Transfer operator analysis of the parallel dynamics of disordered Ising chains

TL;DR

This paper introduces a transfer operator approach to analyze the parallel stochastic dynamics of disordered Ising chains, providing a new formalism that simplifies the study of macroscopic behavior and aligns well with numerical simulations.

Contribution

It develops a transfer operator formalism for the dynamics of disordered Ising chains, offering a more transparent and numerically accessible way to analyze their macroscopic behavior.

Findings

The transfer operator formalism accurately captures the macroscopic dynamics.

Numerical simulations confirm the theoretical predictions.

The approach simplifies the analysis of complex disordered systems.

Abstract

We study the synchronous stochastic dynamics of the random field and random bond Ising chain. For this model the generating functional analysis methods of De Dominicis leads to a formalism with transfer operators, similar to transfer matrices in equilibrium studies, but with dynamical paths of spins and (conjugate) fields as arguments, as opposed to replicated spins. In the thermodynamic limit the macroscopic dynamics is captured by the dominant eigenspace of the transfer operator, leading to a relative simple and transparent set of equations that are easy to solve numerically. Our results are supported excellently by numerical simulations.