                       Plasma

"Plasma is sometimes called the "fourth state of matter". This kind of "matter" needs to fulfill the following two pre-requisites:

- there must be electromagnetic interactions between charged particles

- the number of positively and negatively charged particles per considered volume element may be arbitrarily small oder arbitrarily large, but both numbers need to be approximately identical. The number of neutral particles (atomes or molecules) is irrelevant for the definition of a plasma.

Therefore, plasma is a conductor. As electric current generates magnetic fields, and as electric charged particles are influenced by electric and magnetic fields a plasma is influenced by external electrical and magnetical fields, creating itself such fields. This means that plasma can interact with itself. In electrodynamic plasma is a anisotropic non-linear dispersive conductor.

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 apart", (CaP) p. 1.

From 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 H) and the axiom of observables (each physical observable A 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 H. 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 continuity 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 kinetic theory of gases w/o and with “molecular chaos“

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 Fokker-Planck (Landau) equations (with plasma particle collisions)
- the Vlasov equation (w/o plasma particle collisions).

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 conclusion from the below is that the proposed Hilbert scale model replaces the model for "gas-surface interactions", where the Boltzmann equation in the corresponding variational framework form the adequate kinetic and potential theory of gases, and where the corresponding linearized Boltzmann collision operator forms the quanta kinematical operator. In other words, the proposed variational framework avoids the expounded model equations above, while providing a well defined variational PDE system accompanied with corresponding finite energy norm estimates.

(*) (LiP1): "The Boltzmann and Landau equations provide a mathematical model for the statistical evolution of a large number of particles interacting through "collisions". The unknown function f corresponds at each time t to the density of particles at the point x with velocity v. If the collision operator were zero, the equations would mean that the particles do not interact and f would be constant along particle path. If collisions occur, in which case the rate of changes of f has to be specified such a description was introduced by Maxwell and Boltzmann and involves an integral operator. This model is derived under the assumption of stochastic independence of pairs of particles at (x,t) with different velocities (molecular collision assumption)", (LiP), (LiP1).

The Fokker-Planck equation

The following is a collection from (RiH):

The Fokker-Planck equation was first used by Fokker and Planck to describe the Brownian motion of particles. If a small particle of mass is immersed in a fluid a friction force will act on the particle. The physics behind the friction is that the molecules of the fluid collide with the particle. The momentum of the particle is transferred to the molecules of the fluid and the velocity of the momentum of the particle therefore decreases to zero. If the mass of the particle is large so that its velocity due to thermal fluctuations is neglible, then the related PDE is a deterministic equation governed by the mean energy of the particle. If the mass of a small particle is still large compared to the mass of the molecules the equation needs to be modified so that it leads to a correspondingly adapted thermal energy equation based on the corresponding thermal velocity observable by adding a fluctuation force. In other words, the total forces of the molecules acting on the small particle is decomposed into a continuous damping force and a fluctuating force. The latter one is a stochastic or random force, the properties of which are given only in the average (the fluctuation force per unit mass is called the Langevin force). The mathematical concept is about a stochastic differential equation.

Asking for the probability to find the velocity in the interval (v,v+dv), or in other words, asking for the number of systems of the ensemble whose velocities are in the interval (v,v+dv) divided by the total number of systems in the ensemble results into Fokker-Planck type equations. Its solution is a probability density function W(v), also called probability distribution. The probability density times the length of the interval dv is then the probability of finding the particle in the interval (v,v+dv). This distribution function depends on time t and the initial distribution. Mathematically, the Fokker-Planck equation is a linear second-order PDE of parabolic type.

Roughly speaking, it is a diffusion equation with an additional first-order derivative with respect to the x variable. In mathematical literature the Fokker-Planck equation is also called a forward Kolmogorov equation.

A complete solution of a macroscopic system would consist in solving all microscopic equations of the system. Because one cannot generally do this one uses instead a stochastics description, i.e., one describes the system by macroscopic variables which fluctuate in a stochastic way. The Fokker-Planck equation is just an equation of motion for the distribution function of fluctuating macroscopic variables. For a deterministic treatment one neglects the fluctuations of the macroscopic variables. Therefore, for the Fokker-Planck equations one neglects the diffusion term. Mathematically speaking, one neglects the defining term for the kinematical energy norm (one dimensional case).

The N stochastic variables case requires the concepts of a probability current and joint probability functions (Markov processes). The latter one can be expressed by a transition probability function for small times and a distribution function at the time t. The diffusion coefficients become a (positive definite) diffusion matrix. Roughly speaking, the drift and the diffusion coefficients become the Nabla and the Laplace operators, allowing a reformulation of the Fokker-Planck equation in covariant form.

Magnetohydrodynamic (MHD)

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.

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.

The “mother" of all hydrodynamic models is the continuity 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 is interpreted as a mathematical point.

An ideal plasma is a non-dissipative flow of the incompressible charged particles (CaF).

The MHD equations are derived from continuum theory of non-polar fluids with three kinds of balance laws:

(1) conservation of mass

(2) balance of linear momentum

(3) balance of angular momentum (Ampere law and Faraday law).

The MHD equations consists of 10 equations with 10 parameters accompanied with appropriate boundary conditions from the underlying Maxwell equations (CaF).

In (EyG) it is proven that smooth solutions of non-ideal (viscous and resistive) incompressible magneto-hydrodynamic (plasma fluid) equations satisfy a stochastic (conservation) law of flux. It is shown that the magnetic flux through the fixed Plasma is an ionized gas consisting of approximately equal numbers of positively charged ions and negatively charged electrons.

References

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