Expectation values of general observables in the Vlasov formalism

Collisionless plasmas in an arbitrary dynamical state are described by the Vlasov equation, which gives the time evolution of the probability density rho(x, v). In this work we introduce a new analytical procedure to generate particular partial differential equations (PDEs) for an arbitrary macroscopic observable w(x, v) that can be expressed as a function of positions and velocities, without solving the time evolution of the probability itself. This technique, which we will call the "Ehrenfest procedure" (as it produces relations that are analogous to Ehrenfest's theorem in Quantum Mechanics), is based on the iterative application of the fluctuation-dissipation theorem and the recently proposed conjugate variables theorem (CVT) in order to eliminate the explicit dependence on rho. In particular, we show how this formalism is applied to the Vlasov equation for collisionless plasmas, and derive a general evolution equation for the fluctuations of any macroscopic property w in this kind of plasma.