The non-equilibrium solvent response force: What happens if you push a Brownian particle

TL;DR

This paper derives a generalized Langevin equation incorporating external forces for a particle in a many-body system, revealing an additional term and providing insights into non-equilibrium solvent responses.

Contribution

It introduces a method to include external forces in the generalized Langevin equation, identifying an overlooked term and maintaining the same drift, memory kernel, and fluctuations as the unforced system.

Findings

Derived the exact generalized Langevin equation with external forces.

Identified an additional term in the equation previously overlooked.

Validated the approach with an exemplary system.

Abstract

In this letter we discuss how to add forces to the Langevin equation. We derive the exact generalized Langevin equation for the dynamics of one particle subject to an external force embedded in a system of many interacting particles. The external force may depend on time and/or on the phase-space coordinates of the system. We construct a projection operator such that the drift coefficient, the memory kernel, and the fluctuating force of the generalized Langevin equation are the same as for the system without external driving. We show that the external force then enters the generalized Langevin equation additively. In addition we obtain one term which, to our knowledge, has up to now been overlooked. We analyze this additional term for an exemplary system.