Collisional gyrokinetics teases the existence of metriplectic reduction

In purely non-dissipative systems, Lagrangian and Hamiltonian reduction have been proven to be powerful tools for deriving physical models with exact conservation laws. We have discovered a hint that an analogous reduction method exists also for dissipative systems that respect the first and second laws of thermodynamics. In this paper, we show that modern electrostatic gyrokinetics, a reduced plasma turbulence model, exhibits a serendipitous metriplectic structure. Metriplectic dynamics, in general, is a well developed formalism for extending the concept of Poisson brackets to dissipative systems. Better yet, our discovery enables an intuitive particle-in-cell discretization of the collision operator that also satisfies the first and second laws of thermodynamics. These results suggest that collisional gyrokinetics, and other dissipative physical models that obey the laws of thermodynamics, could be obtained using an as-yet undiscovered metriplectic reduction theory and that numerical methods could benefit from such theory significantly. Once uncovered, the theory would generalize Lagrangian and Hamiltonian reduction in a substantial manner.