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

TL;DR

This paper investigates how the potential of mean force (PMF) influences the Langevin equation in systems with anchored baths, revealing conditions under which the equation remains solvable and illustrating this with a harmonic chain model.

Contribution

It demonstrates that the PMF affects dissipation and noise in Langevin equations, and identifies scenarios where the equation simplifies to a standard form with a quadratic PMF.

Findings

PMF modifies dissipation kernel and noise statistics depending on system position.

For linear forces, the position dependence introduced by PMF cancels out.

In a harmonic chain model, the Langevin equation reduces to a standard form with a quadratic PMF.

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 determines the correction to the equilibrium Gibbs distribution, but it is generally unclear how to implement the concept for time-dependent phenomena 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 is 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.