Correlation between the first-reaction time and the acquired boundary local time

TL;DR

This paper develops a universal theoretical framework to analyze the correlation between a diffusing particle's first-reaction time and boundary local time, with explicit solutions for basic domains and simulations for complex media.

Contribution

It introduces a novel, universal method to derive the joint probability density and correlation of first-reaction time and boundary local time, including explicit solutions and simulation validation.

Findings

Correlation depends on boundary reactivity and shape

Explicit analytical solutions for simple domains

Monte Carlo simulations show effects of obstacles

Abstract

We investigate the statistical correlation between the first-reaction time of a diffusing particle and its boundary local time accumulated until the reaction event. Since the reaction event occurs after multiple encounters of the particle with a partially reactive boundary, the boundary local time as a proxy for the number of such encounters is not independent of, but intrinsically linked to, the first-reaction time. We propose a universal theoretical framework to derive their joint probability density and, in particular, the correlation coefficient. To illustrate the dependence of these correlations on the boundary reactivity and shape, we obtain explicit analytical solutions for several basic domains. The analytical results are complemented by Monte Carlo simulations, which we employ to examine the role of interior obstacles on correlations in disordered media. Applications of these statistical results in chemical physics are discussed