English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study the position distribution $P(\vec{R},N)$ of a run-and-tumbleparticle (RTP) in arbitrary dimension $d$, after $N$ runs. We assume that theconstant speed $v>0$ of the particle during each running phase is independentlydrawn from a probability distribution $W(v)$ and that the direction of theparticle is chosen isotropically after each tumbling. The position distributionis clearly isotropic, $P(\vec{R},N)\to P(R,N)$ where $R=|\vec{R}|$. We showthat, under certain conditions on $d$ and $W(v)$ and for large $N$, acondensation transition occurs at some critical value of $R=R_c\sim O(N)$located in the large deviation regime of $P(R,N)$. For $R<R_c$ (subcriticalfluid phase), all runs are roughly of the same size in a typical trajectory. Incontrast, 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 thetotal displacement (supercritical condensed phase). Focusing on the family ofspeed 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 ratefunction $\psi_{d,\alpha}(z)$ for any $d$ and $\alpha$. We show that thetransition 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 computethe distribution of the condensate size for $R>R_c$. Finally, we study themodel when the total duration $T$ of the RTP, instead of the total number ofruns, is fixed. Our analytical predictions are confirmed by numericalsimulations, performed using a constrained Markov chain Monte Carlo technique,with precision $\sim 10^{-100}$.

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