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

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