Non-Thermal Einstein Relations

TL;DR

This paper investigates a generalized approach to determining the exponential decay rate of stationary probability distributions for particles driven by non-thermal random forces, revealing deviations from classical Einstein relations.

Contribution

It introduces a novel method to compute the decay rate lpharom the particles equation of motion, highlighting cases where Einstein's relation does not hold.

Findings

lphaiffers from Einstein's relation in non-thermal systems.

The approach is illustrated with a Boltzmann-inspired model.

Results show deviations from classical sedimentation equilibrium predictions.

Abstract

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 $P(x)\sim \exp(\alpha x)$. 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 $\alpha$ 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.