Dynamics of a tracer trapped in a correlated medium in the presence of a wall

Abstract

We describe the random motion of a particle immersed in a thermally fluctuating medium and harmonically trapped at a certain distance from a wall. The medium, modeled by a Gaussian field with a tunable correlation length $\xi$, is linearly coupled to the particle and evolves according to dissipative relaxational dynamics. Dirichlet boundary conditions imposed on the field at the wall give rise to a repulsive fluctuation-induced force acting on the particle, causing a shift in its average position and a renormalization of the strength of the harmonic trap. We describe the effective overdamped dynamics of the particle, which features a nonlinear memory term depending on the wall-particle separation. We show that the two-time correlation function of the particle position features a memory-induced term that depends on the distance of the particle from the wall. At the critical point, this term decays algebraically upon increasing time and it displays a crossover from the behavior observed in the bulk to that corresponding to having the particle at the wall.