First-order condensation transition in the position distribution of a run-and-tumble particle in one dimension

We consider a single run-and-tumble particle (RTP) moving in one dimension. We assume that the velocity of the particle is drawn independently at each tumbling from a zero-mean Gaussian distribution and that the run times are exponentially distributed. We investigate the probability distribution P(X, N) of the position X of the particle after N runs, with N >> 1. We show that in the regime X ~ N^{3/4} the distribution P(X, N) has a large deviation form with a rate function characterized by a discontinuous derivative at the critical value X = X_c > 0. The same is true for X = −X_c due to the symmetry of P(X, N). We show that this singularity corresponds to a first-order condensation transition: for X > X_c a single large jump dominates the RTP trajectory. We consider the participation ratio of the single-run displacements as the order parameter of the system, showing that this quantity is discontinuous at X = X_c. Our results are supported by numerical simulations performed with a constrained Markov chain Monte Carlo algorithm.