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.

中文翻译：

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不同温度下兰格文动力学的三体问题

布朗粒子在不同温度下的混合物已成为研究由具有不同活性水平的微观成分组成的系统的失衡性质的有用模型。先前已对该模型进行了稀释极限内的两粒子相互作用的分析研究，得出了具有有效温度的玻尔兹曼样两粒子分布。像牛顿二体和三体问题一样，我们在这里询问是否可以将两粒子结果扩展到三粒子相互作用以获得三粒子分布。通过考虑成对二次相互作用的特殊可解情况，我们表明，与两粒子分布不同，在有效温度下，三粒子分布通常不能像玻尔兹曼一样。