Chaotic behavior and damage spreading in the Glauber Ising model - a master equation approach

TL;DR

This paper explores the chaotic dynamics and damage spreading in the Glauber Ising model using a master equation approach, revealing phases of regular and chaotic behavior and their relation to directed percolation.

Contribution

It introduces a master equation framework for analyzing damage spreading in the kinetic Ising model, extending beyond mean-field approximations to include fluctuations.

Findings

Identification of stable fixed points and chaotic phases

Analysis of damage spreading sensitivity to initial conditions

Connection between chaotic behavior and directed percolation

Abstract

We investigate the sensitivity of the time evolution of a kinetic Ising model with Glauber dynamics against the initial conditions. To do so we apply the "damage spreading" method, i.e., we study the simultaneous evolution of two identical systems subjected to the same thermal noise. We derive a master equation for the joint probability distribution of the two systems. We then solve this master equation within an effective-field approximation which goes beyond the usual mean-field approximation by retaining the fluctuations though in a quite simplistic manner. The resulting effective-field theory is applied to different physical situations. It is used to analyze the fixed points of the master equation and their stability and to identify regular and chaotic phases of the Glauber Ising model. We also discuss the relation of our results to directed percolation.