Significance of the effective temperature obtained from the Einstein relation in a system of interacting active Ornstein-Uhlenbeck particles

Systems consisting of athermal active Ornstein-Uhlenbeck particles (AOUPs) are inherently non-equilibrium and generally cannot be characterized by a unique temperature-like variable. Different formulas that are equal to the temperature for equilibrium systems may give different values for the temperature for systems of AOUPs. Here we show that the effective temperature obtained from the Einstein relation between the self-diffusion and mobility coefficients determines the spatial distribution of a tagged particle under the influence of a slowly varying in space external potential. We analytically show that the tagged particle distribution has the standard Boltzmann form with the temperature equal to the effective temperature obtained from the Einstein relation. We verify this prediction using computer simulations. Thus, we show that this effective temperature, defined through a fluctuation-dissipation relation, is valid outside the linear response regime for a slowly varying external potential.