Nonequilibrium Statistical Mechanics of the Heat Bath for Two Brownian Particles

We propose a new look at the heat bath for two Brownian particles, in which the heat bath as a system is both perturbed and sensed by the Brownian particles. Nonlocal thermal fluctuations give rise to bath-mediated static forces between the particles. Based on the general sum rule of the linear response theory, we derive an explicit relation linking these forces to the friction kernel describing the particles' dynamics. The relation is analytically confirmed in the case of two solvable models and could be experimentally challenged. Our results point out that the inclusion of the environment as a part of the whole system is important for micron- or nanoscale physics.