Tracer particle in a confined correlated medium: an adiabatic elimination method

Abstract

We present a simple and systematic procedure to determine the effective dynamics of a Brownian particle coupled to a rapidly fluctuating correlated medium, modeled as a scalar Gaussian field, under spatial confinement. The method allows us, in particular, to address the case in which the fluctuations of the medium are suppressed in the vicinity of the particle, as described by a quadratic coupling in the underlying Hamiltonian. As a consequence of the confinement of the correlated medium, the resulting effective Fokker–Planck equation features spatially dependent drift and diffusion coefficients. We apply our method to simplified fluid models of binary mixtures and microemulsions near criticality containing a colloidal particle, and we analyze the corrections to the stationary distribution of the particle position and the diffusion coefficient.

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

“…also off-criticality, due to the presence of the conservation law [42]. These long-wavelength modes are always present in the bulk, while they are cut-off in a confined geometry such as that considered in [22,26].…”

“…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.…”

Stochastic thermodynamics of a probe in a fluctuating correlated field

“…The case a = 1 of conserved dynamics involves the inverse Laplace operator (∇ 2 ) −1 , which is non-local in space, and has thus to be interpreted in terms of its Green's function [64]. In confined geometries, the latter depends on the boundary conditions [65,66], but here we will only consider the system in the bulk.…”