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Liouvillian theory of magnetic fluctuations

Published online by Cambridge University Press:  13 March 2009

E. Vanden Eijnden  and
R. Balescu
E. Vanden Eijnden
Affiliation:
Association EURATOM—Etat Belge, Service de Physique Statistique, Plasma et Optique Non-Linéaire CP 231, Université Libre de Bruxelles, Campus Plaine, B-1050 Bruxelles, Belgium
R. Balescu
Affiliation:
Association EURATOM—Etat Belge, Service de Physique Statistique, Plasma et Optique Non-Linéaire CP 231, Université Libre de Bruxelles, Campus Plaine, B-1050 Bruxelles, Belgium
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Abstract

The study of the diffusion of magnetic field lines is based on a stochastic model with a Hamiltonian structure. A Liouville equation can then be associated with Hamilton's equations representing the magnetic line. This allows a statistical description of all observable quantities. In particular, the mean-square displacements (MSD) of a field line and the mean-square relative distance (MSRD) of two lines are considered. Equations for these quantities are obtained by applying the direct-interaction approximation (DIA) to the stochastic Liouville equation. The running diffusion coefficient is derived from the MSD equation. The status of the temporal and spatial Markovian approximations in the DIA is discussed. The MSRD's evolution establishes the existence of a clumping effect between two lines, which eventually leads to gyrotropization of the system. A clump life length is defined. In both cases the equations of motion are shown to be identical to the equations obtained within the Langevin treatment of magnetic fluctuations. This shows the equivalence between the DIA and the Corrsin approximation performed in the Langevin approach.

Type
Research Article
Information
Journal of Plasma Physics , Volume 54 , Issue 2 , October 1995 , pp. 185 - 199
Copyright
Copyright © Cambridge University Press 1995

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