Universal properties of a run-and-tumble particle in arbitrary dimension – Archive ouverte HAL

Francesco Mori 1 Pierre Le Doussal 2 Satya N. Majumdar 1 Grégory Schehr 1

Francesco Mori, Pierre Le Doussal, Satya N. Majumdar, Grégory Schehr. Universal properties of a run-and-tumble particle in arbitrary dimension. Physical Review E , American Physical Society (APS), 2020, 102 (4), ⟨10.1103/PhysRevE.102.042133⟩. ⟨hal-03010271⟩

We consider an active run-and-tumble particle (RTP) in $d$ dimensions, starting from the origin and evolving over a time interval $[0,t]$. We examine three different models for the dynamics of the RTP: the standard RTP model with instantaneous tumblings, a variant with instantaneous runs and a general model in which both the tumblings and the runs are non-instantaneous. For each of these models, we use the Sparre Andersen theorem for discrete-time random walks to compute exactly the probability that the $x$ component does not change sign up to time $t$, showing that it does not depend on $d$. As a consequence of this result, we compute exactly other $x$-component properties, namely the distribution of the time of the maximum and the record statistics, showing that they are universal, i.e. they do not depend on $d$. Moreover, we show that these universal results hold also if the speed $v$ of the particle after each tumbling is random, drawn from a generic probability distribution. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 124, 090603 (2020)].

  • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
  • 2. LPENS (UMR_8023) - Laboratoire de physique de l'ENS - ENS Paris