Inequivalence of time and ensemble averages in ergodic systems: exponential versus power-law relaxation in confinement

TL;DR

This paper demonstrates that in ergodic systems, time averages can differ significantly from ensemble averages, especially in confined diffusion, affecting the interpretation of single-particle tracking data.

Contribution

It reveals the fundamental difference between time and ensemble averages in ergodic systems with confinement, highlighting implications for data analysis in biological and physical systems.

Findings

Time averages differ from ensemble averages in confined diffusion.

Behavior persists regardless of measurement duration.

Impacts interpretation of relaxation times in experiments.

Abstract

Single particle tracking has become a standard tool to investigate diffusive properties, especially in small systems such as biological cells. Usually the resulting time series are analyzed in terms of time averages over individual trajectories. Here we study confined normal as well as anomalous diffusion modeled by fractional Brownian motion and the fractional Langevin equation, and show that even for such ergodic systems time-averaged quantities behave differently from their ensemble averaged counterparts, irrespective of how long the measurement time becomes. Knowledge of the exact behavior of time averages is therefore fundamental for the proper physical interpretation of measured time series, in particular, for extraction of the relaxation time scale from data.