Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials

TL;DR

This paper analyzes the dynamics of a Brownian particle in a logarithmic potential, revealing aging and ergodicity breaking phenomena through analytical and simulation methods.

Contribution

It provides analytical expressions for correlation functions and fluctuations, linking non-normalizable densities to super-aging in a confining logarithmic potential.

Findings

Identification of aging and nonergodic regimes based on potential depth

Analytical derivation of two-time correlation functions

Validation of predictions with Langevin simulations

Abstract

We consider an overdamped Brownian particle moving in a confining asymptotically logarithmic potential, which supports a normalized Boltzmann equilibrium density. We derive analytical expressions for the two-time correlation function and the fluctuations of the time-averaged position of the particle for large but finite times. We characterize the occurrence of aging and nonergodic behavior as a function of the depth of the potential, and support our predictions with extensive Langevin simulations. While the Boltzmann measure is used to obtain stationary correlation functions, we show how the non-normalizable infinite covariant density is related to the super-aging behavior.