Effective temperature and Einstein relation for particles in active matter flows

Active matter are a collection of units with intrinsic supply of energy that is utilized for self-propelled motion. Recent studies have confirmed that these active systems can exist in exotic phases, such as swarming, laning, jamming, and even turbulence, based on the size and density of the constituent units. An interesting question that naturally arises is whether one can identify an effective temperature for particles advected by such an active flow that is far from equilibrium. In this paper, we report using a continuum model of a dense bacterial suspension, an exact expression of the effective temperature for a distribution of interacting particles that are immersed in this suspension. We observe that this effective temperature is linear in particle diffusivity with the slope defining the particle mobility that is higher when the background fluid exhibits global polar ordering and lower when the fluid is in isotropic equilibrium. We believe our paper is a direct verification of the Einstein relation-the simplest fluctuation dissipation relation for interacting particles advected in an active matter flow.