Information theory and variational properties of a collisionless inhomogeneous plasma with a magnetic field

Abstract

The equations describing the electric and magnetic fields of a collisionless one-dimensional plasma at stationary equilibrium in a box (with B parallel to the walls and E perpendicular to B ) can be derived by assuming that their solutions give an extremum value to a certain functional which describes the information necessary for completely fixing the electric and magnetic properties of the physical system for a fixed value of the electric and magnetic energy. According to information theory this functional can be assumed as a definition of an entropy which describes the irreversible behaviour of the collisionless plasma, also when situations very far from the statistical equilibrium are considered. Moreover, it can be shown, using this definition of the entropy, that a functional exists which characterizes the plasma equilibrium with the same formal properties as the Helmholtz function. When the absolute value of all field components has a minimum in the box, the entropy is maximum at equilibrium with respect to any perturbation, for a given electric and magnetic energy. In the special case when only an electric field exists, and for not too inhomogeneous plasmas, it is possible to establish in general, using Penrose's stability criterion, that a minimum value for the entropy in an equilibrium configuration (with fixed electrostatic energy) constitutes a sufficient condition for instability. Examples with magnetic field as the plane sheet pinch or the inhomogeneous plasma in uniform gravitational field with B = e z B 0(1 + ϵχ) are also considered. In these examples the instability of the equilibrium is associated with a minimum of the corresponding entropy. It is also shown that the equilibrium of a neutral polytropic isothermal gas sphere confined by its own central gravitational field is associated with a maximum of the here defined entropy.