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

• Published in: Journal of statistical mechanics 2021-10, Vol.2021 (10), p.103208
• Main Authors: Mori, Francesco, Gradenigo, Giacomo, Majumdar, Satya N
• Format: Article
• Language: English
• Subjects: extreme value; large deviations in non-equilibrium systems; metastable states; Physics; zero-range processes
• ISSN: 1742-5468; 1742-5468
• DOI: 10.1088/1742-5468/ac2899

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 $N\gg 1$. We show that in the regime $ X \sim 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.