Exit time distribution in spherically symmetric two-dimensional domains

TL;DR

This paper analytically solves the exit time distribution for Brownian particles in various symmetric two-dimensional domains, providing exact and approximate formulas applicable to multiple fields like microfluidics and acoustics.

Contribution

It introduces a novel analytical approach to compute exit time distributions in symmetric domains, including explicit formulas and approximations for different geometries.

Findings

Exact solutions for exit time distributions in disks, sectors, and annuli.

Approximate formulas for mean exit times that are highly accurate.

Extension of methods to biased diffusion scenarios.

Abstract

The distribution of exit times is computed for a Brownian particle in spherically symmetric two- dimensional domains (disks, angular sectors, annuli) and in rectangles that contain an exit on their boundary. The governing partial differential equation of Helmholtz type with mixed Dirichlet- Neumann boundary conditions is solved analytically. We propose both an exact solution relying on a matrix inversion, and an approximate explicit solution. The approximate solution is shown to be exact for an exit of vanishing size and to be accurate even for large exits. For angular sectors, we also derive exact explicit formulas for the moments of the exit time. For annuli and rectangles, the approximate expression of the mean exit time is shown to be very accurate even for large exits. The analysis is also extended to biased diffusion. Since the Helmholtz equation with mixed boundary conditions is encountered in microfluidics, heat propagation, quantum billiards, and acoustics, the developed method can find numerous applications beyond exit processes.