Weakly non-ergodic Statistical Physics

TL;DR

This paper derives a general formula for the distribution of time-averaged observables in weakly non-ergodic systems, explaining their anomalous fluctuations and connecting to classical ergodic behavior.

Contribution

It introduces a universal formula for time-averaged observables in weakly non-ergodic systems, bridging anomalous and normal fluctuation regimes.

Findings

Derived a general distribution formula for weakly non-ergodic systems

Applied the formula to fractional dynamics in binding fields

Discussed implications for single particle experimental observations

Abstract

We find a general formula for the distribution of time averaged observables for weakly non-ergodic systems. Such type of ergodicity breaking is known to describe certain systems which exhibit anomalous fluctuations, e.g. blinking quantum dots and the sub-diffusive continuous time random walk model. When the fluctuations become normal we recover usual ergodic statistical mechanics. Examples of a particle undergoing fractional dynamics in a binding force field are worked out in detail. We briefly discuss possible physical applications in single particle experiments.