In case of low density plasma "plasma particles" collisions are ignored. 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.
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.
„ 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 “, (AnJ) p. 83. 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)„ The Landau theory is an
approximation of the partition function in the “, (BiJ) p. 188. Ginzburg-Landau modelRegarding 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.
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.“
(CaF) p. 390 ff:
" In the context of the statistical theory the Landau damping phenomenon is about “
August 2018
| ||||||||||||||||||||