How rare are diffusive rare events?

Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México - Apartado Postal 48-3, 62551 Cuernavaca, Morelos, Mexico

Corresponding author: dps@fciencias.unam.mx

Received: 11 February 2008
Accepted: 5 April 2008

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.