First Passage in Conical Geometry and Ordering of Brownian Particles

TL;DR

This paper reviews recent findings on first-passage times in conical geometries and their application to the ordering of Brownian particles, highlighting the cone approximation's accuracy and its asymptotic exactness for large N.

Contribution

It introduces a cone approximation method for analyzing first-passage problems involving multiple Brownian particles and demonstrates its accuracy and asymptotic exactness.

Findings

Survival probability decays algebraically with an exponent depending on cone angle and dimension.

The cone approximation provides bounds and estimates for first-passage exponents.

As N approaches infinity, the cone approximation becomes asymptotically exact.

Abstract

We survey recent results on first-passage processes in unbounded cones and their applications to ordering of particles undergoing Brownian motion in one dimension. We first discuss the survival probability S(t) that a diffusing particle, in arbitrary spatial dimension, remains inside a conical domain up to time t. In general, this quantity decays algebraically S ~ t^{-beta} in the long-time limit. The exponent beta depends on the opening angle of the cone and the spatial dimension, and it is root of a transcendental equation involving the associated Legendre functions. The exponent becomes a function of a single scaling variable in the limit of large spatial dimension. We then describe two first-passage problems involving the order of N independent Brownian particles in one dimension where survival probabilities decay algebraically as well. To analyze these problems, we identify the trajectories of the N particles with the trajectory of one particle in N dimensions, confined to within a certain boundary, and we use a circular cone with matching solid angle as a replacement for the confining boundary. For N=3, the confining boundary is a wedge and the approach is exact. In general, this "cone approximation" gives strict lower bounds as well as useful estimates or the first-passage exponents. Interestingly, the cone approximation becomes asymptotically exact when N-->infinity as it predicts the exact scaling function that governs the spectrum of first-passage exponents.