Bayesian derivation of plasma equilibrium distribution function for tokamak scenarios and the associated Landau collision operator

Abstract: A class of parametric distribution functions has been proposed in [C.DiTroia, Plasma Physics and Controlled Fusion,54,2012] as equilibrium distribution functions (EDFs) for charged particles in fusion plasmas, representing supra-thermal particles in anisotropic equilibria for Neutral Beam Injection, Ion Cyclotron Heating scenarios. Moreover, the EDFs can also represent nearly isotropic equilibria for Slowing-Down $alpha$ particles and core thermal plasma populations. These EDFs depend on constants of motion (COMs). Assuming an axisymmetric system with no equilibrium electric field, the EDF depends on the toroidal canonical momentum $P_\phi$, the kinetic energy $w$ and the magnetic moment \mu. In the present work, the EDFs are obtained from first principles and general hypothesis. The derivation is probabilistic and makes use of the Bayes' Theorem. The bayesian argument allows us to describe how far from the prior probability distribution function (pdf), e.g. Maxwellian, the plasma is, based on the information obtained from magnetic moment and GC velocity pdf. Once the general functional form of the EDF has been settled, it is shown how to associate a Landau collision operator and a Fokker-Planck equation that ensures the system relaxation towards the proposed EDF.