# Kinetics of First Passage in a Cone

Source: [Kinetics of First Passage in a Cone](<https://www.sota2.com/research/paper/68fa29bb1e209131ce837198>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T13:05:44.158Z

Authors: E. Ben-Naim, P.L. Krapivsky

Published: 2010-09-01T18:22:52.000Z

## 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.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1009.0238>)

[PDF](<https://arxiv.org/pdf/1009.0238.pdf>)