A Random Walk to a Non-Ergodic Equilibrium Concept

TL;DR

This paper explores a new non-ergodic equilibrium concept for continuous time random walks in potential fields, revealing how occupation time distributions differ from classical predictions and relate to statistical mechanics.

Contribution

It introduces a non-ergodic equilibrium framework linking occupation time distributions to the partition function in systems with weak ergodicity breaking.

Findings

Occupation time distributions approach U or W shapes in the non-ergodic phase.

Distributions depend on the partition function, connecting to canonical statistical mechanics.

In the ergodic phase, occupation times follow a delta function consistent with Boltzmann-Gibbs statistics.

Abstract

Random walk models, such as the trap model, continuous time random walks, and comb models exhibit weak ergodicity breaking, when the average waiting time is infinite. The open question is: what statistical mechanical theory replaces the canonical Boltzmann-Gibbs theory for such systems? In this manuscript a non-ergodic equilibrium concept is investigated, for a continuous time random walk model in a potential field. In particular we show that in the non-ergodic phase the distribution of the occupation time of the particle on a given lattice point, approaches U or W shaped distributions related to the arcsin law. We show that when conditions of detailed balance are applied, these distributions depend on the partition function of the problem, thus establishing a relation between the non-ergodic dynamics and canonical statistical mechanics. In the ergodic phase the distribution function of the occupation times approaches a delta function centered on the value predicted based on standard Boltzmann-Gibbs statistics. Relation of our work with single molecule experiments is briefly discussed.