Nonequilibrium solvent response force: What happens if you push a Brownian particle

Abstract

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. Published by the American Physical Society 2024

Memory effects in colloidal motion under confinement and driving

“…The pronounced non-linear response of the model investigated here appears as an ideal benchmark for this situation, in particular, since detailed analytic predictions are available for the transport properties (sections 4.1 and 4.2). Indeed, these results corroborate the previously predicted [21] cubic increase of the random force bias, ξ f→0 ∼ f 3 (figure 1). Yet, we note that the GLE studied here and the one derived in [21] differ in the details.…”

“…which is negative due to v D (f) ⩽ ζ −1 0 f for all f. Below the critical force f cr , the bias ξ f increases apparently linearly as f is increased (figure 1(b)). However, inspection of the numerical data for weak driving suggests that ξ f→0 ∼ f 3 (inset of figure 1(b)), which is in line with recent theoretical predictions for the coarse-grained dynamics of a Hamiltonian many-particle system, obtained within the Mori projection formalism [21].…”

“…The starting point was the linear GLE for the particle velocity, amended by a constant, external force (equation ( 3)). While it has been clear that the memory function in this approach depends on the external force, it has been recognised only recently [11,21] that, outside of the linear response regime, the random force must be biased (equation ( 7)). The pronounced non-linear response of the model investigated here appears as an ideal benchmark for this situation, in particular, since detailed analytic predictions are available for the transport properties (sections 4.1 and 4.2).…”

“…The key ingredients of equation ( 3) are the memory function ζ(t; f ) and the random force ξ f (t), both of which depend on f. The random force may be biased for strong driving, i.e. in the non-linear response regime, and one expects a non-vanishing mean [11,21]. Therefore, we define…”

“…the VACF or the MSD), whereas a complementary, bottom-up approach to the GLE is rooted in statistical mechanics and employs the projection operator techniques due to Zwanzig, Mori and others [7][8][9], see [10] for a review. A typical application is the projection on a suspended particle and thereby coarse-graining its interactions with a complex solvent [11,12]; recent progress was made on the non-linear GLE [13][14][15][16] and on GLEs under non-equilibrium conditions [17][18][19][20][21][22]. In essence, the GLE may be seen as a means to encapsulate a large number of unresolved degrees of freedom in the memory function.…”