Inverse Problem for Ising Connection Matrix with Long-Range Interaction

Leonid Litinskii,Boris Kryzhanovsky

doi:10.3390/math9141624

Abstract

In the present paper, we examine Ising systems on d-dimensional hypercube lattices and solve an inverse problem where we have to determine interaction constants of an Ising connection matrix when we know a spectrum of its eigenvalues. In addition, we define restrictions allowing a random number sequence to be a connection matrix spectrum. We use the previously obtained analytical expressions for the eigenvalues of Ising connection matrices accounting for an arbitrary long-range interaction and supposing periodic boundary conditions.

Highlights

The vectors Fαβγ constitute a full set of the eigenvectors of any connection matrix of the three-dimensional Ising system and they do not depend on the type of the interaction constants {w(n, m, k )}ln,k,m=0
Connection matrices define the most important characteristics of Ising systems—such as the energies of the states and their distribution, the free energy, and all the macroscopic properties defined by the free energy
All these functions are crucially dependent on the connection matrix whose main characteristics are its eigenvalues and eigenvectors

Summary

Introduction

In papers [1,2,3], we calculated eigenvalues of Ising connection matrices defined on d-dimensional hypercube lattices We hope that the same is true for larger values of k These arguments show that the eigenvalues of the Ising connection matrix may be useful when calculating the partition function. Onsager et al found an exact solution for the planar Ising system, when the spins were at the nodes of the plane square lattice and only the nearest spins interacted Sometimes such a short-range interaction describes real systems. When solving the inverse Ising problems, the authors examine how with the aid of the statistical inference method they can estimate the parameters of the Ising system—interaction constants and external magnetic fields—when they know empirical characteristics of a large number of random spin configurations. They are the Boltzmann equilibrium distribution, the principle of the maximal likelihood, the Bayes theorem, etc

One-Dimensional Ising Model

Discussion and Conclusions