square-shaped Young diagrams, we compare the vacuum expectation value obtained from the holographic procedure with the result obtained from the field theory, which is given by \(\langle \mathcal{O}^{(\Delta =2)}\rangle =\sqrt{k}N^{\frac{3}{2}}f_{(\Delta =2)}+\mathcal{O}(N)\), where \(f_{\Delta }\) is independent of N. Based on this result, we examine the gauge/gravity duality in the large N limit and finite k. We also show that the vacuum expectation values of the massive KK graviton modes are vanishing as expected by the supersymmetry.     

Translation Invariant Extensions of Finite Volume Measures

We investigate the following questions: Given a measure \(\mu _\Lambda \) on configurations on a subset \(\Lambda \) of a lattice \(\mathbb {L}\), where a configuration is an element of \(\Omega ^\Lambda \) for some fixed set \(\Omega \), does there exist a measure \(\mu \) on configurations on all of \(\mathbb {L}\), invariant under some specified symmetry group of \(\mathbb {L}\), such that \(\mu _\Lambda \) is its marginal on configurations on \(\Lambda \)? When the answer is yes, what are the properties, e.g., the entropies, of such measures? Our primary focus is the case in which \(\mathbb {L}=\mathbb {Z}^d\) and the symmetries are the translations. For the case in which \(\Lambda \) is an interval in \(\mathbb {Z}\) we give a simple necessary and sufficient condition, local translation invariance (LTI), for extendibility. For LTI measures we construct extensions having maximal entropy, which we show are Gibbs measures; this construction extends to the case in which \(\mathbb {L}\) is the Bethe lattice. On \(\mathbb {Z}\) we also consider extensions supported on periodic configurations, which are analyzed using de Bruijn graphs and which include the extensions with minimal entropy. When \(\Lambda \subset \mathbb {Z}\) is not an interval, or when \(\Lambda \subset \mathbb {Z}^d\) with \(d>1\), the LTI condition is necessary but not sufficient for extendibility. For \(\mathbb {Z}^d\) with \(d>1\), extendibility is in some sense undecidable.     

Thin Static Charged Dust Majumdar–Papapetrou Shells with High Symmetry in <Emphasis Type="Italic">D</Emphasis>≥4

We present a systematical study of static D≥4 space-times of high symmetry with the matter source being a thin charged dust hypersurface shell. The shell manifold is assumed to have the following structure \(\mathbb{S}_{\beta}\times\mathbb{R}^{D-2-\beta}\), β∈{0,…,D?2} is dimension of a sphere \(\mathbb{S}_{\beta}\). In case of β=0, we assume that there are two parallel hyper–plane shells instead of only one.The space-time has Majumdar–Papapetrou form and it inherits the symmetries of the shell manifold—it is invariant under both rotations of the \(\mathbb {S}_{\beta}\) and translations along ? ^D?2?β .We find a general solution to the Einstein–Maxwell equations with a given shell. Then, we examine some flat interior solutions with special attention paid to D=4. A connection to D=4 non-relativistic theory is pointed out. We also comment on a straightforward generalisation to the case of Kastor–Traschen space-time, i.e. adding a non-negative cosmological constant to the charged dust matter source.     

Heuristic Relative Entropy Principles with Complex Measures: Large-Degree Asymptotics of a Family of Multi-variate Normal Random Polynomials

Let \(z\in \mathbb {C}\), let \(\sigma ^2>0\) be a variance, and for \(N\in \mathbb {N}\) define the integrals

$$\begin{aligned} E_N^{}(z;\sigma ) := \left\{ \begin{array}{ll} {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}}\! (x^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x^2}}{\sqrt{2\pi }}dx&{}\quad \text{ if }\, N=1,\\ {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}^N}\! \prod \prod \limits _{1\le k<l\le N}\!\! e^{-\frac{1}{2N}(1-\sigma ^{-2}) (x_k-x_l)^2} \prod _{1\le n\le N}\!\!\!\!(x_n^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x_n^2}}{\sqrt{2\pi }}dx_n &{}\quad \text{ if }\, N>1. \end{array}\right. \!\!\! \end{aligned}$$

