Effective ergodicity in single-spin-flip dynamics

TL;DR

This paper uses the Thirumalai-Mountain metric to analyze how quickly single-spin-flip dynamics in the 1D Ising model reach effective ergodicity, revealing diffusion regimes and cautioning about strong correlations.

Contribution

It introduces a quantitative approach to measure effective ergodicity in spin systems using the TM metric with exact and Monte Carlo methods.

Findings

Identification of diffusion regimes in ergodic convergence

Effect of system size, temperature, and external field on convergence

Caution advised when strong correlations are present

Abstract

A quantitative measure of convergence to effective ergodicity, the Thirumalai-Mountain (TM) metric, is applied to Metropolis and Glauber single-spin-flip dynamics. In computing this measure, finite lattice ensemble averages are obtained using the exact solution for a one dimensional Ising model, whereas the time averages are computed with Monte Carlo simulations. The time evolution of the effective ergodic convergence of Ising magnetization is monitored. By this approach, diffusion regimes of the effective ergodic convergence of magnetization are identified for different lattice sizes, nonzero temperature, and nonzero external field values. Results show that caution should be taken when using the TM metric at system parameters that give rise to strong correlations.