Journal of Statistical Physics

, Volume 160, Issue 5, pp 1294–1335

# Asymptotic Derivation of Langevin-like Equation with Non-Gaussian Noise and Its Analytical Solution

• Kiyoshi Kanazawa
• Tomohiko G. Sano
• Takahiro Sagawa
• Hisao Hayakawa
Article

## Abstract

We asymptotically derive a non-linear Langevin-like equation with non-Gaussian white noise for a wide class of stochastic systems associated with multiple stochastic environments, by developing the expansion method in our previous paper (Kanazawa et al. in Phys Rev Lett 114:090601–090606, 2015). We further obtain a full-order asymptotic formula of the steady distribution function in terms of a large friction coefficient for a non-Gaussian Langevin equation with an arbitrary non-linear frictional force. The first-order truncation of our formula leads to the independent-kick model and the higher-order correction terms directly correspond to the multiple-kicks effect during relaxation. We introduce a diagrammatic representation to illustrate the physical meaning of the high-order correction terms. As a demonstration, we apply our formula to a granular motor under Coulombic friction and get good agreement with our numerical simulations.

## Keywords

Stochastic processes Non-Gaussian noise Langevin equation  Non-linear friction Granular motor

## Notes

### Acknowledgments

We are grateful for the useful discussions between N. Nakagawa and A. Puglisi. A part of the numerical calculations was carried out on SR16000 at YITP in Kyoto University. This work was supported by the JSPS Core-to-Core Program “Non-equilibrium dynamics of soft matter and information,” Grants-in-Aid for the Japan Society for Promotion of Science (JSPS) Fellows (Grant Nos. 24$$\cdot$$3751 and 26$$\cdot$$2906), and JSPS KAKENHI Grant Nos. 25287098, and 25800217.

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## Authors and Affiliations

• Kiyoshi Kanazawa
• 1
• Tomohiko G. Sano
• 1
• Takahiro Sagawa
• 2
• Hisao Hayakawa
• 1
1. 1.Yukawa Institute for Theoretical PhysicsKyoto UniversityKyotoJapan
2. 2.Department of Basic ScienceThe University of TokyoMeguro-kuJapan