How rare are diffusive rare events?

TL;DR

This paper investigates the timing of rare density fluctuations in diffusive systems, using a mean-field model and numerical analysis to approximate first-passage times, and explores how these results vary with spatial correlations and different fluctuation definitions.

Contribution

It introduces a mean-field approximation for first-passage times of rare events in diffusive systems and compares it with numerical results for systems with spatial correlations.

Findings

Mean-field approximation provides a good estimate for large k in higher dimensions.

First-passage times are affected by spatial correlations and the definition of fluctuations.

Results extend to fluctuations occurring anywhere in the system, not just at a specific site.

Abstract

We study the time until first occurrence, the first-passage time, of rare density fluctuations in diffusive systems. We approach the problem using a model consisting of many independent random walkers on a lattice. The existence of spatial correlations makes this problem analytically intractable. However, for a mean-field approximation in which the walkers can jump anywhere in the system, we obtain a simple asymptotic form for the mean first-passage time to have a given number k of particles at a distinguished site. We show numerically, and argue heuristically, that for large enough k, the mean-field results give a good approximation for first-passage times for systems with nearest-neighbour dynamics, especially for two and higher spatial dimensions. Finally, we show how the results change when density fluctuations anywhere in the system, rather than at a specific distinguished site, are considered.