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

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.