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 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 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

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 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).

Wavelets: a mathematical microscope tool


Wavelets, a mathematical microscope tool

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.


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

Related papers

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

(BrK1) Braun K.,Unusual Hilbert or Hoelder space frames for the elementary particles transport (Vlasov) equation(JoR) Jordan R., et. al., The variational formulation of the Fokker-Planck equation 

(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

(MoC) Mouhot C., Villani C., On Landau Damping

(NiJ) Nitsche J. A., lecture notes I, Approximation Theory in Hilbert Scales

(NiJ1) Nitsche J. A., lecture notes II, Extensions and Generalizations

(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

(StE) Stein E. M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970

(ViI) Vidav I., Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator, J. Mat. Anal. Appl., Vol 22, Issue 1, (1968) 144-155