How rare are diffusive rare events?

David P. Sanders; Hernán Larralde

doi:10.1209/0295-5075/82/40005

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arXiv · 0804.1165

How rare are diffusive rare events?

David P. Sanders·Hernán Larralde

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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.

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David P. Sanders, Hernán Larralde. 2008-04-08. How rare are diffusive rare events?. https://doi.org/10.1209/0295-5075/82/40005

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