2021 SOLUTIONS
2021 LOOK BACK
2020 SOLUTIONS
RH 2010-2018
MIE 2018
GUT 2011-2017
NSE 2016
PLASMA PHYSICS
PLASMA HEATING
DISCR.&CONT'S ENTROPY
WHO I AM
LITERATURE


In case of low density plasma "plasma particles" collisions are ignored.

In case of high density plasma "plasma particles" collisions cannot be ignored anymore.

The key differentiator between plasma to neutral gas or neutral fluid is the fact that its electrically positively and negatively charged particles are strongly influenced by electric and magnetic fields, while neutral gas is not.

A characteristics of the plasma "matter" state is the fact that the number of positively charged and negatively charged "particles" are approximately equal over time. In the related standard (cold plasma) PDE model (Vlasov equation) this is about a two media plasma fluid of negatively charged electrons and positively charged ions without "neutralization" over time, where each of the considered particle class needs to be governed by different Vlasov equations (CaF) p. 66.

The "plasma particles collisions" related mathematical model is about a Landau type PDE. This model is basically about friction forces ocurring when the molecules of the fluid collide with the considered particle. In case of the plasma heating phenomenon this requires a two media plasma fluid PDE system with interactions of same type and different type particles. The latter interaction type then is the model for the hot plasma specific plasma heating phenomenon.

The claims are

1. that all three classical plasma theory types, fluid PDE models, statistical PDE theory and MHD, provide inappropriate models for the (micro and macro world) plasma heating and Landau damping phenomena for several reasons, and

2. that the proposed H(1) kinematical Hilbert space decompostion into two positively and negatively charged kinematical energy sub-spaces (with same cardinality governed by an indefinite inner product) in combination with a multiple (at least a binary, but proposed tertiary ("positron", "electron", "neutron")) particle type fluid/statistical/MHD variational theory overcomes the several handicaps of the current models. The "+/- fluid" media are governed by the complex Lorentz group with its underlying two connected components L(+) and L(-), (StR).

Claim 2 is also related to the theory of equilibrium critical phenomena:

"The main aim of the theory of critical phenomena is about to be the (implicitly or explicitly) calculation of the observable properties of a system from first principles using the full microscopic quantum-mechanical description of the constituent electrons, protons and neutrons. Such a calculation, however, even if feasible for a many-particle system which undergoes a phase transition need not and, in all probability, would not increase one’s understanding of the observed behaviour of the system. Rather, the aim of the theory of a complex phenomenon should be to elucidate which general features of the Hamiltonian of the system lead to the most characteristic and typical observed properties. Initially one should aim at a broad qualitative understanding, successively refining one’s quantitative grasp of the problem when it becomes clear that the main features have been found", (FiM) p. 619.

In this context, (FiM) p. 639:

"binary fluids undergo,

i) phase separation, when AA and BB contacts are favoured energetically over AB contacts

ii) ordering, when AB contacts are most favourable.

In case i) the mole fraction, of, say, the A component is analogous to the density in a one-component fluid system. Below the critical point T(c) the mixture will separate into a A-rich and a B-rich phase.

Case ii) of an ordering crystalline binary alloy such as beta-brass is most directly analogous to an antiferromagnet since the phase transition is signalled by the appearance below T(c) of a coherent superlattice scattering line".

The most striking handicap is the fact that obviously the two "plasma particles" of the two +/- charged media are on an equal footing concerning their physical properties and their "parallel existences" over time induce the heating phenomenon. At the same time their existences are "a priori" given before the related fields can "act". The "collisions-less" (Vlasov) model is only about Coloumb forces. The "collisions" (Landau type) models are about "heat energy generation" by collisions without "neutralization".

The proposed theory can be supported by the theory of "vortex dynamics", which is (in the classical PDE case) about coherent structures in turbulence. It is a natural paradigm for the field of chaotic motion and modern dynamical system theory, e.g., accompanied with singular distributions of vorticity, vortex momentum, related creation processes, and the dynamics of line vortices, (SaP).

