Homogenization for a Class of Generalized Langevin Equations with an Application to Thermophoresis

Abstract

We study a class of systems whose dynamics are described by generalized Langevin equations with state-dependent coefficients. We find that in the limit, in which all the characteristic time scales vanish at the same rate, the position variable of the system converges to a homogenized process, described by an equation containing additional drift terms induced by the noise. The convergence results are obtained using the main result in \cite{hottovy2015smoluchowski}, whose version is proven here under a weaker spectral assumption on the damping matrix. We apply our results to study thermophoresis of a Brownian particle in a non-equilibrium heat bath.

The contents and results of the paper have been revised and corrected from a previous version

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• Ergodic properties of quasi-Markovian generalized Langevin equations with configuration dependent noise and non-conservative force