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

TL;DR

This paper analyzes the position distribution of a one-dimensional run-and-tumble particle, revealing a first-order condensation transition characterized by a discontinuous rate function and participation ratio at a critical position.

Contribution

It uncovers a first-order condensation transition in the position distribution of RTP, with a discontinuous rate function and participation ratio, supported by numerical simulations.

Findings

Discontinuous derivative in the rate function at critical position

Condensation transition with a single large jump dominates

Participation ratio is discontinuous at the transition

Abstract

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.