Anomalous diffusion in convergence to effective ergodicity

TL;DR

This paper investigates how the approach to ergodicity in an Ising model exhibits anomalous diffusion behavior, revealing nonlinear dynamics that depend on temperature and external field, and offers insights into nonequilibrium thermodynamics.

Contribution

It introduces the concept of functional-diffusion in the context of ergodicity convergence and characterizes its anomalous behavior across different parameters in the Ising model.

Findings

Power-law behavior indicates anomalous diffusion in ergodicity approach.

Nonlinear dynamics vary with temperature and external magnetic field.

Meta-trajectory analysis enhances understanding of nonequilibrium thermodynamics.

Abstract

The nature of diffusion is usually studied for particles or time-evolving systems. Similar in principle, such studies can be conducted by tracking how a given function of observable properties evolves over time-akin to the evolution of observable functions-referred to as functional-diffusion. This is not the same as the system's individual trajectories, but can be regarded as a meta-trajectory. Following this idea, we measure how the approach to ergodicity evolves over time for the observed magnetization of a full Ising model with an external field. We compute the diffusive behavior of the functional across a range of temperatures via Metropolis and Glauber single-spin-flip dynamics. The system's ensemble-averaged dynamics are computed using expressions from the exact solution. Power-law behavior in the approach to ergodicity provides a classification of anomalies in functional-diffusion, demonstrating nonlinear anomalous behavior over different temperature and field ranges. Studying the ergodicity convergence of these meta-trajectories can help validate and enhance the pedagogical understanding of nonequilibrium thermodynamic systems.