Extremal statistics of a one dimensional run and tumble particle with an absorbing wall

TL;DR

This paper analytically investigates the extremal statistics of a one-dimensional run and tumble particle with absorbing boundaries, revealing distinct behaviors in the distribution of the maximum displacement and its timing, with implications for biological motion modeling.

Contribution

The study provides exact analytical expressions for the joint and marginal distributions of maximum displacement and its timing for a run and tumble particle with absorbing boundaries, including novel asymptotic behaviors.

Findings

Distribution P_M(t_m) varies with initial velocity for small t_m.

Large t_m decay of P_M(t_m) matches Brownian motion.

Analytical results are confirmed by numerical simulations.

Abstract

We study the extreme value statistics of a run and tumble particle (RTP) in one dimension till its first passage to the origin starting from the position $x_0~(>0)$. This model has recently drawn a lot of interest due to its biological application in modelling the motion of certain species of bacteria. Herein, we analytically study the exact time-dependent propagators for a single RTP in a finite interval with absorbing conditions at its two ends. By exploiting a path decomposition technique, we use these propagators appropriately to compute the joint distribution $\mathscr{P}(M,t_m)$ of the maximum displacement $M$ till first-passage and the time $t_m$ at which this maximum is achieved exactly. The corresponding marginal distributions $\mathbb{P}_M(M)$ and $P_M(t_m)$ are studied separately and verified numerically. In particular, we find that the marginal distribution $P_M(t_m)$ has interesting asymptotic forms for large and small $t_m$. While for small $t_m$, the distribution $P_M(t_m)$ depends sensitively on the initial velocity direction $\sigma _i$ and is completely different from the Brownian motion, the large $t_m$ decay of $P_M(t_m)$ is same as that of the Brownian motion although the amplitude crucially depends on the initial conditions $x_0$ and $\sigma _i$. We verify all our analytical results to high precision by numerical simulations.