Chapman-Enskog derivation of the generalized Smoluchowski equation

Abstract

We use the Chapman-Enskog method to derive the Smoluchowski equation from the Kramers equation in a high friction limit. We consider two main extensions of this problem: we take into account a uniform rotation of the background medium and we consider a generalized class of Kramers equations associated with generalized free energy functionals. We mention applications of these results to systems with long-range interactions (self-gravitating systems, 2D vortices, bacterial populations,...). In that case, the Smoluchowski equation is non-local. In the limit of short-range interactions, it reduces to a generalized form of the Cahn-Hilliard equation. These equations are associated with an effective generalized thermodynamical formalism.

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References in corpus (10)

• Non Linear Kinetics underlying Generalized Statistics
• Anomalous diffusion in the presence of external forces: exact time-dependent solutions and entropy
• Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter's great red spot
• Thermodynamics of self-gravitating systems
• Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions
• Gravitational instability of isothermal and polytropic spheres
• Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions
• Generalized thermodynamics and kinetic equations: Boltzmann, Landau, Kramers and Smoluchowski
• Dynamics and Thermodynamics of Systems with Long Range Interactions: an Introduction
• Generalized thermostatistics and mean-field theory

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