Nonequilibrium statistics of harmonically trapped run-and-tumble particles: An exact convolution approach

Abstract

We study one-dimensional run-and-tumble particles confined by a harmonic potential and coupled to an equilibrium thermal bath. Exploiting the coupling of active and thermal degrees of freedom through the Ornstein-Uhlenbeck propagator, we show that the position distribution factorizes, as a convolution, into the Ornstein-Uhlenbeck distribution and the distribution of the athermal run-and-tumble problem. This yields closed-form results, correcting expressions from earlier treatments by identifying the Ornstein-Uhlenbeck propagator, rather than the free-diffusion one, as the correct kernel. Results are confirmed both in Fourier space and by direct Langevin simulation. We further focus our analysis on the stationary regime, characterized by two dimensionless parameters--the ratio of trapping to persistence length, and the ratio of thermal to active diffusion--which control the crossover from a non-Gaussian, boundary-peaked distribution to the Gaussian, equilibrium-like limit. Energy fluctuations and the Kullback-Leibler divergence from the equilibrium distribution quantify the resulting non-equilibrium character of the confined active particle.