Vlasov equation in the stochastic magnetic field

We investigate the Vlasov equation in the stochastic magnetic field as a stochastic Li ouville equation and derive the equation for the ensemble-averaged distribution function. The term resulting from the stochastic magnetic field has the derivatives with respect to both the velocity and the real space coordinates, which is a contrast to both the real space diffusion as seen in the guiding center picture and the velocity space diffusion as in the quasi-linear theory of the Vlasov equation including the electric field fluctuations. We find that this term retains the mass and energy conservation properties of the original Lorentz force due to the stochastic magnetic field and yields the additional force in the momentum equation. This additional force produced by the stochastic field gives the drift velocity which corresponds to the familiar real space diffusion of the guiding center in the stochastic field. The finite Larmor radius effect on the diffusion is also estimated. (author)