The instability criteria of MHD are based on the two methods, the normal modes method and the (kinematical) energy principle method. Regarding the first method we note that there are also existing non-normal modes, (FiM). Regarding the latter method we recall the issue of generated virtual plasma waves "out of the blue" in the context of "explaining" the plasma damping phenomeon. Regarding the micro (quanta) and macro (galaxy) "instability" phenomena scope of plasma, anticipating that the macro plasma world makes 99% of the wole matter universe in the context of a common related field based theory we note the paper (RoK). It is about a plasma slab of infinite extent in the z-direction, with finite resistivity and finite shear, accompanied with twisted slicing modes, i.e. there are modes which are neither localized near a particular horizontal surface, nor dependent on a boundary layer.


The Landau equation type case (with "plasma particles" collisons)

In case of high density plasma corresponding particle collisions cannot be ignored. 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 method (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 Vlasov equation case (w/o "plasma particles" collisions)

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.
This following is basically taken from (ShF) p. 390 ff:

Much of the complication of plasma physics stems from the dominance of internal electromagnetic fields in affecting the motions of charged particles. These fields have a dynamics self-consistent with the dynamics of the particles. An example of such a fully dynamic phenomeon is the high-frequency oscillations that can take place if charge separation occurs with periodic structure, in particular the problem of longitudinal plasma oscillations when electrons possess nonzero random motions. For cold plasma in which only the electrons are mobile, spacial displacement of the electrons from the ions introduces restoring electric forces that set up oscillations at a certain frequency. In case of hot plasma the question arises, how does finite temperature for the electrons modify this specific frequency.

The corresponding perturbational Vlasov equation treatment continues to adapt the model of a two-component plasma in which the ions provide only a positively charged background of uniform density, while the electrons have a phase space description given by a distribution function f(x,v,t). In the absense of magnetic fields or collisions, f(x,v,t) satisfies the Vlasov kinetic equation, where the self-consistent electric field arises from the charge distribution of electrons and ions, and where the equilibrium state of the electric field is assumed to be unifirm and electrically neutral. With the ions immobile, and only considering perturbances that generate no magnetic fields the purely electric oscillations are purely longitudinal.

The related Landau damping decrement depends only on the derivative of the electron distribution function evaluated at the exactly resonant value, (ShF) p. 401. The resonant wave-particle interaction is claimed to underline the physical mechanism behind the Landau damping. Electrons with random velocities substantially different from the phase speed of the wave, drfit in and out oft he crests and troughs oft he wave. Sometime they are accelerated by the collective electric field, sometime decelerated; but integrated over time, no net interaction results (in the linear approximation!) because the time spent in crests and troughs averages out for a sinusoidal disturbance.

In its purest form, Landau damping represents a phase-space behavior peculiar to collisionsless systems. Analogs to Landau damping exist, for example, in the interactions of stars in a galaxy at the Lindblad resonances of a spiral density wave. Such resonance in an inhomogeneous medium can produce wave absorptions (in space rather than in time), which does not usually happen in fluid systems in the absense of dissipative forces.


                   

Longitudinal plasma oscillations and Landau damping

                


The theories of thermodynamics and critical phenomena and the proposed Hilbert scale based gravity and quantum field model

The standard theory of thermodynamics tells us that knowledge of a thermodynamic potential as a function of its natural variables completely specifies the thermodynamics of a system. Several potentials have this property of encoding complete thermodynamic systems, e.g., the Helmholtz free energy F and the Gibbs free energy G. F’s natural variables are the temperature and whatever macroscopic parameters determine the system’s energy levels. … The Gibbs free energy ist he Legendre transform oft he Helmholtz free energy with respect to its second variable. Thus we have

                             G(T,p)=F+pV  and G(T,M)=F+BM

for fluid and magnetic systems, respectively“, (BiJ) p. 21.

The Ginzburg-Landau model is a kind of „metamodel“ in the theory of critical phenomena“, (BiJ) p. 179.

The great beauty of the Ginzburg-Landau model is that it allows one to solve many difficult problems in superconductivity (e.g. surfaces of superconductors, the GL theory of inhomogeneous systems, or the GL theory in a magnetic field)“, (AnJ) p. 83.

