Ultraslow Convergence to Ergodicity in Transient Subdiffusion

TL;DR

This paper studies how certain random walks with heavy-tailed trapping times exhibit extremely slow convergence to ergodic behavior, with a long-lasting distributional ergodicity characterized by Mittag-Leffler distributions, and identifies a crossover to standard ergodicity.

Contribution

It proves distributional ergodicity for truncated alpha-stable trapping times and characterizes the slow convergence and crossover to ordinary ergodicity in subdiffusive processes.

Findings

Distributional ergodicity follows Mittag-Leffler distribution.

Convergence to ergodicity is significantly slower than in typical cases.

A crossover from distributional to ordinary ergodic behavior is observed.

Abstract

We investigate continuous time random walks with truncated $\alpha$-stable trapping times. We prove distributional ergodicity for a class of observables; namely, the time-averaged observables follow the probability density function called the Mittag--Leffler distribution. This distributional ergodic behavior persists for a long time, and thus the convergence to the ordinary ergodicity is considerably slower than in the case in which the trapping-time distribution is given by common distributions. We also find a crossover from the distributional ergodic behavior to the ordinary ergodic behavior.