Plasma is the fourth state of matter, where from general relativity and quantum theory it is known that all of them are fakes resp. interim specific mathematical model items. An adequate model needs to take into account the axiom of (quantum) state (physical states are described by vectors of a separable Hilbert space H) and the axiom of observables (each physical observable A is represented as a linear Hermitian operator of the state Hilbert space). The corresponding mathematical model and its solutions are governed by the Heisenberg uncertainty inequality. As the observable space needs to support statistical analysis the Hilbert space, this Hilbert space needs to be at least a subspace of H. At the same point in time, if plasma is considered as sufficiently collisional, then it can be well-described by fluid-mechanical equations. There is a hierarchy of such hydrodynamic models, where the magnetic field lines (or magneto-vortex lines) at the limit of infinite conductivity is “frozen-in” to the plasma. The “mother of all hydrodynamic models is the continuity equation treating observations with macroscopic character, where fluids and gases are considered as continua. The corresponding infinitesimal volume “element” is a volume, which is small compared to the considered overall (volume) space, and large compared to the distances of the molecules. The displacement of such a volume (a fluid particle) then is a not a displacement of a molecule, but the whole volume element containing multiple molecules, whereby in hydrodynamics this fluid is interpreted as a mathematical point.

In fluid description of plasmas (MHD) one does not consider velocity distributions. It is about number density, flow velocity and pressure. This is about moment or fluid equations (as NSE and Boltzmann/Landau equations). The corresponding situation of the fluid flux of an incompressible viscous fluid leads to the Navier-Stokes equations. They are derived from continuum theory of non-polar fluids with three kinds of balance laws: (1) conservation of mass, (2) balance of linear momentum, (3) balance of angular momentum.

The Boltzmann equation is a (non-linear) integro-differential equation which forms the basis for the kinetic theory of gases. This not only covers classical gases, but also electron /neutron /photon transport in solids & plasmas / in nuclear reactors / in super-fluids and radiative transfer in planetary and stellar atmospheres. The Boltzmann equation is derived from the Liouville equation for a gas of rigid spheres, without the assumption of “molecular chaos”; the basic properties of the Boltzmann equation are then expounded and the idea of model equations introduced. Related equations are e.g. the Boltzmann equations for polyatomic gases, mixtures, neutrons, radiative transfer as well as the Fokker-Planck (or Landau) and Vlasov equations. The treatment of corresponding boundary conditions leads to the discussion of the phenomena of gas-surface interactions and the related role played by proof of the Boltzmann H-theorem.

The Landau equation (a model describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction) is about the Boltzmann equation with a corresponding Boltzmann collision operator where almost all collisions are grazing. The mathematical tool set is about Fourier multiplier representations with Oseen kernels (LiP), Laplace and Fourier analysis techniques (e.g. [LeN]) and scattering problem analysis techniques based on Garding type (energy norm) inequalities (like the Korn inequality). Its solutions enjoy a rather striking compactness property, which is main result of P. Lions ((LiP) (LiP1)).

The Landau damping (physical, observed) phenomenon is about “wave damping w/o energy dissipation by collisions in plasma”, because electrons are faster or slower than the wave and a Maxwellian distribution has a higher number of slower than faster electrons as the wave. As a consequence, there are more particles taking energy from the wave than vice versa, while the wave is damped. The (kinetic) Vlasov equation is collisions-less.

The common distributional Hilbert space framework is also proposed for a proof of the Landau damping alternatively to the approach from C. Villani. Our approach basically replaces an analysis of the classical (strong) partial differential (Vlasov) equation (PDE) in a corresponding Banach space framework by a quantum field theory adequate (weak) variational representation of the concerned PDE system. This goes along with a corresponding replacement of the “hybrid” and “gliding” analytical norms (taking into account the transfer of regularity to small velocity scales) by problem adequate Hilbert space norms H(-1/2) resp. H(1/2). The latter ones enable a "fermions quantum state" Hilbert space H(0), which is dense in H(-1/2) with respect to the H(-1/2) norm, and its related (orthogonal) "bosons quantum state" Hilbert space H(-1/2)-H(0), which is a closed subspace of H(-1/2).

    

Braun K., A distributional Hilbert space framework to prove the Landau damping phenomenon

 

    

Braun K., An integrated electro-magnetic plasma field model

As a shortcut reference to the underlying mathematical principles of classical fluid mechanics we refer to(SeJ).

Earliest examples of complementary variational principles are provided by the energy principle of Dirichlet in the theory of electrostatics, together with the Thomson principles of complementary energy. As a short cut reference in the context of the considered Maxwell equations we refer to (ShM1).

A central concept of the proposed solution Hilbert space frame is the alternative normal derivative concept of Plemelj. It is built for the logarithimc potential case based on the Cauchy-Riemann differential equations with its underlying concept of conjugate harmonic functions. Its generalization to several variables is provided in the paper below. It is based on the equivalence to the statement that a vector u is the gradient of a harmonic function H, that is u=gradH. Studying other systems than this, which are also in a natural sense generalizations of the Cauchy-Riemann differential equations, leads to representations of the rotation group (StE).

References

(LiP) Lions P. L., On Boltzmann and Landau equations, Phil. Trans. R. Soc. Lond. A, 346, 191-204, 1994

(LiP1) Lions P. L., Compactness in Boltzmann’s equation via Fourier integral operators and applications. III, J. Math. Kyoto Univ., 34-3, 539-584, 1994 

(SeJ) Serrin J., Mathematical Principles of Classical Fluid Mechanics

(ShM1) Shimoji M., Complementary variational formulation of Maxwell s equations in power series form

(StE) Stein E. M., Conjugate harmonic functions in several variables