On the Einstein relation between mobility and diffusion coefficient in an active bath

TL;DR

This paper investigates the validity of the Einstein relation in an active bath, showing it approximately holds when using a mechanically defined temperature related to pressure.

Contribution

It demonstrates that the Einstein relation can be approximately restored in an active bath by defining a mechanical temperature via pressure measurements.

Findings

Einstein relation approximately holds in active baths with a mechanical temperature.

Correlation functions can relate mobility and diffusion in nonequilibrium conditions.

Mechanical temperature differs from thermodynamic temperature in active systems.

Abstract

An active bath, made of self-propelling units, is a nonequilibrium medium in which the Einstein relation $D=\mu k_B T$ between the mobility $\mu$ and the diffusivity $D$ of a tracer particle cannot be expected to hold a priori. We consider here heavy tracers for which these coefficients can be related to correlation functions which we estimate. We show that, to a good approximation, an Einstein relation does hold in an active bath upon using a different temperature which is defined mechanically, through the pressure exerted on the tracer.