Dynamics of a colloidal particle coupled to a Gaussian field: from a confinement-dependent to a non-linear memory

Abstract

The effective dynamics of a colloidal particle immersed in a complex medium is often described in terms of an overdamped linear Langevin equation for its velocity with a memory kernel which determines the effective (time-dependent) friction and the correlations of fluctuations. Recently, it has been shown in experiments and numerical simulations that this memory may depend on the possible optical confinement the particle is subject to, suggesting that this description does not capture faithfully the actual dynamics of the colloid, even at equilibrium. Here, we propose a different approach in which we model the medium as a Gaussian field linearly coupled to the colloid. The resulting effective evolution equation of the colloidal particle features a non-linear memory term which extends previous models and which explains qualitatively the experimental and numerical evidence in the presence of confinement. This non-linear term is related to the correlations of the effective noise via a novel fluctuation-dissipation relation which we derive.

Memory-induced oscillations of a driven particle in a dissipative correlated medium

“…The presence of thermal noise represents an obstacle to the analytical derivation of the time-dependent relaxation of the particle, because it modifies the steady-state average of both the particle position Z ss in equation (20), and the field profile φ q ss in equation (21). Once incorporated into the effective equation of motion of the particle, the field-induced fluctuations turn out to be non-Gaussian, as we will verify shortly; in order to account for them, we shall resort below to a perturbative expansion in the coupling constant λ (as previously done in related investigations of this model [22][23][24][25]44]). We emphasize that the (noiseless) effective equation ( 25) is actually non-perturbative in λ, and so is its solution in equation (34).…”

“…where ξ = r −1/2 ⩾ 0 is the correlation length, which controls the spatial range of the field correlations at equilibrium, and diverges upon approaching the critical point r = 0. The interaction between the particle and the field is chosen as [23][24][25]44]…”

“…Due to the coupling between the field φ q and the particle coordinate Z, it is difficult in general to evaluate the terms φ q e i q•Z ss and e −i q•Z ss which appear in equations ( 20) and (21). For example, in [23][24][25] this has been achieved by a perturbative expansion in increasing powers of the weak coupling λ. In particular, the properties of the nonequilibrium stationary state of the system investigated here and predicted with that approach are analyzed quantitatively in [44], also based on numerical simulations.…”

“…In the present work we are primarily interested in exploring the effect of the field when the coupling λ is relatively strong, and thus we will adopt a different approach compared to that used in [23][24][25]44]. In particular, we will focus first on the noiseless limit of the dynamics, i.e.…”

“…Memory terms in the effective evolution equation of a particle actually appear quite naturally in many physical systems, after integrating the slow degrees of freedom out of the original, microscopic dynamics in which they are coupled to those describing the tracer particle [13][14][15]. For example, a minimal model for diffusion in a thermally fluctuating correlated medium can be formulated in terms of the joint overdamped dynamics of a particle and of a scalar Gaussian field ϕ(x, t), the latter being characterized by a correlation length ξ and a finite relaxation time [16][17][18][19][20][21][22][23][24][25][26]. If the coupling between the field and the particle is chosen to be linear, then the field can be integrated out exactly, resulting into an effective evolution equation for the particle.…”