Time asymptotic statistics of the Vlasov equation

A statistical description of the Vlasov equation is made possible by truncation of phase space in the velocity co-ordinates and writing the equation in terms of Fourier components in configuration and velocity space. Invariants of the resulting nonlinear turbulence equations are discussed. Expectation values and in particular an electric field spectrum of the form (β+ ακ²)^-1 are derived. α and β are constants; α is always positive; β may be negative, depending on the initial conditions of the plasma. The spectrum is in reasonable agreement with available experiments and simulations.