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

TL;DR

This paper derives a distribution function for a tagged active particle in a slowly varying external potential, showing it is governed by an effective temperature consistent with the Einstein relation, validated by simulations.

Contribution

It introduces a theoretical framework linking the tagged particle distribution to an effective temperature defined via fluctuation-dissipation, applicable beyond linear response.

Findings

Tagged particle distribution follows a Boltzmann form with an effective temperature.

Effective temperature equals the ratio of self-diffusion to mobility coefficients.

Theory breaks down near motility-induced phase separation due to large-scale fluctuations.

Abstract

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