These are expected values of the polynomials \(P_N^{}(z)=\prod _{1\le n\le N}(X_n^2+z^2)\) whose 2N zeros \(\{\pm i X_k\}^{}_{k=1,\ldots ,N}\) are generated by N identically distributed multi-variate mean-zero normal random variables \(\{X_k\}^{N}_{k=1}\) with co-variance \(\mathrm{{Cov}}_N^{}(X_k,X_l)=(1+\frac{\sigma ^2-1}{N})\delta _{k,l}+\frac{\sigma ^2-1}{N}(1-\delta _{k,l})\). The \(E_N^{}(z;\sigma )\) are polynomials in \(z^2\), explicitly computable for arbitrary N, yet a list of the first three \(E_N^{}(z;\sigma )\) shows that the expressions become unwieldy already for moderate N—unless \(\sigma = 1\), in which case \(E_N^{}(z;1) = (1+z^2)^N\) for all \(z\in \mathbb {C}\) and \(N\in \mathbb {N}\). (Incidentally, commonly available computer algebra evaluates the integrals \(E_N^{}(z;\sigma )\) only for N up to a dozen, due to memory constraints). Asymptotic evaluations are needed for the large-N regime. For general complex z these have traditionally been limited to analytic expansion techniques; several rigorous results are proved for complex z near 0. Yet if \(z\in \mathbb {R}\) one can also compute this “infinite-degree” limit with the help of the familiar relative entropy principle for probability measures; a rigorous proof of this fact is supplied. Computer algebra-generated evidence is presented in support of a conjecture that a generalization of the relative entropy principle to signed or complex measures governs the \(N\rightarrow \infty \) asymptotics of the regime \(iz\in \mathbb {R}\). Potential generalizations, in particular to point vortex ensembles and the prescribed Gauss curvature problem, and to random matrix ensembles, are emphasized.

     

Scaling exponents of forced polymer translocation through a nanopore

We investigate several properties of a translocating homopolymer through a thin pore driven by an external field present inside the pore only using Langevin Dynamics (LD) simulations in three dimensions (3D). Motivated by several recent theoretical and numerical studies that are apparently at odds with each other, we estimate the exponents describing the scaling with chain length (Nof the average translocation time \(\ensuremath \langle\tau\rangle\) , the average velocity of the center of mass \(\ensuremath \langle v_{{\rm CM}}\rangle\) , and the effective radius of gyration \(\ensuremath \langle {R}_g\rangle\) during the translocation process defined as \(\ensuremath \langle\tau\rangle \sim N^{\alpha}\) , \(\ensuremath \langle v_{{\rm CM}} \rangle \sim N^{-\delta}\) , and \(\ensuremath {R}_g \sim N^{\bar{\nu}}\) respectively, and the exponent of the translocation coordinate (s -coordinate) as a function of the translocation time \(\ensuremath \langle s^2(t)\rangle\sim t^{\beta}\) . We find \(\ensuremath \alpha=1.36 \pm 0.01\) , \(\ensuremath \beta=1.60 \pm 0.01\) for \(\ensuremath \langle s^2(t)\rangle\sim \tau^{\beta}\) and \(\ensuremath \bar{\beta}=1.44 \pm 0.02\) for \(\ensuremath \langle\Delta s^2(t)\rangle\sim\tau^{\bar{\beta}}\) , \(\ensuremath \delta=0.81 \pm 0.04\) , and \(\ensuremath \bar{\nu}\simeq\nu=0.59 \pm 0.01\) , where \( \nu\) is the equilibrium Flory exponent in 3D. Therefore, we find that \(\ensuremath \langle\tau\rangle\sim N^{1.36}\) is consistent with the estimate of \(\ensuremath \langle\tau\rangle\sim\langle R_g \rangle/\langle v_{{\rm CM}} \rangle\) . However, as observed previously in Monte Carlo (MC) calculations by Kantor and Kardar (Y. Kantor, M. Kardar, Phys. Rev. E 69, 021806 (2004)) we also find the exponent α = 1.36 ± 0.01 < 1 + ν. Further, we find that the parallel and perpendicular components of the gyration radii, where one considers the “cis” and “trans” parts of the chain separately, exhibit distinct out-of-equilibrium effects. We also discuss the dependence of the effective exponents on the pore geometry for the range of N studied here.     

