Statistical thermodynamics of collisionless plasma. I. The thermodynamical potentials

It is shown that the equilibria and the stability of non-Maxwellian inhomogeneous plasma can be described by a theory with a thermodynamical structure, provided that, in the linearly unstable cases, the perturbation modes under consideration are not too far from marginal stability. On the one hand, variational functionals prove to exist which can characterize the equilibrium and its stability by means of their extremum properties; on the other hand, a statistical model is proposed from which the same functionals can be derived in an independent way. In this model the plasma is considered as a statistical assembly of identifiable volume elements Delta V which are functions of relevant random variables (called information variables) that describe the plasma configuration (e.g. the charge or the current density). Two independent constraints are introduced in the statistical system in such a way that it is characterized by the given values of two quantities, viz. the mean square deviation of the information variable which is essentially connected with individual fluctuations inside each Delta V (namely fluctuations of non-interacting particles), and a collective energy related to the inhomogeneous equilibrium defined by means of the Vlasov model.