General perturbative framework for kinetics of rare transitions in 1-dimensional active particle systems

TL;DR

This paper develops a perturbative theoretical framework to accurately compute rare transition rates in one-dimensional active particle systems across different persistence time regimes, validated by numerical simulations.

Contribution

It introduces a unified analytical approach for transition rates in active particles valid for all persistence times, bridging previous asymptotic regimes.

Findings

Derived analytical expressions for transition rates in both small and large persistence time regimes.

Created a rational approximation that accurately predicts rates at intermediate persistence times.

Validated the theoretical predictions with numerical simulations showing excellent agreement.

Abstract

We present a theoretical framework that enables investigating rare transitions in a general model of an active particle in an external potential, with the thermal Active Ornstein-Uhlenbeck Particle (AOUP) appearing as a special case. Using a projection-operator formalism, we compute transition rates perturbatively in two distinct asymptotic regimes. In the regime of small persistence times-where the activity evolves much faster than the particle's position-integrating out the activity reproduces the rates previously reported in the literature. In the opposite regime of large persistence times, we instead integrate out the position and obtain the corresponding rates analytically. Together, these asymptotic expansions uniquely specify a rational approximation that remains accurate across intermediate persistence times. As a result, we obtain an analytic expression for the rate valid across all persistence times and activity strengths in the rare-event limit, which are in excellent agreement with numerical simulations. The presented framework applies to rare transitions in a broad class of driven systems.