Slow Kinetics of Brownian Maxima

TL;DR

This paper investigates the extreme-value statistics of Brownian motion, revealing algebraic decay in the probability that maxima of two particles remain ordered, with results extending to multiple particles and higher dimensions.

Contribution

It introduces a detailed analysis of the ordering probability of maxima in Brownian motion, including the dependence on diffusion constants and extensions to multiple particles and higher dimensions.

Findings

Probability P(t) decays as t^(-1/4) for two particles

Exponent depends on diffusion constants as (1/pi)arctan[sqrt(D2/D1)]

Numerical simulations extend results to multiple particles and higher dimensions

Abstract

We study extreme-value statistics of Brownian trajectories in one dimension. We define the maximum as the largest position to date and compare maxima of two particles undergoing independent Brownian motion. We focus on the probability P(t) that the two maxima remain ordered up to time t, and find the algebraic decay P ~ t^(-beta) with exponent beta=1/4. When the two particles have diffusion constants D1 and D2, the exponent depends on the mobilities, beta=(1/pi)arctan[sqrt(D2/D1)]. We also use numerical simulations to investigate maxima of multiple particles in one dimension and the largest extension of particles in higher dimensions.