Journal of Statistical Physics

, Volume 174, Issue 3, pp 656–691

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

• Soon Hoe Lim
• Jan Wehr
Article

## 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 Hottovy et al. (Commun Math Phys 336(3):1259–1283, 2015), 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.

## Keywords

Generalized Langevin equation Small mass limit Multiscale analysis Noise-induced drift Thermophoresis

## Notes

### Acknowledgements

The authors were partially supported by NSF grant DMS-1615045. S. Lim is grateful for the support provided by the Michael Tabor Fellowship from the Program in Applied Mathematics at the University of Arizona during the academic year 2017-2018. The authors learned the method of introducing additional variables to eliminate the memory term from E. Vanden-Eijnden. They would like to thank Maciej Lewenstein for insightful discussion on the GLEs and one of the referees for the constructive comments.

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## Authors and Affiliations

• Soon Hoe Lim
• 1
• 2
• Jan Wehr
• 3
• 4
1. 1.Program in Applied MathematicsUniversity of ArizonaTucsonUSA
2. 2.Nordita, KTH Royal Institute of Technology and Stockholm UniversityStockholmSweden
3. 3.Department of Mathematics and Program in Applied MathematicsUniversity of ArizonaTucsonUSA
4. 4.Department of MathematicsUniversity of ArizonaTucsonUSA