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arXiv · 1009.2530

First Passage in Infinite Paraboloidal Domains

P. L. Krapivsky·S. Redner
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Abstract

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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P. L. Krapivsky, S. Redner. 2010-10-25. First Passage in Infinite Paraboloidal Domains. https://doi.org/10.1088/1742-5468%2F2010%2F11%2Fp11028

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