First passages in bounded domains: When is the mean first passage time meaningful?

TL;DR

This study investigates the distribution of first passage times for Brownian motion in bounded 2D domains, revealing when the mean first passage time is a meaningful measure and highlighting the significance of fluctuations.

Contribution

It introduces a probability distribution analysis of first passage times, showing how domain shape influences the validity of the mean first passage time as a characteristic.

Findings

P(ω) distribution can be unimodal or bimodal depending on domain shape

Large fluctuations in first passage times are common in 2D domains

MFPT may be insufficient to characterize first passage behavior

Abstract

We study the first passage statistics to adsorbing boundaries of a Brownian motion in bounded two-dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries. From extensive numerical analysis we obtain the probability P(\omega) distribution of the random variable \omega=\tau_1/(\tau_1+\tau_2), which is a measure for how similar the first passage times \tau_1 and \tau_2 are of two independent realisations of a Brownian walk starting at the same location. We construct a chart for each domain, determining whether P(\omega) represents a unimodal, bell-shaped form, or a bimodal, M-shaped behaviour. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behaviour, in the latter case it is an insufficient measure for the process. Strikingly we find a distinct turnover between the two modes of P(\omega), characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that large fluctuations of the first passage times may occur frequently in two-dimensional domains, rendering quite vague the general use of the MFPT as a robust measure of the actual behaviour even in bounded domains, in which all moments of the first passage distribution exist.