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

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.