Extreme Narrow Escape: shortest paths for the first particles to escape through a small window

TL;DR

This paper develops a variational principle to identify the most probable paths of the fastest Brownian particles escaping through a small window in higher dimensions, revealing that these paths are geodesics.

Contribution

It introduces a novel variational framework based on Wiener path-integral to determine the optimal escape trajectories in higher-dimensional domains.

Findings

Fastest escape paths are geodesics.

Optimal trajectories concentrate along geodesics.

Simulations confirm the theoretical predictions.

Abstract

What is the path associated with the fastest Brownian particle that reaches a narrow window located on the boundary of a domain? Although the distribution of the fastest arrival times has been well studied in dimension 1, much less is known in higher dimensions. Based on the Wiener path-integral, we derive a variational principle for the path associated with the fastest arrival particle. Specifically, we show that in a large ensemble of independent Brownian trajectories, the first moment of the shortest arrival time is associated with the minimization of the energy-action and the optimal trajectories are geodesics. Escape trajectories concentrate along these geodesics, as confirmed by stochastic simulations when an obstacle is positioned in front of the narrow window. To conclude paths in stochastic dynamics and their time scale can differ significantly from the mean properties, usually at the basis of the Smoluchowski's theory of chemical reactions.