Statistical Mechanics of Turbulent Plasmas. I

A statistical mechanical theory for the investigation of the effects of nonlinear fluctuations in a plasma is developed. The starting equation is the Klimontovich equation for the “fine-grained” single particle distribution function and the quantities of interest are the space-time correlation functions, which are certain ensemble averages of the products of these fine-grained distribution function. In this paper, the ensemble average is specified in essentially the same way as in Prigogine's theory of non-equilibrium statistical mechanics for a spatially homogeneous system. As in the usual theory of turbulent plasmas, the nonlinear terms in the Klimontovich equation are treated as a perturbation. Various types of secular terms are found in the perturbational solution. Of these, only those which contribute secularly to the physical quantities are discussed in detail. A diagram technique is introduced to classify various terms in the perturbation series. It is then found that the secular terms of importance can be attributed to a certain class of diagrams (the diagonal part) and that the latter can be summed up so as to yield a well-defined time-dependence of the physical quantities. The total contribution of the diagrams is then expanded in an infinite series in terms of the contributions of the diagonal parts and appropriately defined vertex parts. The general results obtained are illustrated using the quasi-linear approximation. Although the main discussion is restricted to an electron plasma in zero external field, it is shown that little modification is needed in order to generalize the results to an electron-ion plasma and to a plasma in the presence of a uniform external magnetic field.