Non-Thermal Einstein Relations

We consider a particle moving with equation of motion $\dot x=f(t)$, where $f(t)$ is a random function with statistics which are independent of $x$ and $t$, with a finite drift velocity $v=\langle f\rangle$ and in the presence of a reflecting wall. Far away from the wall, translational invariance implies that the stationary probability distribution is $P(x)\sim \exp(αx)$. A classical example of a problem of this type is sedimentation equilibrium, where $α$ is determined by temperature. In this work we do not introduce a thermal reservoir and $α$ is determined from the equation of motion. We consider a general approach to determining $α$ 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.

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