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