How rare are diffusive rare events?D. P. Sanders and H. Larralde
Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México - Apartado Postal 48-3, 62551 Cuernavaca, Morelos, Mexico
received 11 February 2008; accepted in final form 5 April 2008; published May 2008
published online 14 May 2008
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.
05.40.-a - Fluctuation phenomena, random processes, noise, and Brownian motion.
05.40.Fb - Random walks and Levy flights.
05.60.-k - Transport processes.
© EPLA 2008