Noncrossing run-and-tumble particles on a line

Pierre Le Doussal (1, 2) , Satya N. Majumdar (3) , Gregory Schehr (3)

1 LPTENS - Laboratoire de Physique Théorique de l'ENS [École Normale Supérieure]
2 Champs Aléatoires et Systèmes hors d'Équilibre
3 LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques

We study active particles performing independent run and tumble motion on an infinite line with velocities $v_0 \sigma(t)$, where $\sigma(t) = \pm 1$ is a dichotomous telegraphic noise with constant flipping rate $\gamma$. We first consider one particle in the presence of an absorbing wall at $x=0$ and calculate the probability that it has survived up to time $t$ and is at position $x$ at time $t$. We then consider two particles with independent telegraphic noises and compute exactly the probability that they do not cross up to time $t$. Contrarily to the case of passive (Brownian) particles this two-RTP problem can not be reduced to a single RTP with an absorbing wall. Nevertheless, we are able to compute exactly the probability of no-crossing of two independent RTP's up to time $t$ and find that it decays at large time as $t^{-1/2}$ with an amplitude that depends on the initial condition. The latter allows to define an effective length scale, analogous to the so called `` Milne extrapolation length'' in neutron scattering, which we demonstrate to be a fingerprint of the active dynamics.

Domaines: Physique [physics]

Pierre Le Doussal, Satya N. Majumdar, Gregory Schehr. Noncrossing run-and-tumble particles on a line. Physical Review E , 2019, 100 (1), pp.012113. ⟨10.1103/PhysRevE.100.012113⟩. ⟨hal-02291902⟩