Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension

Kanaya Malakar 1 V. Jemseena 2 Anupam Kundu 2 K. Vijay Kumar 2 Sanjib Sabhapandit 3 Satya N. Majumdar 4 S. Redner 5 Abhishek Dhar 2

Journal of Statistical Mechanics: Theory and Experiment, IOP Publishing, 2018

We investigate the motion of a run-and-tumble particle (RTP) in one dimension. We find the exact probability distribution of the particle with and without diffusion on the infinite line, as well as in a finite interval. In the infinite domain, this probability distribution approaches a Gaussian form in the long-time limit, as in the case of a regular Brownian particle. At intermediate times, this distribution exhibits unexpected multi-modal forms. In a finite domain, the probability distribution reaches a steady state form with peaks at the boundaries, in contrast to a Brownian particle. We also study the relaxation to the steady state analytically. Finally we compute the survival probability of the RTP in a semi-infinite domain. In the finite interval, we compute the exit probability and the associated exit times. We provide numerical verifications of our analytical results.

  • 1. Presidency University
  • 2. International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore
  • 3. Raman Research Institute
  • 4. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
  • 5. Santa Fe Institute