  The Brownian motion of a single particle is a paradigmatic model of the
nonequilibrium dynamics of dissipative systems. In the system-plus-reservoir
approach, one can derive the particle's equations of motion from the reversible
dynamics of the system coupled to a bath of oscillators representing its
thermal environment. However, extending the system-plus-reservoir approach to
multiple particles in a collective environment is not straightforward, and
conflicting models have been proposed to that end. Here, we set out to
reconcile some aspects of the nonlinear and the bilinear models of two Brownian
particles. We show how the nonlinear dissipation originally derived from
exponential system-reservoir couplings can alternatively be obtained from the
bilinear Lagrangian, with a modified spectral function that explicitly depends
on the distance between the particles. As applications, we discuss how to avoid
the anomalous diffusion from the standard nonlinear model, as well as how to
phenomenologically model a hydrodynamic interaction between a pair of Brownian
particles in a viscous fluid.

Statistical Mechanics;Quantum Physics