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LITERATURE


The three laws of thermodynamics


- The first law distinguishes two kinds of transfers of energy, (kinematical) heat and thermodynamic „work“ (called „internal energy“), governed by the principle of „conservation of law“.

- The second „law“ only describes an observed phenomenon. It is about the concept of „entropy“ predicting the direction of spontaneous irreversible processes, despite obeying the principle of „conservation of law“; the corresponding (continuous) Boltzmann entropy cannot derived from the model parameters.

- The third (Nernst distribution) law governs the distribution of a solute between two non miscible solvents.  

The proposed model enabling a truly second law of thermodynamics

The third (Nernst distribution) law stays untouched governed by the kinematical Hilbert space H(1).

The first law governs the energy transfer between a kinematical (heat) energy and an „internal energy“. The two energy concepts ("heat" and "internal energy") are now reflected by the decomposition of the Hilbert space H(1/2) into H(1) and its complementary sub-space in H(1/2).

The kinematical energy Hilbert space H(1) is now governed by the (discrete) Shannon entropy. 

The second law is now about the two probabilities for such an energy transfer.
This is determined by the ratio of the cardinalities of both spaces, where the H(1) Hilbert space is compactly embedded into the overall Hilbert space H(1/2), i.e., the sub-space H(1) in H(1/2) is a zero set only.

Considering hermitian operators with either domain H(1) or its complementary sapce this results into either discrete spectra or purely continuous spectra.
In other words, the cardinalities ratio determines the probability of energy transfers between both spaces. This probability is „zero“ for a transfer from the internal energy space into the heat space; it can be interpreted as the probability to generate a matter particle out of the „internal (ether) energy“ space. The probability into the other direction is measured by an exponential decay norm (in line with the Boltzmann probability distribution), which governs all polynomial decay norms of the considered Hilbert scales defined by eigenpair solutions of hermitian operators.  

Plasma

Plasma is the fourth state of matter, where 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 is interpreted as a mathematical point.

Regarding the concept "entropy" we note that plasma is characterized by a constant entropy.

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
      - the Vlasov equation.  

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).


Schrödinger’s statistical (classical & quanta) thermodynamics

A thermodynamic state of a system is not a sharply defined state of the system, because it corresponds to a large number of dynamical states. This consideration led to the Boltzmann entropy relation S = k*log(p), where p is the (infinite) number of dynamical states that correspond to the given thermodynamic state. The value of p, and therefore the value of the entropy also, depend on the arbitrarily chosen size of the cells by which the phase space is divided of which having the same hyper-volume s. If the volume of the cells is made vanishing small, both p and S become infinite. It can be shown, however, that if one changes s, p is altered by a factor. But from the Boltzmann relation it follows that an undetermined factor in p gives rise to an undetermined additive constant in S. Therefore the classical statistical mechanics cannot lead to a determination of the entropy constant. This arbitrariness associated with p can be removed by making use of the principles of quantum theory (providing discrete quantum state without making use of the arbitrary division of the phase space into cells). According to the Boltzmann relation, the value of p which corresponds to S=0 is p=1.


Thermo-kinematical discrete L(2) based entropy & complementary thermo-dynamical entropy

The discrete Shannon entropy in information theory is analogous to the entropy in Thermostatistics.  The analogy results when the values of the random variable designate energies of microstates. For a continuous random variable, differential entropy is analogous to the „continuous“ Boltzmann entropy. However, the continuous (Boltzmann) entropy cannot be derived from the Shannon (discrete) entropy in the limit of n, which is the number of symbols in distribution P(x) of a discrete random variable X, (MaC1). In other words, the Boltzmann entropy cannot be derived from the underlying model parameters. Therefore, the second theorem of thermodynamics states only an observation, which cannot be derived from the underlying model parameters.

The central notion in Schrödinger’s thermostatistics which makes the difference between the classical and the quanta world is the „vapour-pressure formula of an ideal gas“ for computing the so-called entropy constant or chemical constant. The crucial „auxiliary“ term to build the vapour-pressure formula is the „thermodynamical potential“, from which then the entropy itself is derived. The essential physical law it the third theorem of thermodynamics (Nernst), which states, that the ground state energy level is always a constant in any considered system, i.e. there is a part of the entropy, which does not vanish at T=0, and which is independent from all system parameters. The only mathematical relevant assumption is that the considered particles are energy quanta without individuality (ScE) pp. 16, 43.

In the physics of plasma the entropy is constant, information is conserved and the initial state data is always known, caused by the so-called Landau damping.

Regarding the notion „vapour-pressure“ we note that „pressure“ is nothing else than a potential difference. Therefore, the proposed coarse-grained kinematical H(1) energy Hilbert space model and its complementary closed („potential“) the subspace in H(1/2) accompanied with model intrinsic concepts of a potential function and a potential barrior enable an alternative model for the „vapour-pressure“, resulting in a corresponding entropy concept between both spaces. We note that H(1) is compactly embedded into H(1/2), i.e. from a probability theory perspective it is a zero set with discrete spectrum of the corresponding „energy operator“.

From a mathematical perspective we note that the distributional Hilbert scales (accompanied with polynomial degree norms) are governed by a weaker norm with exponential degree. This enables norm weighted estimates with two parts, a statistical L(2) based part and a corresponding „exponential degree“ part. For a corresponding apprximation theory we refer to (NiJ), (NiJ1).



References

(LiP) Lions P. L., Boltzmann and Landau equations  

(LiP1) Lions P. L., Compactness in Boltzmann’s Fourier integral operators and applications  

(MaC1) Marsh C. Introduction to Continuous Entropy

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

(NiJ1) Nitsche J. A., Extensions and Generalizations

(ScE) Schrödinger E., Statistical Thermodynamics, Dover Publications, Inc., New York, 1989