Lattice Statistical Mechanics Research Articles

Effect of Bjerrum pairs on the electrostatic properties of an electrolyte solution near charged surfaces: a mean-field approach.

In this paper, we investigate the consequences of ion association, coupled with the considerations of finite size effects and orientational ordering of Bjerrum pairs as well as ions and water molecules, on the electric double layer near charged surfaces. Based on the lattice statistical mechanics accounting for finite sizes and dipole moments of ions, Bjerrum pairs and solvent molecules, we consider the formation of Bjerrum pairs and derive the mathematical expressions for Bjerrum pair number density as well as cation/anion number density and water molecule number density. We reveal several significant phenomena. Firstly, it is shown that our approach naturally yields the equilibrium constant for dissociation-association equilibrium between Bjerrum pairs and ions. Secondly, at low surface charge densities, an increase in the bulk concentration of Bjerrum pairs enhances the permittivity and decreases the differential capacitance. Next, for the cases where Bjerrum pairs in an alcohol electrolyte solution have a high value of dipole moment, the Bjerrum pair number density increases with decreasing distance from the charged surface, and the differential capacitance and permittivity are high compared to those for the cases with lower values of Bjerrum-pair dipole moments. Finally, we show that the difference in the concentration and dipole moment of Bjerrum pairs can lead to some variation in osmotic pressure between two similarly charged surfaces.

Inhomogeneous Gaussian free field inside the interacting arctic curve

The six-vertex model with domain-wall boundary conditions is one representative of a class of two-dimensional lattice statistical mechanics models that exhibit a phase separation known as the arctic curve phenomenon. In the thermodynamic limit, the degrees of freedom are completely frozen in a region near the boundary, while they are critically fluctuating in a central region. The arctic curve is the phase boundary that separates those two regions. Critical fluctuations inside the arctic curve have been studied extensively, both in physics and in mathematics, in free models (i.e. models that map to free fermions, or equivalently to determinantal point processes). Here we study those critical fluctuations in the interacting (i.e. not free, not determinantal) six-vertex model, and provide evidence for the following two claims:(i) the critical fluctuations are given by a Gaussian free field (GFF), as in the free case, but(ii) contrarily to the free case, the GFF is inhomogeneous, meaning that its coupling constant becomes position-dependent, .The evidence is mainly based on the numerical solution of appropriate Bethe ansatz equations with an imaginary extensive twist, and on transfer matrix computations, but the second claim is also supported by the analytic calculation of and its first two derivatives in selected points. Contrarily to the usual GFF, this inhomogeneous GFF is not defined in terms of the Green’s function of the Laplacian inside the critical domain, but instead, of the Green’s function of a generalized Laplacian parametrized by the function . Surprisingly, we also find that there is a change of regime when the anisotropy parameter Δ is smaller than , with becoming singular at one point.

Diagonals of rational functions, pullbacked hypergeometric functions and modular forms

We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. We find that the diagonal of a seven parameter rational function of three variables with a numerator equal to one and a denominator which is a polynomial of degree at most two, can be expressed as a pullbacked hypergeometric function. This result can be seen as the simplest non-trivial family of diagonals of rational functions. We focus on some subcases such that the diagonals of the corresponding rational functions can be written as a pullbacked hypergeometric function with two possible rational functions pullbacks algebraically related by modular equations, thus showing explicitely that the diagonal is a modular form. We then generalize this result to nine and ten parameter families adding some selected cubic terms at the denominator of the rational function defining the diagonal. We show that each of these rational functions yields an infinite number of rational functions whose diagonals are also pullbacked hypergeometric functions and modular forms.

Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics**Dedicated to A J Guttmann, for his 70th birthday.

We recall that the full susceptibility series of the Ising model, modulo powers of the prime , reduce to algebraic functions. We also recall the nonlinear polynomial differential equation obtained by Tutte for the generating function of the q-coloured rooted triangulations by vertices, which is known to have algebraic solutions for all the numbers of the form , the holonomic status of being unclear. We focus on the analysis of the case, showing that the corresponding series is quite certainly non-holonomic. Along the line of a previous work on the susceptibility of the Ising model, we consider this series modulo the first eight primes , , ... , and show that this (probably non-holonomic) function reduces, modulo these primes, to algebraic functions. We conjecture that this probably non-holonomic function reduces to algebraic functions modulo (almost) every prime, or power of prime numbers. This raises the question of whether such remarkable non-holonomic functions can be seen as a ratio of diagonals of rational functions, or even algebraic functions of diagonals of rational functions.

