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Analytical Solution to Nonlinear Non-Gaussian Langevin Equation

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Statistical Mechanics for Athermal Fluctuation

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Abstract

The non-Gaussian Langevin equation under nonlinear friction is derived from microscopic dynamics by generalizing the expansion in Chap. 7. We also present an analytical solution to non-Gaussian Langevin equation in the presence of nonlinear frictions; a full-order asymptotic formula for the steady PDF is derived for large frictional limit. The first-order approximation of our formula leads to the independent-kick model, and higher-order approximation corresponds to multiple kicks during relaxation. As a demonstration, a granular motor under dry friction is finally addressed.

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Notes

  1. 1.

    For example, in the case with the cubic friction \(f(\mathcal {V})=a\mathcal {V}+b\mathcal {V}^3\), the characteristic velocity scale of the friction function \(f(\mathcal {V})\) is given by \(\mathcal {V}^*\equiv \sqrt{a/b}\).

  2. 2.

    This assumption is valid for the first-order approximation. Modification due to higher order corrections is discussed in Sect. 8.3.6.

  3. 3.

    We note that the definition of \(\beta ^{-1}\) is a little different from that in Refs. [2,3,4], where \(\beta ^{-1}\) is defined by \(\beta ^{-1}\equiv \varepsilon ^{1/2}\rho Sv_0^2/\sqrt{2}\pi \gamma R_{\mathrm {I}}\).

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Authors and Affiliations

  1. Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan

    Kiyoshi Kanazawa

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  1. Kiyoshi Kanazawa
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Correspondence to Kiyoshi Kanazawa .

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Kanazawa, K. (2017). Analytical Solution to Nonlinear Non-Gaussian Langevin Equation. In: Statistical Mechanics for Athermal Fluctuation. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-10-6332-9_8

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  • DOI: https://doi.org/10.1007/978-981-10-6332-9_8

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