Life and Death at the Edge of a Windy Cliff

Abstract

The survival probability of a particle diffusing in the two dimensional domain $x>0$ near a ``windy cliff'' at $x=0$ is investigated. The particle dies upon reaching the edge of the cliff. In addition to diffusion, the particle is influenced by a steady ``wind shear'' with velocity $\vec{v}(x,y)=v\mathrm{sign}(y)\hat{x}$, \ie, no average bias either toward or away from the cliff. For this semi-infinite system, the particle survival probability decays with time as $t^{-1/4}$, compared to $t^{-1/2}$ in the absence of wind. Scaling descriptions are developed to elucidate this behavior, as well as the survival probability within a semi-infinite strip of finite width $|y|<w$ with particle absorption at $x=0$. The behavior in the strip geometry can be described in terms of Taylor diffusion, an approach which accounts for the crossover to the $t^{-1/4}$ decay when the width of the strip diverges. Supporting numerical simulations of our analytical results are presented.

arXiv comment13 pages, plain TeX, 5 figures available upon request to SR (submitted to J. Stat. Phys.)

References in corpus (2)

• Anisotropic critical phenomena in parabolic geometries: the directed self-avoiding walk
• First Passage Time in a Two-Layer System

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