We propose a new look at the heat bath for two Brownian particles, in whichthe heat bath as a `system' is both perturbed and sensed by the Brownianparticles. Non-local thermal fluctuation give rise to bath-mediated staticforces between the particles. Based on the general sum-rule of the linearresponse theory, we derive an explicit relation linking these forces to thefriction kernel describing the particles' dynamics. The relation isanalytically confirmed in the case of two solvable models and could beexperimentally challenged. Our results point out that the inclusion of theenvironment as a part of the whole system is important for micron- ornano-scale physics.
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