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Generalized Langevin Equation

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Part of the book series: Developments in Mathematics ((DEVM,volume 61))

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Abstract

The Langevin equation is connected to the Brownian motion formulated by Einstein and Smoluchowski. The Langevin equation for a free particle with mass m is given by Langevin (CR Acad Sci Paris 146:530, 1908)

$$\displaystyle \begin{aligned} \begin{array}{rcl} m\dot{v}(t)+\gamma v(t)&\displaystyle =&\displaystyle \xi(t),\\ \dot{x}(t)&\displaystyle =&\displaystyle v(t), \end{array} \end{aligned} $$

where x(t) is the particle displacement, v(t) is its velocity, γ is the friction coefficient and ξ(t) is a Gaussian random noise with zero mean 〈ξ(t)〉 = 0 (so-called white noise).

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Authors and Affiliations

  1. Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Skopje, North Macedonia

    Trifce Sandev

  2. Faculty of Natural Sciences and Mathematics, Institute of Mathematics, Ss. Cyril and Methodius University in Skopje, Skopje, North Macedonia

    Živorad Tomovski

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  1. Trifce Sandev
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Sandev, T., Tomovski, Ž. (2019). Generalized Langevin Equation. In: Fractional Equations and Models. Developments in Mathematics, vol 61. Springer, Cham. https://doi.org/10.1007/978-3-030-29614-8_6

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