Fluctuations of time averages for Langevin dynamics in a binding force field

TL;DR

This paper derives a formula for the fluctuations of time averages in Langevin dynamics within binding fields, highlighting ergodic and non-ergodic behaviors and their relation to the Boltzmann measure and super-aging correlations.

Contribution

It introduces a simple formula for fluctuations of time averages in overdamped Brownian motion and explores ergodicity breaking in logarithmic potentials.

Findings

Finite measurement fluctuations are determined by the Boltzmann measure in ergodic processes.

Ergodicity is broken in logarithmic potentials, leading to large non-ergodic fluctuations.

Non-ergodic fluctuations relate to super-aging correlation functions.

Abstract

We derive a simple formula for the fluctuations of the time average around the thermal mean for overdamped Brownian motion in a binding potential U(x). Using a backward Fokker-Planck equation, introduced by Szabo, et al. in the context of reaction kinetics, we show that for ergodic processes these finite measurement time fluctuations are determined by the Boltzmann measure. For the widely applicable logarithmic potential, ergodicity is broken. We quantify the large non-ergodic fluctuations and show how they are related to a super-aging correlation function.