Three-body problem for Brownian particles at different temperatures

A mixture of Brownian particles at different temperatures has been a useful model for studying out-of-equilibrium systems of many components with differing levels of activity (e.g. phase separation of mixtures of passive and active particles). This model was previously studied analytically up to the second virial coefficients using pair distributions in the dilute limit. We are interested whether or not the two-particle results can be extended to understand the three-particle distributions. By considering the solvable case of pairwise quadratic interactions, we show that, unlike the two-particle distribution, the three-particle distribution takes on an interesting form that is not in general Boltzmann-like with an effective temperature. We summarize some of our results here and briefly discuss an interesting connection with the Newtonian three-body problem.