A theory of fluctuations in plasmas

Abstract

A theory of thermal fluctuations in plasmas is developed based on a probability ensemble for one-particle distribution functions ƒ(r, ν). The probability for a specific ƒ(r, ν) is obtained from the canonical ensemble with the aid of the continuum approximation. Subsequently the probability distribution for deviations from the Maxwell-Boltzmann distribution is established. The exponent of this Gaussian distribution is diagonalized by expansion into normal mode solutions of the linearized Vlasov equation. To that purpose van Kampen's normal mode analysis is extended to threedimensional velocity space. Since the development in time of a fluctuation ƒ1(r, ν) follows from the linearized Vlasov equation a general expression can be given for the joint probability distribution of quantities depending linearly on ƒ1(r, ν) and taken at different points of space and time. By way of example joint probability distributions involving density, potential and electric field, current density and pressure are calculated. The influence of an external force on the fluctuations is investigated. Finally the non-equilibrium case is discussed.