Energy-density spectrum of the electric field in a turbulent plasma

The Liouville equation describing the phase-space density in the multidimensional space formed by the Fourier amplitudes of the canonical-conjugate fluctuation fields in a conservative electron plasma is applied to the turbulence problem. The distributive function of the longitudinal velocity and electric field fluctuations in fully developed stationary turbulence is derived as that solution of the Liouville equation which maximizes the entropy of turbulent randomness. The spectral energy density of the electric field which represents a second-order moment of the phase-space density is evaluated by analytical integration for the general case of three-dimensional turbulence. The theory is applicable in the small- and medium-wave-number regions but not in the large-wave-number region where dissipation by Landau damping or collisional interaction is significant.