Threshold surface reactions and local time resetting

TL;DR

This paper develops a mathematical framework for analyzing threshold surface reactions involving local time resetting in diffusing particles, introducing a joint probability density approach and a stochastic resetting protocol, with applications to simple geometries.

Contribution

It introduces a new method to analyze threshold surface absorption using joint probability densities and incorporates a stochastic resetting protocol affecting both position and internal state.

Findings

The joint propagator $P_0( extbf{x}, ext{ell},t| extbf{x}_0)$ can be used to analyze absorption.

Resetting both position and internal state simplifies survival probability calculations.

Applications to diffusion on the half-line and spherical domains demonstrate the theory.

Abstract

In this paper we consider a threshold surface absorption mechanism for a particle diffusing in a domain containing a single target $\calU $. The target boundary $\partial \calU$ is taken to be a reactive surface that modifies an internal state $Z_t$ of the particle when in contact with the surface at time $t$, with $Z_0=h$. The state $Z_t$ is taken to be a monotonically decreasing function of the so-called boundary local time, and absorption occurs as soon as $Z_t$ reaches zero. (The boundary local time is a Brownian functional that determines the amount of time that the particle spends in a neighborhood of $\partial \calU$.) We first show how to analyze threshold surface absorption in terms of the joint probability density or generalized propagator $P_0(\x,\ell,t|\x_0)$ for the pair $(\X_t,\ell_t)$ in the case of a perfectly reflecting surface, where $\X_t$ and $\ell_t$ denote the particle position and local time at time $t$, respectively, and $\x_0$ is the initial position. We then introduce a generalized stochastic resetting protocol in which both the position $\X_t$ and the internal state $Z_t$ are reset to their initial values, $\X_t\rightarrow \x_0$ and $Z_t\rightarrow h$, at a Poisson rate $r$. The latter is mathematically equivalent to resetting the boundary local time, $\ell_t\rightarrow 0$. Since resetting is governed by a renewal process, the survival probability with resetting can be expressed in terms of the survival probability without resetting, which means that the statistics of absorption can be determined by calculating the Laplace transform of $P_0(\x,\ell,t|\x_0)$ with respect to $t$. We contrast this with the case where only particle position is reset, which is not governed by a renewal process. We illustrate the theory using the simple examples of diffusion on the half-line and a spherical target in a spherical domain.