First Passage in Infinite Paraboloidal Domains

Details First Passage in Infinite Paraboloidal Domains First Passage in Infinite Paraboloidal Domains

2010-09-13

arxiv.org/abs/1009.2530v2

J. Stat. Mech. P11028, (2010)

Physics

Condensed Matter

Statistical Mechanics

7 pages, 2 figure, revtex4 format. Version 2 contains minor changes and one added figure in response to referee comments. For

Scientific paper

10.1088/1742-5468/2010/11/P11028

We study first-passage properties for a particle that diffuses either inside or outside of generalized paraboloids, defined by y=a(x_1^2+...+x_{d-1}^2)^{p/2} where p>1, with absorbing boundaries. When the particle is inside the paraboloid, the survival probability S(t) generically decays as a stretched exponential, ln(S) ~ -t^{(p-1)/(p+1)}, independent of the spatial dimensional. For a particle outside the paraboloid, the dimensionality governs the asymptotic decay, while the exponent p specifying the paraboloid is irrelevant. In two and three dimensions, S ~ t^{-1/4} and S ~(ln t)^{-1}, respectively, while in higher dimensions the particle survives with a finite probability. We also investigate the situation where the interior of a paraboloid is uniformly filled with non-interacting diffusing particles and estimate the distance between the closest surviving particle and the apex of the paraboloid.

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