# Archive ouverte HAL – Noncrossing run-and-tumble particles on a line

### Pierre Le Doussal 1 Satya N. Majumdar 2 Pierre Le Doussal 1 Satya Majumdar 2 Gregory Schehr 2

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

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.

• 1. LPTENS - Laboratoire de Physique Théorique de l'ENS
• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques