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Fokker–Planck Theory of Nonequilibrium Systems Governed by Hierarchical Dynamics

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Abstract

Dynamics of complex systems is often hierarchically organized on different time scales. To understand the physics of such hierarchy, here Brownian motion of a particle moving through a fluctuating medium with slowly varying temperature is studied as an analytically tractable example, and a kinetic theory is formulated for describing the states of the particle. What is peculiar here is that the (inverse) temperature is treated as a dynamical variable. Dynamical hierarchy is introduced in conformity with the adiabatic scheme. Then, a new analytical method is developed to show how the Fokker–Planck equation admits as a stationary solution the Maxwellian distribution modulated by the temperature fluctuations, the distribution of which turns out to be determined by the drift term. A careful comment is also made on so-called superstatistics.

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Acknowledgments

The author would like to thank C. Beck and E.G.D. Cohen for discussions and comments. He also thanks Péter Ván for drawing his attention to the work in Ref. [9], where the Born–Oppenheimer-like approximation is discussed for a Fokker–Planck equation in the context of superstatistics. This work has been supported in part by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science.

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  1. Department of Physical Engineering, Mie University, Tsu, Mie, 514-8507, Japan

    Sumiyoshi Abe

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  1. Sumiyoshi Abe
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Correspondence to Sumiyoshi Abe.

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Abe, S. Fokker–Planck Theory of Nonequilibrium Systems Governed by Hierarchical Dynamics. Found Phys 44, 175–182 (2014). https://doi.org/10.1007/s10701-014-9775-3

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  • DOI: https://doi.org/10.1007/s10701-014-9775-3

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