Hamiltonian formulations of quasilinear theory for magnetized plasmas

Hamiltonian formulations of quasilinear theory are presented for the cases uniform and nonuniform magnetized plasmas. First, standard Kennel Engelmann (Kennel, Phys. Fluids, 1966, 9, 2377) is reviewed reinterpreted in terms a general formulation. Within this representation, we present transition from two-dimensional diffusion spatially background plasma to three-dimensional based on our previous work (Brizard Chan, Plasmas, 2001, 8, 4762–4771; Brizard 2004, 11, 4220–4229). The resulting plasmas yields 3 × tensor that naturally incorporates radial as well its synergistic connections invariant velocity space (e.g., energy pitch angle).