", (CaP) p. 1.As interstellar gas and all stars consist of ionized gases 99% of the whole matter of the universe is in plasma state with values of state variables far apartFrom 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 ) and the Haxiom of
observables (each physical observable 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 A.
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 Hcontinuity
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 particle is interpreted as a mathematical point.Regarding the concept "entropy" we note that plasma is characterized by a constant entropy and that the entropy is related to the so-called dissipative function. It describes the amount of heat per volume unit and per time unit generated by the friction forces. An ideal plasma is a frictionless plasma without Joule heat and without heat conduction, and therefore with constant entropy, i.e., it is a non-dissipative flow. MHD is about the flow of an incompressible plasma. The mathematical model is about a system of 10 (partially non-linear) PDE. Ideal MHD is about reversible processes. Based on a corresponding classifcation of the different types of plasma states, (CaF) p. 25, there are three types of plasma theories: - the fluid theory of electromagnetic charged particles - the statistical theory with its most prominent examples, the Vlasov and the Fokker-Planck equations - the theory of Magnetohydrodynamics (MHD).
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 for polyatomic gases, mixtures, neutrons, radiative transfer 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: for example
The following is a collection from (RiH):
MHD is concerned with the motion of electriclly conducting fluids in the presence of electric or magnetic fields. In MHD one does not consider velocity distributions. It is about
notions like number density, flow velocity and pressure.
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(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 (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 (SeW) Sears W. R., Resler E. L., Theory of thin airfoils in fluids of high electrical conductivity, Journal of Fluid Mechanics, Vol. 5, Issue 2 (1959), 257-273 (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 | |||||||||||||||||