Non-self-averaging behaviors and ergodicity in quenched trap model with finite system size

TL;DR

This paper investigates non-self-averaging behaviors and ergodicity in finite-sized quenched trap models, revealing significant sample-to-sample fluctuations and conditions under which ergodicity holds or breaks down.

Contribution

It demonstrates the presence of non-self-averaging phenomena in quenched trap models with finite size and analyzes how ergodicity manifests in such disordered systems.

Findings

Sample-to-sample diffusivity fluctuations exceed trajectory-to-trajectory fluctuations.

Time-averaged observables converge to constants, indicating ergodicity.

Dependence of quantities on disorder realization rather than initial conditions.

Abstract

Tracking tracer particles in heterogeneous environments plays an important role in unraveling the material properties. These heterogeneous structures are often static and depend on the sample realizations. Sample-to-sample fluctuations of such disorder realizations sometimes become considerably large. When we investigate the sample-to-sample fluctuations, fundamental averaging procedures are a thermal average for a single disorder realization and the disorder average for different disorder realizations. Here, we report on non-self-averaging phenomena in quenched trap models with finite system sizes, where we consider the periodic and the reflecting boundary conditions. Sample-to-sample fluctuations of diffusivity greatly exceeds trajectory-to-trajectory fluctuations of diffusivity in the corresponding annealed model. For a single disorder realization, the time-averaged mean square displacement and position-dependent observables converge to constants with the aid of the existence of the equilibrium distribution. This is a manifestation of ergodicity. As a result, the time-averaged quantities do not depend on the initial condition nor on the thermal histories but depend crucially on the disorder realization.