Equilibrium and non-equilibrium Ising models by means of PCA

TL;DR

This paper introduces a unified PCA approach to analyze equilibrium and non-equilibrium Ising models, demonstrating how stationary measures can be computed and closely approximate Gibbs measures in both reversible and irreversible cases.

Contribution

It presents a novel unified framework for reversible and irreversible PCA dynamics applied to Ising models, with methods to compute stationary measures and analyze their relation to Gibbs measures.

Findings

Stationary measures can be computed for 1D and 2D Ising models with periodic boundaries.

Stationary measures are very close to Gibbs measures for suitable parameters.

Numerical methods and parallel implementation strategies are discussed.

Abstract

We propose a unified approach to reversible and irreversible PCA dynamics, and we show that in the case of 1D and 2D nearest neighbour Ising systems with periodic boundary conditions we are able to compute the stationary measure of the dynamics also when the latter is irreversible. We also show how, according to [DPSS12], the stationary measure is very close to the Gibbs for a suitable choice of the parameters of the PCA dynamics, both in the reversible and in the irreversible cases. We discuss some numerical aspects regarding this topic, including a possible parallel implementation.