Quasilinear theory for inhomogeneous collisional plasma

The classic formulation of quasilinear theory (QLT) for plasmas interacting with weak turbulence fails to conserve the action of noresonant waves and is inapplicable to inhomogeneous plasmas. We present a reformulation of QLT as a local theory [I. Y. Dodin, J. Plasma Phys. 88, 905880407 (2022)] where these issues are fixed by replacing shortcuts of the classic theory with rigorous calculations. 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. 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 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 [Phys. Fluids 16, 1102 (1973)] is reproduced as a particular case.