Three-body problem for Langevin dynamics with different temperatures

TL;DR

This paper investigates the three-body distribution in a system of Langevin particles at different temperatures, revealing that unlike two-particle cases, the three-particle distribution cannot be simplified to a Boltzmann-like form and depends on all three particles.

Contribution

The study extends the analysis of non-equilibrium particle systems to three-body interactions, showing the limitations of effective temperature descriptions in triplet distributions.

Findings

Three-particle distributions are not Boltzmann-like in general.

Triplet distributions depend on all three particles and their interactions.

Two-particle effective temperature models do not extend straightforwardly to three particles.

Abstract

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.