Ergodic properties of occupation times in heterogeneous media

TL;DR

This paper analytically studies the ergodic behavior of Brownian motion in media with spatially varying diffusion, revealing transitions between ergodic and non-ergodic regimes through occupation time distributions.

Contribution

It derives explicit analytical expressions for occupation time distributions in heterogeneous media and demonstrates the transition from non-ergodic to ergodic behavior.

Findings

Distributions follow arcsine, Gaussian, Lamperti, and Mittag-Leffler forms.

Transition from non-ergodic to ergodic dynamics with parameter variation.

Numerical simulations confirm analytical predictions.

Abstract

We investigate the ergodic properties of Brownian motion in heterogeneous media through the statistics of occupation times. Using the Feynman-Kac formalism, we derive analytical expressions for the distributions, moments, and ergodicity breaking parameters of occupation times in two models with spatially varying diffusion coefficient: a piecewise-constant profile and a power-law profile. In the piecewise model, the half occupation time and the occupation time within an interval follow asymmetric arcsine and half-Gaussian distributions, respectively, indicating non-ergodic behavior. For the power-law case, the corresponding distributions are the Lamperti and Mittag-Leffler. In both models, we identify a transition from non-ergodic to ergodic dynamics as the exponent vary. Numerical simulations fully corroborate the analytical results, demonstrating the effectiveness of the Feynman-Kac approach for quantifying ergodicity in heterogeneous diffusion processes.