Kinetic theory with angle–action variables

Pierre-Henri Chavanis

Physica A: Statistical Mechanics and its Applications, 2007, vol. 377, issue 2, 469-486

Abstract: We present a kinetic theory for one-dimensional inhomogeneous systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory valid at order 1/N in a proper thermodynamic limit N→+∞, we obtain a closed kinetic equation describing the relaxation of the distribution function of the system as a whole due to resonances between different orbits. This equation is written in angle–action variables. It conserves mass and energy and increases the Boltzmann entropy (H-theorem). Using a thermal bath approximation, we derive a Fokker–Planck equation describing the relaxation of a test particle towards the Boltzmann distribution under the combined effects of diffusion and friction. We mention some analogies with the kinetic theory of point vortices in two-dimensional hydrodynamics. We also stress the limitations of our approach and the connection with recent works.

Keywords: Kinetic theory; Long-range interactions; Fokker–Planck equations (search for similar items in EconPapers)
Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:377:y:2007:i:2:p:469-486

DOI: 10.1016/j.physa.2006.11.078

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