Long range hot plasma particles interactions between
electrons and ions

In case there are no plasma particles interactions the Liouville
distribution function becomes for each considered particle a specific distribution
function (separation approach), which is characterized by the Vlasov equation.

In case of
plasma with high density plasma corresponding particle collisions cannot be ignored anymore. The
corresponding equations also need to be derived from the underling Liouville equation.

In case there are different types of plasma particles (typical case: plasma heating, which is about a two media plasma fluid of negatively charged electrons and positively charged ions without "neutralization" over time)
each of the considered particle class needs to be governed by different Vlasov equations (CaF) p. 66.

There are two classes of plasma collision
processes:

(1) binary collisions with smaller densities and interacting
forces and short range (like intermolecular forces) This is the situation in a
diluted, neutral gas as modelled by the Boltzmann equation, which has been adapted
resp. improved by Landau in case of plasma particles

(2) because with a long range Coulomb forces play an
essential role and it will occur multiple collisions. The long range of Coulomb
forces causes collisions with small scattering angle.

There are three methods available to derive appropriate
equations adresseing (1) and (2), (CaP) p. 67:

a) the BGKBY method resulting into the Boltzmann-Landau
equation

b) the method of Klimontovitsch and Deupree
resulting into the Boltzmann-Landau equation and the Fokker-Planck equation

c) the Balescu ethod (solving the Liouville
equation applying the Green function concept) resulting into the Fokker-Planck equation
and the Lenard-Balescu equations.

The main objective of all methods is to derive the Boltzmann
collision equation and the corresponding Boltzmann collision integral from the Liouville
equation. The common idea of all methods is to replace the "one-particle distribution function" first by a "two-particle distribution function" in the framework of the Liouville equation. This corresponds to a twofold (pair) correlation
function in the framework of the Boltzmann equation. This conceptual approach results into a
hierarchical ordered PDE system, which is equivalent to the Liouville
equation, but only solvable for a finite number of those PDEs.

The reduction to
only binary collision in small density gases and neglecting all other than
2-particle short distance interactions results into the Vlasov equation.

The case of neglected 3- (and higher) particle interactions allows the
definition of a third distribution function (stochastic independent from the already existing 2-particle distribution functions) resulting into the
Vlasov-Liouville equation.

For plasma gases Landau has modified the collision integral of
the Boltzmann equation in case of collisions with small scattering angles. His
collision integral can be derived from the Fokker-Landau equations, as well as from
the Lenard-Balescu equations. This shows that in the context of the several possible
approximation equations above all those equations are in a certain sense equivalent.
The Fokker-Planck equation is mainly used one in plasma physics.

We emphasis that from a physical modelling perspective the 1-plasma
particle Vlasov equation is inappropriate for collision processes of class (2) above, i.e., it is an inappropriate physical model for the non-linear Landau phenomenon in the context of long range Coulomb
forces.

The Boltzmann equation

The Boltzmann equationis
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 Boltzmann equation is a
nonlinear integro-differential equation with a linear first-order operator. The
nonlinearity comes from the quadratic integral (collision) operator that is
decomposed into two parts (usually called the gain and the loss terms). In (LiP)
it is proven that the gain term enjoys striking compactness properties. The
Boltzmann equation and the Fokker-Planck (Landau) equation are concerned with
the Kullback information, which is about a differential entropy. It
plays a key role in the mathematical expression of the entropy principle. The
existence of global solutions of the Boltzmann and Landau equations depends
heavily on the structure of the collision operators (LiP1). The corresponding
variational representation of B=A+K with a H(a)-coercive operator A and
a compact disturbance K fulfills a Garding type coerciveness condition (KaY).

In (ViI) the existence and
uniqueness of nonnegative eigenfunction is analyzed.

In (MoB) the eigenvalue
spectrum of the linear neutron transport (Boltzmann) operator has been studied.
The spectrum turns out to be quite different from that obtained according to
the classical theory. The two theories about related physical aspects have one
aspect in common: namely that there exists a region of the spectral plane which
filled up by the spectrum.

