Statistical Frameworks for Weak Plasma Turbulence

A hierarchy is constructed describing the time evolution of correlations due to collective interactions in a weakly turbulent, spatially uniform ensemble of Vlasov plasmas. It is shown that this ``Vlasov hierarchy'' is identical to the Bogoliubov—Born—Green—Kirkwood—Yvon (BBGKY) hierarchy for irreducible correlations in a spatially homogeneous plasma, provided the effects of single particle encounters are deleted from the BBGKY framework. In addition, the random phase approximation used in conventional treatments of weak plasma turbulence is shown to be automatically embodied by such a formalism. Upon ordering the system of equations in the weak turbulence parameter, 〈E2〉/nmvav2, the questions of closure and time validity are examined.