Condensation transition in the late-time position of a Run-and-Tumble particle

TL;DR

This paper investigates a phase transition in the position distribution of a run-and-tumble particle, revealing a condensation phenomenon where a single large run dominates the displacement in the large deviation regime.

Contribution

It provides an exact analysis of the condensation transition in the position distribution of RTPs for arbitrary dimensions and speed distributions, including the rate function and condensate size distribution.

Findings

Identification of a condensation transition at a critical displacement R_c

Explicit computation of the rate function ψ_{d,α}(z) for large N

Confirmation of theoretical predictions through high-precision simulations

Abstract

We study the position distribution $P(\vec{R},N)$ of a run-and-tumble particle (RTP) in arbitrary dimension $d$, after $N$ runs. We assume that the constant speed $v>0$ of the particle during each running phase is independently drawn from a probability distribution $W(v)$ and that the direction of the particle is chosen isotropically after each tumbling. The position distribution is clearly isotropic, $P(\vec{R},N)\to P(R,N)$ where $R=|\vec{R}|$. We show that, under certain conditions on $d$ and $W(v)$ and for large $N$, a condensation transition occurs at some critical value of $R=R_c\sim O(N)$ located in the large deviation regime of $P(R,N)$. For $R<R_c$ (subcritical fluid phase), all runs are roughly of the same size in a typical trajectory. In contrast, an RTP trajectory with $R>R_c$ is typically dominated by a `condensate', i.e., a large single run that subsumes a finite fraction of the total displacement (supercritical condensed phase). Focusing on the family of speed distributions $W(v)=\alpha(1-v/v_0)^{\alpha-1}/v_0$, parametrized by $\alpha>0$, we show that, for large $N$, $P(R,N)\sim \exp\left[-N\psi_{d,\alpha}(R/N)\right]$ and we compute exactly the rate function $\psi_{d,\alpha}(z)$ for any $d$ and $\alpha$. We show that the transition manifests itself as a singularity of this rate function at $R=R_c$ and that its order depends continuously on $d$ and $\alpha$. We also compute the distribution of the condensate size for $R>R_c$. Finally, we study the model when the total duration $T$ of the RTP, instead of the total number of runs, is fixed. Our analytical predictions are confirmed by numerical simulations, performed using a constrained Markov chain Monte Carlo technique, with precision $\sim 10^{-100}$.