Transition path theory for diffusive search with stochastic resetting

TL;DR

This paper extends transition path theory to analyze diffusive search processes with stochastic resetting, deriving formulas for reaction rates and identifying optimal resetting rates that maximize resource delivery.

Contribution

It introduces a novel application of TPT to diffusive search with resetting, addressing non-equilibrium stationary states and flux calculations across dividing surfaces.

Findings

Derived a general expression for the resource accumulation rate $k_{AB}$.

Showed that $k_{AB}$ is independent of the dividing surface choice.

Identified an optimal resetting rate that maximizes $k_{AB}$ in a finite interval.

Abstract

Many chemical reactions can be formulated in terms of particle diffusion in a complex energy landscape. Transition path theory (TPT) is a theoretical framework for describing the direct (reaction) pathways from reactant to product states within this energy landscape, and calculating the effective reaction rate. It is now the standard method for analyzing rare events between long lived states. In this paper, we consider a completely different application of TPT, namely, a dual-aspect diffusive search process in which a particle alternates between collecting cargo from a source domain $A$ and then delivering it to a target domain $B$. The rate of resource accumulation at the target, $k_{AB}$, is determined by the statistics of direct (reactive or transport) paths from A to B. Rather than considering diffusion in a complex energy landscape, we focus on pure diffusion with stochastic resetting. Resetting introduces two non-trivial problems in the application of TPT. First, the process is not time-reversal invariant, which is reflected by the fact that there exists a unique non-equilibrium stationary state (NESS). Second, calculating $k_{AB}$ involves determining the total probability flux of direct transport paths across a dividing surface $S$ between $A$ and $B$. This requires taking into account discontinuous jumps across $S$ due to resetting. We derive a general expression for $k_{AB}$ and show that it is independent of the choice of dividing surface. Finally, using the example of diffusion in a finite interval, we show that there exists an optimal resetting rate at which $k_{AB}$ is maximized. We explore how this feature depends on model parameters.