In this Letter we discuss how to add forces to the Langevin equation. We derive an 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 next to the external force another force term occurs that is caused by the nonequilibrium response of the solvent. The first contribution to the ensemble average of this force stems from third or higher order terms of the external force and from sixth or higher order terms of time. We also analyze the additional force term numerically for an exemplary system.

• Received 26 September 2023
• Revised 30 November 2023
• Accepted 24 January 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.L012032

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Published by the American Physical Society