Analytic study of the effect of persistence on a one-dimensional biased random walk

An analytic study of a one-dimensional biased random walk with correlations between nearest-neighbour steps is presented, both in a lattice model and in its continuous version. First, the treatment of the unbiased problem is recalled and the effect of correlations on the diffusion coefficient is discussed. Then the study is extended to the biased case. The problem is then completely determined by two independent parameters, the degree of correlations in the motion on the one hand and the value of the bias on the other. Both the velocity of the particle and its diffusion coefficient are computed. As a result, the velocity as well as the diffusion coefficient are enhanced when there are positive correlations (qualified as persistence) in the motion, and reduced in the opposite case.