Langevin equation with potential of mean force: The case of anchored bath

Abstract: The potential of mean force (PMF) is an effective average potential acting on an open system, renormalized due to the interaction with the surrounding thermal bath. The PMF is defined for an equilibrium ensemble, and generally it is not clear how to use it when the system is out of equilibrium and described by a (generalized) Langevin equation. We study a model where the system is a single particle (so there are no complications related to internal forces) and a non-trivial PMF is due to the presence of on-site (anchor) potentials applied to the bath particles. We found that the PMF does not merely replace the external potential, but also makes the dissipation kernel and statistical properties of noise dependent on the system's position. That dependence is determined by the internal bath and system-bath interactions and is a priori unknown. Therefore, in the general case the Langevin equation with the PMF is not closed and thus inoperable. However, for systems with linear forces the aforementioned dependence on the system's position may be canceled. As an example, we consider a model where the bath is formed by the Klein-Gordon chain, i. e. a harmonic chain with on-site harmonic potentials. In that case, the generalized Langevin equation has the standard form with an external potential replaced by a quadratic PMF.