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).
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).
(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 (ShM1) Shimoji M., Complementary variational formulation of Maxwell s equations
in power series form (StE)
Stein E. M., Conjugate harmonic functions in several variables | ||||||||||||||||||||||||