Geometric Kinetic Theory And Simulation Method

Physics can be best described using a geometric language. Essentially, simulationof a physical system is to recover its geometric structure. A generalized framework hasbeen developed to translate the geometric kinetic theory for plasmas into a geometrickinetics simulation algorithm. Its validity is confirmed by a series numerical examples.In the geometric kinetic theory, the kinetics of plasma system is expressed usingthe language of differential geometry. The Poincaré-Cartan-Einstein one-formγin theseven-dimensionalphasespacedeterminesaparticle'sworldlineinthephasespace. Thephase space distribution function of particle is a constant of motion along worldlines.The action of the Vlasov-Maxwell system is the sum of the actions of the particles andthe electromagnetic field. Maxwell's equations are obtained from the variation of theaction. The 4-current density is given by the fiber integral of the distribution function.Using the Lie perturbation method, the multiscale space-time structure of dynamics isdecoupled by introduced a gauge function S.The fiber integral of the phase-space distribution function and its pullback can berepresented as a integral over the phase-space with a Dirac-δfunction. By replacing theδfunction with an analytical shape function, the integral can be numerically calculatedusing the Monte Carlo method. From this point of view, interpolation, difference, andvelocity integral are all results of the discretization of the integral representation. Be-causethe evolution ofelectromagnetic field depends only on the 4-current density of thecharged particles, the kinetic simulation algorithm should be based on the discretizationof the velocity integral, instead of the discretization of the distribution function. Alsobased on the geometric structure of electromagnetic field, the potentials should be sam-pled on the mid-points between the space-time grids. The explicit difference schemefor Maxwell's equations is obtain from the Ampere's law in terms of the potentials. Forcold plasmas, an explicit difference scheme for the Ohm's law is constructed.Gyrocenter-Gauge (G-Gauge) kinetics is the implementation of the geometric ki-netic theory in the magnetized plasma. In G-Gauge kinetic theory, the magnetized plasma system is simulated in the gyrocenter coordinate system. The gyrocenter distri-bution function F is sampled on the gyrocenter, parallel velocity, and magnetic momentcoordinates. The G-Gauge function S is sampled on the Kruskal rings and shares thefirstfivecoordinateswithF. Themomentintegralofpullbacktransformationisdirectlycalculated using the Monte Carlo method. The G-Gauge algorithm has been success-fully applied to the simulation studies of high frequency extraordinary wave, electronBernstein wave, and the mode conversion process between the extraordinary wave andthe electron Bernstein wave in inhomogeneous plasmas.In the geometric kinetic theory, the description of kinetic system is complete andmultiscale. The complex dynamics is separated into several simple sub-systems withoutlosing the physics content. Compared with the classical kinetic theory, the geometric ki-netic theory provides us with a more fundamental viewpoint and a more comprehensiveunderstanding of the complicated dynamics of confined plasma systems.