Non-crossing run-and-tumble particles on a line

Abstract

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.

19 pages, 3 figures; revised version, Refs. added

References in corpus (5)

• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• First-passage time of run-and-tumble particles
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Cited by in corpus (14)

• Universal survival probability for a $d$-dimensional run-and-tumble particle
• Mean area of the convex hull of a run and tumble particle in two dimensions
• Exact position distribution of a harmonically-confined run-and-tumble particle in two dimensions
• Nonequilibirum steady state for harmonically-confined active particles
• Run-and-Tumble particle in inhomogeneous media in one dimension
• Encounter-based model of a run-and-tumble particle II: absorption at sticky boundaries
• Stationary nonequilibrium bound state of a pair of run and tumble particles
• Local time for run and tumble particle
• First-encounter time of two diffusing particles in confinement
• Extremal statistics of a one dimensional run and tumble particle with an absorbing wall
• Encounter-based model of a run-and-tumble particle
• Generating constrained run-and-tumble trajectories
• The one-dimensional telegraphic process with noninstantaneous stochastic resetting
• Trapping of a run-and-tumble particle in an inhomogeneous domain: the weak noise limit