Abstract
A statistical theory of plasma turbulence in a magnetic field is described. This theory makes use of averaging operators to formally solve the Vlasov equation and the stochastic acceleration problem. Sets of turbulence equations are thereby derived for determining the ensemble average of the oneparticle distribution function and the electric field spectrum. The average distribution function is seen to satisfy a diffusion equation to all orders in the perturbation—even in the presence of a magnetic field. The mentioned equations are used to derive a nonlinear dispersion relation for waves in turbulent plasmas in a magnetic field. This dispersion relation is a nonlinear generalization of Bernstein's linear dispersion relation for electrostatic waves. The nonlinearity manifests itself as a damping factor proportional to mean square deviations from mean particle trajectories. A simplifying feature of the present work is the use of cumulant expansions, which also avoid certain heuristic arguments. A previous result, given by Dupree, is shown to be a special case of the present result.
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