Generic Representation of Y(so(3)) Based on the Lie Algebraic Basis of so(3)

We focus on constructing a generic representation of Y(so(3)) based on the Lie algebraic basis of so(3) basis, and further developing transition of Yangian operator \(\hat {\mathbf {Y}}\). As an application of Y(so(3)), we calculate all the matrix elements of unit vector operators \(\hat {\mathbf {n}}\) in angular momentum basis. It is also discovered that the Yangian operator \(\hat {\mathbf {Y}}\) may work in quantum vector space. In addition, some shift operators \(\hat {O}^{(\pm )}_{\mu }\) are naturally built on the basis of the representation of Y(so(3)). As an another application of Y(so(3)), we can derive the CG cofficients of two coupled angular momenta from the down-shift operator \(\hat {O}^{(-)}_{-1}\) acting on a so(3)-coupled tensor basis. This not only explores that Yangian algebras can work in quantum tensor space, but also provides a novel approach to solve CG coefficients for two coupled angular momenta.     

The closure constraint for the hyperbolic tetrahedron as a Bianchi identity

The closure constraint is a central piece of the mathematics of loop quantum gravity. It encodes the gauge invariance of the spin network states of quantum geometry and provides them with a geometrical interpretation: each decorated vertex of a spin network is dual to a quantized polyhedron in \({\mathbb R}^{3}\). For instance, a 4-valent vertex is interpreted as a tetrahedron determined by the four normal vectors of its faces. We develop a framework where the closure constraint is re-interpreted as a Bianchi identity, with the normals defined as holonomies around the polyhedron faces of a connection (constructed from the spinning geometry interpretation of twisted geometries). This allows us to define closure constraints for hyperbolic tetrahedra (living in the 3-hyperboloid of unit future-oriented spacelike vectors in \({\mathbb R}^{3,1}\)) in terms of normals living all in \(\mathrm {SU}(2)\) or in \(\mathrm {SB}(2,{\mathbb C})\). The latter fits perfectly with the classical phase space developed for q-deformed loop quantum gravity supposed to account for a non-vanishing cosmological constant \(\Lambda >0\). This allows the interpretation of q-deformed twisted geometries as actual discrete hyperbolic triangulations for 4d quantum gravity.     

On <Emphasis Type="Italic">τ</Emphasis>-Compactness of Products of <Emphasis Type="Italic">τ</Emphasis>-Measurable Operators

Let \(\mathcal {M}\) be a von Neumann algebra of operators on a Hilbert space \(\mathcal {H}\), τ be a faithful normal semifinite trace on \(\mathcal {M}\). We obtain some new inequalities for rearrangements of τ-measurable operators products. We also establish some sufficient τ-compactness conditions for products of selfadjoint τ-measurable operators. Next we obtain a τ-compactness criterion for product of a nonnegative τ-measurable operator with an arbitrary τ-measurable operator. We construct an example that shows importance of nonnegativity for one of the factors. The similar results are obtained also for elementary operators from \(\mathcal {M}\). We apply our results to symmetric spaces on \((\mathcal {M}, \tau )\). The results are new even for the *-algebra \(\mathcal {B}(\mathcal {H})\) of all linear bounded operators on \(\mathcal {H}\) endowed with the canonical trace τ = tr.     

Onsager’s Ensemble for Point Vortices with Random Circulations on the Sphere

Onsager’s ergodic point vortex (sub-)ensemble is studied for N vortices which move on the 2-sphere \(\mathbb{S}^{2}\) with randomly assigned circulations, picked from an a-priori distribution. It is shown that the typical point vortex distributions obtained from the ensemble in the limit N→∞ are special solutions of the Euler equations of incompressible, inviscid fluid flow on \(\mathbb{S}^{2}\). These typical point vortex distributions satisfy nonlinear mean-field equations which have a remarkable resemblance to those obtained from the Miller-Robert theory. Conditions for their perfect agreement are stated. Also the non-random limit, when all vortices have circulation 1, is discussed in some detail, in which case the ergodic and holodic ensembles are shown to be inequivalent.     

