Kinetics of First Passage in a Cone

TL;DR

This paper analyzes the decay of the probability that a diffusing particle remains inside a cone over time, revealing how the decay exponent depends on cone geometry and dimension, with explicit solutions in four dimensions.

Contribution

It provides explicit formulas and asymptotic analysis for the first passage time exponent in cones across arbitrary dimensions, including special cases and scaling behaviors.

Findings

Decay exponent depends on cone angle and dimension

Explicit expression for four-dimensional case

Scaling behavior in large dimensions

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.