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

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2022-11, Vol.55 (46), p.465004
• Main Authors: Singh, Prashant, Santra, Saikat, Kundu, Anupam
• Format: Article
• Language: English
• Subjects: absorbing wall; extremal statistics; run and tumble particle
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/aca230

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 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.