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

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 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 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 σ 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 σ i . We verify all our analytical results to high precision by numerical simulations.