Diagonals of rational functions and selected differential Galois groups

We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. In all the examples emerging from physics, the minimal linear differential operators annihilating these diagonals of rational functions have been shown to actually possess orthogonal or symplectic differential Galois groups. In order to understand the emergence of such orthogonal or symplectic groups, we analyze exhaustively three sets of diagonals of rational functions, corresponding respectively to rational functions of three variables, four variables and six variables. We impose the constraints that the degree of the denominators in each variable is at most one, and the coefficients of the monomials are 0 or so that the analysis can be exhaustive. We find the minimal linear differential operators annihilating the diagonals of these rational functions of three, four, five and six variables. We find that, even for these sets of examples which, at first sight, have no relation with physics, their differential Galois groups are always orthogonal or symplectic groups. We discuss the conditions on the rational functions such that the operators annihilating their diagonals do not correspond to orthogonal or symplectic differential Galois groups, but rather to generic special linear groups.

Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals

Focusing on examples associated with holonomic functions, we try to bring new ideas on how to look at phase transitions, for which the critical manifolds are not points but curves depending on a spectral variable, or even fill higher dimensional submanifolds. Lattice statistical mechanics often provides a natural (holonomic) framework to perform singularity analysis with several complex variables that would, in the most general mathematical framework, be too complex, or simply could not be defined. In a learn-by-example approach, considering several Picard–Fuchs systems of two-variables ‘above’ Calabi–Yau ODEs, associated with double hypergeometric series, we show that D-finite (holonomic) functions are actually a good framework for finding properly the singular manifolds. The singular manifolds are found to be genus-zero curves. We then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics, the n-fold integrals χ(n), corresponding to the n-particle decomposition of the magnetic susceptibility of the anisotropic square Ising model. In this anisotropic case, we revisit a set of so-called Nickelian singularities that turns out to be a two-parameter family of elliptic curves. We then find the first set of non-Nickelian singularities for χ(3) and χ(4), that also turns out to be rational or elliptic curves. We underline the fact that these singular curves depend on the anisotropy of the Ising model, or, equivalently, that they depend on the spectral parameter of the model. This has important consequences on the physical nature of the anisotropic χ(n)s which appear to be highly composite objects. We address, from a birational viewpoint, the emergence of families of elliptic curves, and that of Calabi–Yau manifolds on such problems. We also address the question of singularities of non-holonomic functions with a discussion on the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility χ.This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.

A Constraint-Based Approach to Fast and Exact Structure Prediction in Three-Dimensional Protein Models

Simplified protein models are used for investigating general properties of proteins and principles of protein folding. Furthermore, they are suited for hierarchical approaches to protein structure prediction. A well known protein model is the HP-model of Lau and Dill [Lau, K. F., & Dill, K. A. (1989)]. A lattice statistical mechanics model of the conformational and sequence spaces of proteins. Macromolecules, 22, 3986---3997) which models the important aspect of hydrophobicity. One can define the HP-model for various lattices, among them two-dimensional and three-dimensional ones. Here, we investigate the three-dimensional case. The main motivation for studying simplified protein models is to be able to predict model structures much more quickly and more accurately than is possible for real proteins. However, up to now there was a dilemma: the algorithmically tractable, simple protein models can not model real protein structures with good quality and introduce strong artifacts. We present a constraint-based method that largely improves this situation. It outperforms all existing approaches for lattice protein folding in HP-models. This approach is the first one that can be applied to two three-dimensional lattices, namely the cubic lattice and the face-centered-cubic (FCC) lattice. Moreover, it is the only exact method for the FCC lattice. The ability to use the FCC lattice is a significant improvement over the cubic lattice. The key to our approach is the ability to compute maximally compact sets of points (used as hydrophobic cores), which we accomplish for the first time for the FCC lattice.

The analytic structure of lattice models — why can’t we solve most models?

We investigate the solvability of a variety of well-known problems in lattice statistical mechanics. We provide a new numerical procedure which enables one to conjecture whether the solution falls into a class of functions calleddifferentiably finite functions. Almost all solved problems fall into this class. The fact that one can conjecture whether a given problem is or is not D-finite then informs one as to whether the solution is likely to be tractable or not. We also show how, for certain problems, it is possible to prove that the solutions are notD-finite, based on the work of Rechnitzer [1–3].

Transfer Matrix Functional Relations for the Generalized 2(tq) Model

The $N$-state chiral Potts model in lattice statistical mechanics can be obtained as a ``descendant'' of the six-vertex model, via an intermediate ``$Q$'' or ``$\tau_2 (t_q)$'' model. Here we generalize this to obtain a column-inhomogeneous $\tau_2 (t_q)$ model, and derive the functional relations satisfied by its row-to-row transfer matrix. We do {\em not} need the usual chiral Potts relations between the $N$th powers of the rapidity parameters $a_p, b_p, c_p, d_p$ of each column. This enables us to readily consider the case of fixed-spin boundary conditions on the left and right-most columns. We thereby re-derive the simple direct product structure of the transfer matrix eigenvalues of this model, which is closely related to the superintegrable chiral Potts model with fixed-spin boundary conditions.

