Quasilinear theory for inhomogeneous plasma

This paper presents quasilinear theory (QLT) for classical plasma interacting with inhomogeneous turbulence. The particle Hamiltonian is kept general; for example, relativistic, electromagnetic, and gravitational effects are subsumed. A Fokker--Planck equation for the dressed 'oscillation-center' distribution is derived from the Klimontovich equation and captures quasilinear diffusion, interaction with the background fields, and ponderomotive effects simultaneously. The local diffusion coefficient is manifestly positive-semidefinite. Waves are allowed to be off-shell (i.e. not constrained by a dispersion relation), and a collision integral of the Balescu--Lenard type emerges in a form that is not restricted to any particular Hamiltonian. This operator conserves particles, momentum, and energy, and it also satisfies the H-theorem, as usual. As a spin-off, a general expression for the spectrum of microscopic fluctuations is derived. For on-shell waves, which satisfy a quasilinear wave-kinetic equation, the theory conserves the momentum and energy of the wave--plasma system. The action of nonresonant waves is also conserved, unlike in the standard version of QLT. Dewar's oscillation-center QLT of electrostatic turbulence (1973, Phys. Fluids 16, 1102) is proven formally as a particular case and given a concise formulation. Also discussed as examples are relativistic electromagnetic and gravitational interactions, and QLT for gravitational waves is proposed.