Spatial density, first passage times, and entropy of run and tumble particles in a bistable potential

Abstract

We study the dynamics of a run and tumble particle (RTP) in a bistable potential, in contact with a heat bath. We find the exact form of the steady state position distribution as well as its time-dependent form in Laplace space. We use this result to study relaxation properties of the spatial probability density and find that it converges to the steady state from the center outwards, with the relaxation front moving ballistically in time. Physically this ballistic behavior arises when observing relaxation properties at the tails of the density, where the distance of the particle from the origin is larger than the length-scale describing the steady state of the RTP. We also find the exact form of the mean first passage time, and the long-time behavior of the first passage time distribution. This allows us to coarse-grain the dynamics as a dichotomous Markov process, a paradigm for a wide class of bistable processes. Coarse-graining, however, comes at the cost of a loss of information: while the RTP possesses a finite entropy production even at steady state, it identically vanishes for its coarse-grained description.