Effective drift and diffusion of a particle jumping between mobile and immobile states

We study propagation of a particle that jumps between two states, in which it moves with different velocities and diffusion coefficients. To simplify analysis, in the main part of the paper we derive formulas assuming that in one of the states the particle is immobile. A generalization to the case when the particle is mobile in both states is given at the end of the paper. The formulas show how the effective drift velocity and effective diffusion coefficient depend on jump rates between the two states as well as on the particle velocities and diffusion coefficients in these states. Specifically, we find that the effective diffusion coefficient can exhibit a non-monotonic behavior as a function of the ratio of the jump rates.