Kinetics of First Passage in a Cone

Abstract

We study statistics of first passage inside a cone in arbitrary spatial dimension. The probability that a diffusing particle avoids the cone boundary decays algebraically with time. The decay exponent depends on two variables: the opening angle of the cone and the spatial dimension. In four dimensions, we find an explicit expression for the exponent, and in general, we obtain it as a root of a transcendental equation involving associated Legendre functions. At large dimensions, the decay exponent depends on a single scaling variable, while roots of the parabolic cylinder function specify the scaling function. Consequently, the exponent is of order one only if the cone surface is very close to a plane. We also perform asymptotic analysis for extremely thin and extremely wide cones.

9 pages, 5 figures, 2 tables

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Cited by in corpus (16)

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• First-Passage Exponents of Multiple Random Walks
• Polymer-mediated entropic forces between scale-free objects
• Entropic force of polymers on a cone tip
• Ideal Polymers near Scale-Free Surfaces
• Slow Kinetics of Brownian Maxima
• First passage time and change of entropy
• Diffusion in the Presence of Scale-Free Absorbing Boundaries
• On the Mixing of Diffusing Particles
• On sequences of records generated by planar random walks
• Long Polymers Near Wedges and Cones
• Attractive and repulsive polymer-mediated forces between scale-free surfaces
• Scaling Exponents for Ordered Maxima
• Infinite System of Random Walkers: Winners and Losers
• Diffusion in a fluid flow generated by a source at the apex of a wedge
• Statistical properties of sites visited by independent random walks