Residence Time Statistics for Normal and Fractional Diffusion in a Force Field

TL;DR

This paper analyzes occupation time statistics for particles in force fields, revealing ergodic behavior in normal diffusion and weak ergodicity breaking in fractional diffusion, linking fractional calculus to ergodicity properties.

Contribution

It provides a general relation between occupation times and first passage times applicable to both normal and fractional diffusion, and classifies occupation time behaviors based on recurrence properties.

Findings

Normal diffusion exhibits ergodic behavior in binding potentials.

Fractional diffusion shows weak ergodicity breaking.

Classification of occupation times depends on recurrence and return times.

Abstract

We investigate statistics of occupation times for an over-damped Brownian particle in an external force field. A backward Fokker-Planck equation introduced by Majumdar and Comtet describing the distribution of occupation times is solved. The solution gives a general relation between occupation time statistics and probability currents which are found from solutions of the corresponding problem of first passage time. This general relationship between occupation times and first passage times, is valid for normal Markovian diffusion and for non-Markovian sub-diffusion, the latter modeled using the fractional Fokker-Planck equation. For binding potential fields we find in the long time limit ergodic behavior for normal diffusion, while for the fractional framework weak ergodicity breaking is found, in agreement with previous results of Bel and Barkai on the continuous time random walk on a lattice. For non-binding potential rich physical behaviors are obtained, and classification of occupation time statistics is made possible according to whether or not the underlying random walk is recurrent and the averaged first return time to the origin is finite. Our work establishes a link between fractional calculus and ergodicity breaking.