Journal of Statistical Physics

, Volume 160, Issue 5, pp 1294–1335 | Cite as

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

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


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.


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



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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Copyright information

© Springer Science+Business Media New York 2015

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

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