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.

Nonequilibrium relaxation of a trapped particle in a near-critical Gaussian field

“…( 18), and similarly to what was done in Refs. [36,37]. Plugging these expansions into the equations of motion of the particles, Eqs.…”

“…One way to proceed (which has been successfully pursued in Refs. [36,37] in the case of a single particle) is to formally expand the field and the particle coordinates as…”

“…We can then use the cumulant generating function in Eq. (36) to compute the expectation value of the position and the variance of the particle Y, which read…”

“…( 35)). In fact, the specific choice of the interaction potential is in general largely irrelevant [36,37] and what really matters is its characteristic lengthscale R, which sets a UV cutoff on the field fluctuations (see also Appendix F).…”

“…A series of works [23][24][25][26] focused on the dynamics of an unconfined particle stochastically diffusing in contact with a scalar Gaussian field, studying the resulting effective diffusion constant. Two recent works [36,37], instead, considered a harmonically trapped particle immersed in a field, and explored how its dynamics is affected by the presence of the latter. In particular, they focused on the average particle position during its relaxation to equilibrium, and on the autocorrelation function of the particle as it diffuses in the trap, both of which can be determined within the weak-coupling approximation.…”