Velocity and diffusion constant of an active particle in a one-dimensional force field

¹ Laboratoire de Physique de l'Ecole Normale Supérieure, PSL University, CNRS, Sorbonne Universités 24 rue Lhomond, 75231 Paris, France
² LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay - 91405 Orsay, France

Received: 8 April 2020
Accepted: 25 May 2020

Abstract

We consider a run-and-tumble particle with two velocity states , in an inhomogeneous force field f(x) in one dimension. We obtain exact formulae for its velocity V_L and diffusion constant D_L for arbitrary periodic f(x) of period L. They involve the “active potential” which allows to define a global bias. Upon varying parameters, such as an external force F, the dynamics undergoes transitions from non-ergodic trapped states, to various moving states, some with non-analyticities in the V_L vs. F curve. A random landscape in the presence of a bias leads, for large L, to anomalous diffusion , , or to a phase with a finite velocity that we calculate.