Stochastic advection and landau damping in a collisionless plasma

We consider a simple model of passive chaotic advection of a kinetic field in a collisionless plasma. We test the fluctuation dissipation theorem for a kinetic field, without stochastic advection as a first result. We then present results discussing the effect of stochastic advection on the evolution of the kinetic field-- specifically, the efficiency of linear Landau damping in this simple model. We observe that the free energy is scattered through phase space, allowing a possibility of return of energy from small velocity scales to large. This is interpreted as a manifestation of the ``plasma echo'' phenomenon in a 3-d nonlinear stochastic system.