The Landau theory is an approximation of the partition function in the Ginzburg-Landau model“, (BiJ) p. 188.

Regarding the proposed H(1/2) energy model we note that only the Hamiltonian formalism is valid, as the Legendre transform is not defined due to the reduced regularity assumptions; therefore

-          the Lagrange formalism

-          the Ehrenfest theorem

-          the GL (meta-) model for critcial („turbulence“) phenomena

are only valid in the compactly embedded kinematical H(1) framework.

                                    

Legendre transforms



With respect to the application of probabilistic considerations to the study of turbulence we refer to (SwH) 2.3:

"there is one feature of the initial distribution which is generally agreed to be universal: Any specific set of initial states of zero state space volume (i.e., zero Lebesgue measure) will also have probability zero with respect to the initial distribution. In other words, a phenomenon which occurs only for a set of initial states of zero volume will never be seen. .... For what sorts of differential equations are all but a neglegible set of solution curves statistical regular? One answer provides by the Birkhoff pointwise ergodic theorem, is that Hamiltonian equations of motion have this property provided that the microcanonical measure is finite on each energy surface."

In this context we note that the compactly embedded kinematical sub-Hilbert space H(1) of H(1/2) is a "zero set" with respect to the H(1/2) inner product.

Regarding the notion „spontaneous symmetry break down“ in the context of a Hamiltonian, we recall from (BiJ) p. 48:

When an exact symmetry of the laws governing a system is not manifest in the state of the system the symmetry is said to be spontaneously broken. Since the symmetry of the laws is not actually broken it would perhaps be better described as „hidden“, but the term „spontaneously broken symmetry“ has stuck.“


The Landau damping phenomenon and the proposed Hilbert scale based gravity and quantum field model

(CaF) p. 390 ff: "The turbulence of plasma differs from the hydrodynamic turbulence by the action of the magnetic field. A more relevant difference is due to the hydrodynamic interaction between the plasma particles, the interaction with the magnetic fields, and the interaction between the electromagnetic waves. ... All of them are the root cause of electromagnetic plasma turbulence. ... The case of interactions between quasi-stationary electromagnetic waves is called weak turbulence. ... The case of non-linear Landau damping (strong plasma turbulence) leads to the generation of virtual waves, which transfer their energy to the affected particles asymptotically with 1/t; the plasma is heated (turbulence heating) faster than this may happen by purely particles collisions", (TsV).

In the context of the statistical theory the Landau damping 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.

Regarding the above mentioned two phenoemona area, the micro-plasma-particles world and the macro-galaxy-stars-world, we note that in the latter case no SRT validated relevant physical parameters are taken into account, while in the latter case additional purely mathematical assumptions are required (micro turbulance relevant Penrose stability condition) to govern the considered electric (Coloumb) force.

Mathematical speaking the Landau damping is a specific behavior of linear waves in plasma governed by the non-linear term of the considered PDE system. In this context we note that in (RoK) it is shown that there exist modes influencing plasma, which are not of the form exp(iwt).

In case of the statistical theory the core defining element of a plasma (i.e. a system of nearly equal numbers of positive and negative charged plasma particles staying over time with constant entropy) requires a two-component plasma fluid media. The Vlasov equation is about a positively charged ions background of uniform density, and negatively charged electrons having a phase space description given by a distribution function f(x,v,t).

The electromagnetic turbulence of plasma is caused by

1. the action of the magnetic field

2. the hydrodynamic interaction between the plasma particles

3. the interaction of the plasma particles with the magnetic fields

4. the (quasi-stationary resp. non-linear) interactions between the electromagnetic waves (weak resp. strong plasma (heating) turbulence).

The proposed modelling framework governs 1.-3., while the quasi-linear or non-linear term of the considered PDE governs 4.

In the proposed H(1/2) energy Hilbert space framework the strong plasma heating energy is provided from the total of potential differences (~ "pressure", ~ potential operator) of the colliding plasma quanta particles resp. from the related energy norms.

