New Approach to the Theory of Fluctuations in a Plasma

A rigorous but simple method is presented for calculating autocorrelation functions (fluctuations) of nonequilibrium plasmas—including inhomogeneous and nonstationary plasmas—in external fields. This method is based upon the derivation of an exact and remarkably simple formal relation between autocorrelation functions and the usual one-particle distribution function $f_1(\mathrm{R}, \mathrm{v}, t)$ for an explicitly defined initial value. This relation explicitly reduces the problem of calculating fluctuation spectra to the problem of solving the usual kinetic equations for $f_1(\mathrm{R}, \mathrm{v}, t)$. Consequently, the central quantity of fluctuation theory is one and the same with the central quantity of kinetic theory, and the two theories are completely and explicitly united. To first order in the plasma parameter it is shown that one need only solve the linearized Vlasov equation for $f_1(\mathrm{R}, \mathrm{v}, t)$, and when this solution is substituted into our formal relation we obtain a general formula for autocorrelation functions which is valid for nonstationary systems and includes the effects of the transverse motion of the plasma in addition to the longitudinal motion. For stationary systems this formula approaches the numerous calculations of previous authors in the limits where the transverse terms vanish. As a result of the transverse terms, the spectrum of scattered light can have resonant peaks at frequencies which correspond to transverse modes of oscillation as well as to the well-known longitudinal modes.