Hamiltonian Formulations of Quasilinear Theory for Magnetized Plasmas

Hamiltonian formulations of quasilinear theory are presented for the cases of uniform and nonuniform magnetized plasmas. First, the standard quasilinear theory of Kennel and Engelmann (1966) is reviewed and reinterpreted in terms of a general Hamiltonian formulation. Within this Hamiltonian representation, we present the transition from two-dimensional quasilinear diffusion in a spatially uniform magnetized background plasma to three-dimensional quasilinear diffusion in a spatially nonuniform magnetized background plasma based on our previous work Brizard_Chan (2001,2004). The resulting quasilinear theory for nonuniform magnetized plasmas yields a $3\times 3$ diffusion tensor that naturally incorporates quasilinear radial diffusion as well as its synergistic connections to diffusion in two-dimensional invariant velocity space (e.g., energy and pitch angle).