Trapping of Planar Brownian Motion: Full First Passage Time Distributions by Kinetic Monte-Carlo, Asymptotic and Boundary Integral Methods

TL;DR

This paper develops and compares methods for calculating full first passage time distributions of planar Brownian motion to complex targets, including asymptotic, boundary integral, and Kinetic Monte Carlo approaches, with applications to practical scenarios.

Contribution

It introduces a combination of analytical and numerical methods, including a novel Kinetic Monte Carlo approach, for obtaining full arrival time distributions in complex geometries.

Findings

Methods accurately compute arrival distributions in complex configurations.

Homogenization theories effectively approximate complex target scenarios.

KMC method efficiently validates theoretical predictions.

Abstract

We consider the problem of determining the arrival statistics of unbiased planar random walkers to complex target configurations. In contrast to problems posed in finite domains, simple moments of the distribution, such as the mean (MFPT) and variance, are not defined and it is necessary to obtain the full arrival statistics. We describe several methods to obtain these distributions and other associated quantities such as splitting probabilities. One approach combines a Laplace transform of the underlying parabolic equation with matched asymptotic analysis followed by numerical transform inversion. The second approach is similar, but uses a boundary integral equation method to solve for the Laplace transformed variable. To validate the results of this theory, and to obtain the arrival time statistics in very general configurations of absorbers, we introduce an efficient Kinetic Monte Carlo (KMC) method that describes trajectories as a combination of large but exactly solvable projection steps. The effectiveness of these methodologies is demonstrated on a variety of challenging examples highlighting the applicability of these methods to a variety of practical scenarios, such as source inference. A particularly useful finding arising from these results is that homogenization theories, in which complex configurations are replaced by equivalent simple ones, are remarkably effective at describing arrival time statistics.