Steady state and relaxation dynamics of run-and-tumble particles in contact with a heat bath

We study the relaxation dynamics of a run and tumble particle (RTP) in a one-dimensional piecewise linear potential U(x)=b|x|, from delta-function initial conditions at x=0 to steady state. In addition to experiencing active telegraphic noise, the particle is in contact with a heat bath at temperature T that applies white thermal noise. We find that the position distribution of the RTP is described by a sum of two distributions ("modes"), each of which of the form P(x,t→∞)∼e^{-λ_{i}|x|} (i=1,2) at steady state. The two modes are dynamically coupled: At very short times (t→0), each mode stores half of the probability, and exhibits thermal diffusive spreading with a Gaussian profile. With progressing time and evolution toward steady state, the partition of probability between the modes becomes increasingly uneven, and, depending on the model parameters, the mode with the smaller value of λ_{i} may carry an overwhelming majority of the probability. Moreover, we identify that the characteristic relaxation time of each mode is τ_{i}=(λ_{i}^{2}T)^{-1}, which implies that the minority mode also relaxes much faster than the dominant one. A more detailed analysis reveals that τ_{i} is characteristic of the mode relaxation only close to the origin at the core of the distribution, while farther away it increases linearly with |x| as if a relaxation front is propagating at constant speed v_{i}^{*}=2sqrt[T/τ_{i}] in the system. The rate of nonequilibrium entropy production can be related to the two-mode splitting of the probability distribution and be expressed in terms of their correlation lengths λ_{i} and their contributions to the steady-state distribution.