Firstorder condensation transition in the position distribution of a runandtumble particle in one dimension
Abstract
We consider a single runandtumble particle (RTP) moving in one dimension. We assume that the velocity of the particle is drawn independently at each tumbling from a zeromean 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 firstorder condensation transition: for $X>X_c$ a single large jump dominates the RTP trajectory. We consider the participation ratio of the singlerun 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.00338
 Bibcode:
 2021arXiv210700338M
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Soft Condensed Matter;
 Mathematical Physics;
 Mathematics  Probability
 EPrint:
 24 pages, 10 figures