The Landau equation

TheLandau 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 Leray-Hopf operator and the linearized
Landau collision operator

In a weak H(-1/2) Hilbert space framework in the context of the Landau damping
phenomenon the linerarized Landau collision operator can be interpreted as a
compactly disturbed Leray-Hopf operator.

The Leray-Hopf operator plays a key role in
existence and uniqueness proofs of weak solutions of the Navier-Stokes
equations, obtaining weak and strong energy inequalities.

Both operators, the "Leray-Hopf (or Helmholtz-Weyl) operator and the
linearized Landau collision operator are not classical Pseudo-Differential Operators,
but Fourier multipliers with same continuity properties as those of the Riesz
operators (LiP1).

For the related Oseen operators Fourier multiplier we refer to (LeN).

The related hypersingular integral equation theory, including the Prandtl
operator, is provided in (LiI).

The Landau damping phenomenon

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 over time.

Mathematical speaking the Landau damping phenomenon is caused by the non-linear term, i.e. the generation of "virtual waves" to "explain" the plasma heating phenomenon without neutralization of the differently charged carries is governed by this non-kinematical term, while the physical differentiator to normal gas (the nearly equal numbers of positive and negative charged plasma particles stay with constant entropy).

This following is about the mathematical "théoreme
vivant" (MoC), which is about a "proof" of the physical observed (Plasma
physics) Landau damping phenomenon based on the classical (PDE) Vlasov
equation.

To the author´s humble opinion, ... the situation: there
exists now a complex and sophisticated mathematical proof of the Landau
phenomenon based
on the Vlasov equation, which is inappropriate from a physical modelling perspective with respect to the plasma heating phenomenon (as there is only one class of distribution functions). At the same time the proof requires strong mathematical assumptions, like
the Landau-Penrose stability criterion to govern the singularity of the
Coulomb potential and analytical regularity assumptions of the
PDE solution function to enable the defintion of so-called “hybrid” and “gliding”
analytical norms the conclusion: the existence of the mathematical proof provides
now also evidence from a mathematical view that the Vlasov equation is an inappropriate model

the comment: regarding the "two-class distribution functions" modelling requirement we note that the counterpart of the Heisenberg uncertainty inequality in
probability theory is the co-variance concept of two independent random
variables. The special physical effect of the Landau damping phenomenon
context is about plasma heating caused by strong turbulence behavior of
nearly the same number of negatively and positively charged ions and
electrons w/o "neutralization" over time. Any classical
thermostatistical considerations subsuming those two phenomenon enabling
media into a single "fluid" governed by a single class of probability
distributions at least looks inappropriate, especially in a quantum
field framework governed by the Heisenberg inequality.

Below we sketch appropriate H(1/2) energy norm estimates in the context of a variational representation of the
Vlasov equation, where the analytical norms in (MoC) are replaced by an
"exponential decay" Hilber space norm, which is even weaker than any polynomial distributional Hilbert space norm.

(CaF) Cap F., Lehrbuch der Plasmaphysik und Magnetohydrodynamik, Springer-Verlag, Wien, New York, 1994

(CeC) Cercignani C., Theory
and application of the Boltzmann equation, Scottish Academic Press, Edingburgh
and London, 1975

(EyG) Eyink G. L., Stochastic Line-Motion and Stochastic
Conservation Laws for Non-Ideal Hydrodynamic Models. I. Incompressible Fluids
and Isotropic Transport Coefficients, arXiv:0812.0153v1, 30 Nov 2008

(KaY) Kato Y., The
coerciveness for integro-differential quadratic forms and Korn’s inequality,
Nagoya Math. J. 73, 7-28, 1979

LeN) Lerner, N., A note on the Oseen kernels, Advances
in Phase Space Analysis of Partial Differential Equations, pp. 161-170, 2007

(LiI) Lifanov
I. K., Poltavskii L. N., Vainikko G. M., Hypersingular integral equations and
their applications, Chapman & Hall, CRC Press Company, Boca Raton, London,
New York, Washington, 2004

(MoB) Montagnini B., The
eigenvalue spectrum of the linear Boltzmann operator in L(1)(R(6)) and
L(2)(R(6)), Meccanica, Vol 14, issue 3, (1979) 134-144