Non-thermal Einstein relations

We consider a particle moving with equation of motion (x) over dot = f(t), where f(t) is a random function with statistics which are independent of x and t, with a finite drift velocity v = < f > and in the presence of a reflecting wall. Far away from the wall, translational invariance implies that the stationary probability distribution is P(x) similar to exp(alpha x). A classical example of a problem of this type is sedimentation equilibrium, where alpha is determined by temperature. In this work we do not introduce a thermal reservoir and alpha is determined from the equation of motion. We consider a general approach to determining a which is not always in agreement with Einstein's relation between the mean velocity and the diffusion coefficient. We illustrate our results with a model inspired by the Boltzmann equation. Copyright (C) EPLA, 2016