  A mixture of Brownian particles at different temperatures has been a useful
model for studying the out-of-equilibrium properties of systems made up of
microscopic components with differing levels of activity. This model was
previously studied analytically for two-particle interactions in the dilute
limit, yielding a Boltzmann-like two-particle distribution with an effective
temperature. Like the Newtonian two and three-body problems, we ask here
whether the two-particle results can be extended to three-particle interactions
to get the three-particle distributions. By considering the special solvable
case of pairwise quadratic interactions, we show that, unlike the two-particle
distribution, the three-particle distribution cannot in general be
Boltzmann-like with an effective temperature. We instead find that the steady
state distribution of any two particles in a triplet depends on the properties
of and interactions with the third particle, leading to some unexpected
behaviors not present in equilibrium.
