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

The Einstein relation relates the diffusivity and the mobility of a tracer particle immersed in an equilibrium fluid to the bath’s temperature. It is one the major predictions of near-equilibrium linear response theory. Active baths, made of self-propelling units, are driven far from equilibrium and as such we would not expect a similar relation. We show that for large, heavy tracers the diffusivity and mobility can be related to correlation functions, just like for an equilibrium fluid. Moreover, to good approximation, an Einstein relation does hold in an active bath upon using a different temperature defined mechanically through the pressure exerted on the tracer.