Thermodynamics of Vlasov equilibria

Abstract

This paper contains a systematic exposition of a statistical method leading to a characterization of relevant equilibrium and stability properties of col-lisionless Vlasov (collective) plasma configurations according to a formalism similar to that of the classical thermodynamics of Maxwellian systems. We reconsider a statistical model, proposed in earlier works, in which the basic objects of the statistics are volume elements in a configuration space of the charge or current density. The probability distribution in this space is calculated subject to a constraint expressing the existence of static equilibria involving only the smeared-out or collective part of the above densities, while the collective energy is uncorrelated with the fluctuations arising from the single-particle structure. It is one of the aims of this paper to show that the thermodynamic quantities arising automatically in the formalism, for instance the entropy, can be consistently inserted in the physical and conceptual context of classical thermodynamics. This is achieved by studying in detail a reversible energy interaction between the collective system and the external world, in order to identify the entropy variations calculated with the model with those of the entropy as conventionally defined. Our thermodynamic concepts are illustrated by applications to electrostatic Vlasov equilibria (in unstable situations and in the Maxwellian limit) and to magnetic systems, both in a case open to energy interaction with the external world (the tokamak) and in the case of an isolated system (a plasma enclosed in a perfectly conducting shell).