Motion of a Brownian molecule in the presence of reactive boundaries

TL;DR

This paper provides an exact analysis of a Brownian particle's motion within a confined one-dimensional domain with reactive boundaries, deriving key probabilistic quantities and reaction time profiles.

Contribution

It introduces exact formulas for displacement, survival probability, and absorption time considering reactive boundaries, advancing understanding of boundary effects in stochastic processes.

Findings

Exact probability density of particle displacement derived.

Expressions for survival probability and mean absorption time obtained.

Analytical results validated by numerical simulations.

Abstract

We study the one-dimensional motion of a Brownian particle inside a confinement described by two reactive boundaries which can partially reflect or absorb the particle. Understanding the effects of such boundaries is important in physics, chemistry and biology. We compute the probability density of the particle displacement exactly, from which we derive expressions for the survival probability and the mean absorption time as a function of the reactive coefficients. Furthermore, using the Feynman-Kac formalism, we investigate the reaction time profile, which is the fluctuating time spent by the particle at a given location, both till a fixed observation time and till the absorption time. Our analytical results are compared to numerical simulations showing perfect agreement.