Random matrix theory in lattice statistical mechanics

In this short note we collect together known results on the use of random matrix theory (RMT) in lattice statistical mechanics. The purpose here is two fold. Firstly the RMT analysis provides an intrinsic characterization of integrability, and secondly it appears to be an effective tool to find new integrable models. Various examples from quantum and classical statistical mechanics are presented.

Fa-Yueh Wu's contributions in physics

We sketch a subset of Professor F.Y. Wu's contributions in lattice statistical mechanics, solid state physics, graph theory, enumerative combinatorics and other domains of physics and mathematics. We will recall some of F.Y. Wu's most important and well-known classic results and we will also sketch some of his most recent researches dedicated to the connections of lattice statistical mechanical models with deep problems in pure mathematics.

Obtaining Ising-like expansions for binary alloys from first principles

Many measurable properties of crystalline binary A1−xBx alloys, such as phase diagrams and excess thermodynamic functions, could be predicted via lattice statistical mechanics methods if one knew the `configurational energy'. The latter describes the energy at T = 0 for each of the 2N possible occupation patterns of the N lattice sites by an A or a B atom. Traditional approaches described the configurational energy either via empirically fitted, truncated Ising Hamiltonians, or through highly approximated coherent-potential constructs. We illustrate here the alternative approach of `mixed-basis cluster expansion' which extracts from a set of ab initio local density approximation calculations of the total energies of a few ordered A–B compounds a complete configurational energy function. This method includes both pair and multibody terms, whose number and range of interaction are decided by the variational procedure itself, as well as long-range strain terms. In this paper, we describe the computational details of this method, emphasizing methods of construction, interpolations, fits and convergence. This procedure is illustrated for Ni–Pt, Cu–Au and ScS–□S (where □ denotes cation vacancy). The parameters of the final expansions are provided on our webpage (http://www.sst.nrel.gov).

Origins of nonstoichiometry and vacancy ordering in Sc1-x squarexS.

Whereas nearly all compounds A(n)B(m) obey Dalton's rule of integer stoichiometry (n: m, both integer), there is a class of systems, exemplified by the rocksalt structure Sc1-x squarexS, that exhibits large deviations from stoichiometry via vacancies, even at low temperatures. By combining first-principles total energy calculations with lattice statistical mechanics, we scan an astronomical number of possible structures, identifying the stable ground states. Surprisingly, all have the same motifs: (111) planes with (112) vacancy rows arranged in (110) columns. Electronic structure calculations of the ground states (identified out of approximately 3x10(6) structures) reveal the remarkable origins of nonstoichiometry.

Lattice dynamics and statistical mechanics of the Fm3m → I4/m structural phase transition in Rb2KInF6

The static and dynamic properties of cubic Rb2KInF6 crystals with elpasolite structure are calculated using a nonempirical method. Calculations are performed within a microscopic ionic-crystal model taking into account the deformation and polarization of ions. The deformation parameters of ions are determined by minimizing the total energy of the crystal. The calculated equilibrium lattice parameters agree satisfactorily with the experimental data. It is found that in the cubic phase there are vibrational modes that are unstable everywhere in the Brillouin zone. The eigenvectors of the unstablest mode at the center of the Brillouin zone of the cubic phase are associated with the displacements of F ions and correspond to rotations of InF6 octahedra. Condensation of this mode leads to a tetragonal distortion of the structure. In order to describe the Fm3m → I4/m phase transition, an effective Hamiltonian is constructed under the assumption that the soft mode whose eigenvector corresponds to octahedron rotation is local and coupled with homogeneous elastic strains. The parameters of the effective Hamiltonian are determined using the calculated crystal energy for the distorted structures due to soft-mode condensation. The thermodynamic properties of the system with this model Hamiltonian are investigated using the Monte Carlo method. The phase transition temperature is calculated to be 550 K, which is twice its experimental value (283 K). The tetragonal phase remains stable down to T=0 K; the effective Hamiltonian used in this paper thus fails to describe the second phase transition (to the monoclinic phase). Thus, the transition to the tetragonal phase occurs for the most part through octahedron rotations; however, additional degrees of freedom, first of all, the displacements of Rb ions, should be included into the effective Hamiltonian in order to describe the transition to the monoclinic phase.

A statistical mechanics model for receptor clustering.

