Conditional Ergodicity and Universal Fluctuations in Weak Ergodicity Breaking

TL;DR

This paper introduces the concept of conditional ergodicity, showing that conditioning on an internal clock restores self-averaging in weakly ergodic systems, leading to universal fluctuations described by the Mittag-Leffler distribution.

Contribution

It uncovers the principle of conditional ergodicity and demonstrates its implications for universal fluctuation laws in systems with weak ergodicity breaking.

Findings

Time-averaged transport coefficients follow Mittag-Leffler distribution after rescaling.

Conditional ergodicity restores self-averaging in anomalous diffusion.

Universal behavior observed across multiple disordered media models.

Abstract

Time averages extracted from single-particle trajectories in complex media often vary strongly from one trajectory to another, even for long measurement times. Such persistent trajectory-to trajectory scatter is commonly observed in anomalous diffusion and signals weak ergodicity breaking driven by scale-free trapping. Here we identify conditional ergodicity: conditioning on a natural internal clock restores self-averaging of time-averaged observables. Combining conditional ergodicity with the stochastic mapping between the internal clock and physical time implies a universal law: once rescaled by their mean, time-averaged transport coefficients in systems exhibiting weak ergodicity breaking follow the Mittag-Leffler distribution. We demonstrate this universality across multiple models of disordered media displaying anomalous diffusion.