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We derive a distribution function for the position of a tagged activeparticle in a slowly varying in space external potential, in a system ofinteracting active particles. The tagged particle distribution has the form ofthe Boltzmann distribution but with an effective temperature that replaces thetemperature of the heat bath. We show that the effective temperature thatenters the tagged particle distribution is the same as the effectivetemperature defined through the Einstein relation, i.e. it is equal to theratio of the self-diffusion and tagged particle mobility coefficients. Thisshows that this effective temperature, which is defined through afluctuation-dissipation ratio, is relevant beyond the linear response regime.We verify our theoretical findings through computer simulations. Our theoryfails when an additional large length scale appears in our active system. Thislength scale is associated with long-wavelength density fluctuations thatemerge upon approaching motility-induced phase separation.

Tagged active particle: probability distribution in a slowly varying  external potential is determined by effective temperature obtained from the  Einstein relation