We introduce and study a simple lattice statistical mechanics modelfor the clustering of tumor necrosis factor receptor I (TNFR1).Our model explains clustering under over-expression of the cytoplasmicsignal transducer as well as the clustering induced via extracellularligand binding; also we explain why the loss of transducer leads to arapid break-up of the clusters. The basic mechanism at work is a first-order(cooperative) phase transition caused by the multimeric binding capability ofthe receptor-transducer complex. Using cooperativity of this type, the cellsare found to have an enhanced sensitivity and robustness. In general, ourmethod can be applied to other receptor-clustering related signaling system.

Functional relations in lattice statistical mechanics, enumerative combinatorics, and discrete dynamical systems

We recall some non-trivial, non-linear functional relations appearing in various domains of mathematics and physics, such as lattice statistical mechanics, quantum mechanics, or enumerative combinatorics. We focus, more particularly, on the analyticity properties of the solutions of these functional relations. We then consider discrete dynamical systems corresponding to birational transformations. The rational expressions for dynamical zeta functions obtained for a particular two-dimensional birational mapping, depending on two parameters, are recalled, as well as some non-trivial functional relations satisfied by these dynamical zeta functions. We finally give some functional equations corresponding to some singled out orbits of this two-dimensional birational mapping for particular values of the two parameters. This example shows that functional equations associated with curves, for real values of the variables, are actually compatible with a chaotic dynamical system.

Discrete symmetry groups of vertex models in statistical mechanics

We analyze discrete symmetry groups of vertex models in lattice statistical mechanics represented as groups of birational transformations. They can be seen as generated by involutions corresponding respectively to two kinds of transformations onq×q matrices: the inversion of theq×q matrix and an (involutive) permutation of the entries of the matrix. We show that the analysis of the factorizations of the iterations of these transformations is a precious tool in the study of lattice models in statistical mechanics. This approach enables one to analyze two-dimensionalq4-state vertex models as simply as three-dimensional vertex models, or higher-dimensional vertex models. Various examples of birational symmetries of vertex models are analyzed. A particular emphasis is devoted to a three-dimensional vertex model, the 64-state cubic vertex model, which exhibits a polynomial growth of the complexity of the calculations. A subcase of this general model is seen to yield integrable recursion relations. We also concentrate on a specific two-dimensional vertex model to see how the generic exponential growth of the calculations reduces to a polynomial growth when the model becomes Yang-Baxter integrable. It is also underlined that a polynomial growth of the complexity of these iterations can occur even for transformations yielding algebraic surfaces, or higher-dimensional algebraic varieties.

Integrable mappings from matrix transformations and their singularity properties

Various examples of bi-rational transformations having their origin in the theory of exactly solvable vertex models in lattice statistical mechanics, as well as bi-rational transformations originating from spin edge models, are analysed using the singularity confinement method. This method provides results concerning the integrable (or not) character of these bi-rational transformations in complete agreement with the results obtained by visualization methods as well as methods based on the analysis of algebraic invariants.

Integrable mappings and polynomial growth

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds of transformations on $q \times q$ matrices: the inversion of the $q \times q$ matrix and an (involutive) permutation of the entries of the matrix. We concentrate on the case where these permutations are elementary transpositions of two entries. In this case the birational transformations fall into six different classes. For each class we analyze the factorization properties of the iteration of these transformations. These factorization properties enable to define some canonical homogeneous polynomials associated with these factorization properties. Some mappings yield a polynomial growth of the complexity of the iterations. For three classes the successive iterates, for $q=4$, actually lie on elliptic curves. This analysis also provides examples of integrable mappings in arbitrary dimension, even infinite. Moreover, for two classes, the homogeneous polynomials are shown to satisfy non trivial non-linear recurrences. The relations between factorizations of the iterations, the existence of recurrences on one or several variables, as well as the integrability of the mappings are analyzed.

DETERMINANTAL IDENTITIES ON INTEGRABLE MAPPINGS

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds of transformations on q×q matrices: the inversion of the q×q matrix and an (involutive) permutation of the entries of the matrix. In a case where the permutation is a particular elementary transposition of two entries, it is shown that the iteration of this group of birational transformations yield algebraic elliptic curves in the parameter space associated with the (homogeneous) entries of the matrix. It is also shown that the successive iterated matrices do have remarkable factorization properties which yield introducing a series of canonical polynomials corresponding to the greatest common factor in the entries. These polynomials do satisfy a simple nonlinear recurrence which also yields algebraic elliptic curves, associated with biquadratic relations. In fact, these polynomials not only satisfy one recurrence but a whole hierarchy of recurrences. Remarkably these recurrences are universal: they are independent of q, the size of the matrices. This study provides examples of infinite dimensional integrable mappings.