Uniform in N global well-posedness of the time-dependent Hartree–Fock–Bogoliubov equations in $$\mathbb {R}^{1+1}$$

We prove the global well-posedness of the time-dependent Hartree–Fock–Bogoliubov (TDHFB) equations in \(\mathbb {R}^{1+1}\) with two-body interaction potential of the form \(N^{-1}v_N(x) = N^{\beta -1} v(N^\beta x)\) where \(v\ge 0\) is a sufficiently regular radial function, i.e., \(v \in L^1(\mathbb {R})\cap C^\infty (\mathbb {R})\). In particular, using methods of dispersive PDEs similar to the ones used in Grillakis and Machedon (Commun Partial Differ Equ 42:24–67, 2017), we are able to show for any scaling parameter \(\beta >0\) the TDHFB equations are globally well-posed in some Strichartz-type spaces independent of N, cf. (Bach et al. in The time-dependent Hartree–Fock–Bogoliubov equations for Bosons, 2016. arXiv:1602.05171).     

Identification of potential trypanothione reductase inhibitors among commercially available $$\upbeta $$-carboline derivatives using chemical space,lead-like and drug-like filters,pharmacophore models and molecular docking

American trypanosomiasis or Chagas disease caused by the protozoan Trypanosoma cruzi (T. cruzi) is an important endemic trypanosomiasis in Central and South America. This disease was considered to be a priority in the global plan to combat neglected tropical diseases, 2008–2015, which indicates that there is an urgent need to develop more effective drugs. The development of new chemotherapeutic agents against Chagas disease can be related to an important biochemical feature of T. cruzi: its redox defense system. This system is based on trypanothione (\(N^{1}\),\(N^{8}\)-bis(glutathyonil)spermidine) and trypanothione reductase (TR), which are rather unique to trypanosomes and completely absent in mammalian cells. In this regard, tricyclic compounds have been studied extensively due to their ability to inhibit the T. cruzi TR. However, synthetic derivatives of natural products, such as \(\upbeta \)-carboline derivatives (\(\upbeta \)-CDs), as potential TR inhibitors, has received little attention. This study presents an analysis of the structural and physicochemical properties of commercially available \(\upbeta \)-CDs in relation to compounds tested against T. cruzi in previously reported enzymatic assays and shows that \(\upbeta \)-CDs cover chemical space that has not been considered for the design of TR inhibitors. Moreover, this study presents a ligand-based approach to discover potential TR inhibitors among commercially available \(\upbeta \)-CDs, which could lead to the generation of promising \(\upbeta \)-CD candidates.     

$${\mathbb{Z}_N}$$ -Curves Possessing No Thomae Formulae of Bershadsky–Radul Type

A \({\mathbb{Z}_N}\) -curve is one of the form \({y^{N}=(x-\lambda_{1})^{m_{1}}\cdots(x-\lambda_{s})^{m_{s}}}\) . When N = 2 these curves are called hyperelliptic and for them Thomae proved his classical formulae linking the theta functions corresponding to their period matrices to the branching values λ₁, . . . , λ _s . In his work on Fermionic fields on \({\mathbb{Z}_N}\) -curves with arbitrary N, Bershadsky and Radul discovered the existence of generalized Thomae’s formulae for these curves which they wrote down explicitly in the case in which all rotation numbers m _i equal 1. This work was continued by several authors and new Thomae’s type formulae for \({\mathbb{Z}_N}\) -curves with other rotation numbers m _i were found. In this article we prove that for some choices of the rotation numbers the corresponding \({\mathbb{Z}_N}\) -curves do not admit such generalized Thomae’s formulae.     