The Landau damping is a characteristic of collisionsless plasma. In the context of the proposed H(1/2) Hilbert space framework it is a characteristic of this framework, i.e. it is related to the complementary sub-space of the compactly embedded (kinematical energy) Hilbert sub-space H(1) of H(1/2).

(ChF) p. 245: „Landau damping may also have applications in other fields. For instance, in the kinetic treatment of galaxy formation, stars can be considered as atoms of a plasma interacting via gravitational rather than electromagnetic forces. Instabilities of the gas of stars cause spiral arms to form, but this process is limited by Landau damping“.


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: with (MoC) there exists now a complex and sophisticated mathematical proof of the Landau phenomenon based on the classical 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 governing the electron fluid, while the approximately identical number of positively charfed particles is considered as a background field). At the same time the proof requires strong mathematical assumptions, like analytical regularity assumptions of the PDE solution function to enable the defintion of so-called “hybrid” and “gliding” analytical norms, while at the same time the Landau-Penrose stability criterion to govern the singularity of the Coulomb potential governs micro turbulences of the plasma particles.

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.

the conclusion: the existence of a non-linear Landau damping proof based on the statistical "turbulence" classical Vlasov PDE provides evidence that this PDE is an inappropriate physical model for the behind physical phenomena.

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" Hilbert space norm, which is even weaker than any polynomial distributional Hilbert space norm.
Just to anticipate the kinetic treatment of galaxy formation, where stars can be considered as atoms of a plasma interacting via gravitational rather than electromagnetic forces with the "plasma matter" of the universe the considered "one-particle-type" variational representation of the Vlasov equation still remains to be an inappropriate physical model. Clearly some type of Maxwell/Lie-type boundary value conditions with the plasma heating particles are missing. Additionally a global/universal time variable is conflicting already with the SRT.


    

Braun K., A distributional Hilbert space framework to prove the Landau damping phenomenon
 
                                              August 2018


Related papers


(AnJ) Annett J. F., Superconductivity, Superfluids and Condensate, Oxford University Press, Oxford, 2004

(BiJ) Binney J. J., Dowrick N. J., Fisher A. J., Newman M. E. J., The Theory of Critical Phenomena, Oxford Science Publications, Clarendon Press, Oxford, 1992

(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

(ChF) Chen F. F., Introduction to plasma physics and controlled fusion, Vol. 1: Plasma physics, Plenum Presse, New York, London, 1929

(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

(FiM) Fisher M. E., The theory of equilibrium critical phenomena, Rep. Prog. Phys., 30 (1967) 615-730

(HaW) Hayes W. D., An alternative proof of the circulation, Quart. Appl. Math. 7 (1949), 235-236

(HoE) Hopf E., Ergodentheorie, Springer-Verlag, Berlin, Heidelberg, New York, 1070

(JoR) Jordan R., Kinderlehrer D., Otto F., The variational formulation of the Fokker-Planck equation, Research Report No. 96-NA-008, 1996

(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

(RiH) Risken H., The Fokker-Planck Equation, Springer, Berlin, 1989

(RoK) Roberts K. V., Taylor J. B., Gravitational Resistive Instability of an Incompressible Plasma in a Sheared Magnetic Field, The Physics of Fluids, Vol. 8, No. 2 (1965), 315-322

(SaP) Saffman P. G., Vortex Dynamics, Cambridge University Press, Cambridge, 1992

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

(ShF) Shu F. H., The Physics of Astrophysics, Vol. II, Gas Dynamics, University Science Books, Sausalito, California, 1992

(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

(StR) Streater R. F., Wightman A. S., PCT, Spin & Statistics, and all that, W. A. Benjamin, Inc., New York, Amsterdam, 1964

(SwH) Swinney H. L., Gollub J. P., Hydrodynamic Instabilities and the Transition to Turbulence, Springer-Verlag, Berlin, Heidelberg, New York, 1980

(TsV) Tsytovich V., Theory of Turbulent Plasma, Consultants Bereau, New York, 1977

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