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

## TL;DR

This paper investigates the behavior of active run-and-tumble particles on a line, deriving exact probabilities for survival and non-crossing scenarios, revealing unique active dynamics features distinct from passive particles.

## Contribution

It provides the first exact calculations of survival and non-crossing probabilities for run-and-tumble particles, highlighting differences from passive particle behavior.

## Key findings

- Non-crossing probability decays as t^{-1/2} at large times.
- An effective length scale analogous to Milne extrapolation length is identified.
- Exact solutions are obtained for both single and two-particle scenarios.

## 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.