Condensation transition in the late-time position of a run-and-tumble particle

Abstract

We study the position distribution $P(\stackrel{P\vec}{R},N)$ 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, $P(\stackrel{P\vec}{R},N)\ensuremath{\rightarrow}P(R,N)$ where $R=|\stackrel{P\vec}{R}|$. We show that, under certain conditions on $d$ and $W(v)$ and for large $N$, a condensation transition occurs at some critical value of $R={R}_{c}\ensuremath{\sim}O(N)$ 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 $W(v)=\ensuremath{\alpha}{(1\ensuremath{-}v/{v}_{0})}^{\ensuremath{\alpha}\ensuremath{-}1}/{v}_{0}$, parametrized by $\ensuremath{\alpha}>0$, we show that, for large $N, P(R,N)\ensuremath{\sim}exp[\ensuremath{-}N{\ensuremath{\psi}}_{d,\ensuremath{\alpha}}(R/N)]$, and we compute exactly the rate function ${\ensuremath{\psi}}_{d,\ensuremath{\alpha}}(z)$ for any $d$ and $\ensuremath{\alpha}$. 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 $\ensuremath{\alpha}$. 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 $\ensuremath{\sim}{10}^{\ensuremath{-}100}$.

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