The Failure of the Ergodic Assumption

TL;DR

This paper investigates the limitations of using phenomenological ensembles from single time series, revealing that they can be non-stationary and non-ergodic, leading to potential misrepresentations of the underlying dynamics.

Contribution

It demonstrates that phenomenological ensembles from a single time series may not be stationary or ergodic, challenging common assumptions in time series analysis.

Findings

Phenomenological ensemble can be non-stationary.

Probability ensemble averages may differ from time averages.

Simple Langevin process can be mean ergodic but still produce misleading ensembles.

Abstract

The well established procedure of constructing phenomenological ensemble from a single long time series is investigated. It is determined that a time series generated by a simple Uhlenbeck-Ornstein Langevin equation is mean ergodic. However the probability ensemble average yields a variance that is different from that determined using the phenomenological ensemble (time average). We conclude that the latter ensemble is often neither stationary nor ergodic and consequently the probability ensemble averages can misrepresent the underlying dynamic process.