Exact holography of the mass-deformed M2-brane theory

We test the holographic relation between the vacuum expectation values of gauge invariant operators in \({\mathcal {N}} = 6\) U\(_k(N)\times \mathrm{U}_{-k}(N)\) mass-deformed ABJM theory and the LLM geometries with \({\mathbb {Z}}_k\) orbifold in 11-dimensional supergravity. To do so, we apply the Kaluza–Klein reduction to construct a 4-dimensional gravity theory and implement the holographic renormalization procedure. We obtain an exact holographic relation for the vacuum expectation values of the chiral primary operator with conformal dimension \(\Delta = 1\), which is given by \(\langle {\mathcal {O}}^{(\Delta =1)}\rangle = N^{\frac{3}{2}} \, f_{(\Delta =1)}\), for large N and \(k=1\). Here the factor \(f_{(\Delta )}\) is independent of N. Our results involve an infinite number of exact dual relations for all possible supersymmetric Higgs vacua and so provide a non-trivial test of gauge/gravity duality away from the conformal fixed point. We extend our results to the case of \(k\ne 1\) for LLM geometries represented by rectangular-shaped Young diagrams. We also discuss the exact mapping of the gauge/gravity at finite N for classical supersymmetric vacuum solutions in field theory side and corresponding classical solutions in gravity side.     

Quantum Fisher Information for <Emphasis Type="Italic">s</Emphasis><Emphasis Type="Italic">u</Emphasis>(2) Atomic Coherent States and <Emphasis Type="Italic">s</Emphasis><Emphasis Type="Italic">u</Emphasis>(1, 1) Coherent States

We investigate quantum Fisher information (QFI) for s u(2) atomic coherent states and s u(1, 1) coherent states. In this work, we find that for s u(2) atomic coherent states, the QFI with respect to \(\vartheta ~(\mathcal {F}_{\vartheta })\) is independent of φ, the QFI with respect to \(\varphi (\mathcal {F}_{\varphi })\) is governed by ??. Analogously, for s u(1,1) coherent states, \(\mathcal {F}_{\tau }\) is independent of φ, and \(\mathcal {F}_{\varphi }\) is determined by τ. Particularly, our results show that \(\mathcal {F}_{\varphi }\) is symmetric with respect to ?? = π/2 for s u(2) atomic coherent states. And for s u(1,1) coherent states, \(\mathcal {F}_{\varphi }\) also possesses symmetry with respect to τ = 0.     

Hydrodynamic Limit of Multiple SLE

Recently del Monaco and Schleißinger addressed an interesting problem whether one can take the limit of multiple Schramm–Loewner evolution (SLE) as the number of slits N goes to infinity. When the N slits grow from points on the real line \(\mathbb {R}\) in a simultaneous way and go to infinity within the upper half plane \(\mathbb {H}\), an ordinary differential equation describing time evolution of the conformal map \(g_t(z)\) was derived in the \(N \rightarrow \infty \) limit, which is coupled with a complex Burgers equation in the inviscid limit. It is well known that the complex Burgers equation governs the hydrodynamic limit of the Dyson model defined on \(\mathbb {R}\) studied in random matrix theory, and when all particles start from the origin, the solution of this Burgers equation is given by the Stieltjes transformation of the measure which follows a time-dependent version of Wigner’s semicircle law. In the present paper, first we study the hydrodynamic limit of the multiple SLE in the case that all slits start from the origin. We show that the time-dependent version of Wigner’s semicircle law determines the time evolution of the SLE hull, \(K_t \subset \mathbb {H}\cup \mathbb {R}\), in this hydrodynamic limit. Next we consider the situation such that a half number of the slits start from \(a>0\) and another half of slits start from \(-a < 0\), and determine the multiple SLE in the hydrodynamic limit. After reporting these exact solutions, we will discuss the universal long-term behavior of the multiple SLE and its hull \(K_t\) in the hydrodynamic limit.     

Discrete One-Dimensional Coverage Process on a Renewal Process

Consider the following coverage model on \(\mathbb {N}\), for each site \(i \in \mathbb {N}\) associate a pair \((\xi _i, R_i)\) where \((\xi _i)_{i \ge 0}\) is a 1-dimensional undelayed discrete renewal point process and \((R_i)_{i \ge 0}\) is an i.i.d. sequence of \(\mathbb {N}\)-valued random variables. At each site where \(\xi _i=1\) start an interval of length \(R_i\). Coverage occurs if every site of \(\mathbb {N}\) is covered by some interval. We obtain sharp conditions for both, positive and null probability of coverage. As corollaries, we extend results of the literature of rumor processes and discrete one-dimensional Boolean percolation.