Fokker–Planck Theory of Nonequilibrium Systems Governed by Hierarchical Dynamics

Foundations of Physics 44 (2):175-182 (2014)

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
Keywords Hierarchical dynamics  Fluctuating dynamical temperature   Fokker–Planck equation
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DOI 10.1007/s10701-014-9775-3
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