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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.  

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.

The proposed quantum field 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.


Thermo-kinematical discrete L(2) based entropy & complementary L(2)-ortho thermo-kinematical continuous 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).


The Boltzmann equation

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 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. Related equations are e.g. the Boltzmann equations for polyatomic gases, mixtures, neutrons, radiative transfer as well as the Fokker-Planck (or Landau) and Vlasov equations. 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 Boltzmann equation is a nonlinear integro-differential equation with a linear first-order operator. The nonlinearity comes from the quadratic integral (collision) operator that is decomposed into two parts (usually called the gain and the loss terms). In (LiP) it is proven that the gain term enjoys striking compactness properties. The Boltzmann equation and the Fokker-Planck (Landau) equation are concerned with the Kullback information, which is about a differential entropy. It plays a key role in the mathematical expression of the entropy principle. The existence of global solutions of the Boltzmann and Landau equations depends heavily on the structure of the collision operators (LiP1). The corresponding variational representation of B=A+K with a H(a)-coercive operator A and a compact disturbance K fulfills a Garding type coerciveness condition (KaY).  

In (ViI) the existence and uniqueness of nonnegative eigenfunction is analyzed.

In (MoB) the eigenvalue spectrum of the linear neutron transport (Boltzmann) operator has been studied. The spectrum turns out to be quite different from that obtained according to the classical theory. The two theories about related physical aspects have one aspect in common: namely that there exists a region of the spectral plane which filled up by the spectrum.


The Landau equation

The Landau equation (a model describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction) is about the Boltzmann equation with a corresponding Boltzmann collision operator where almost all collisions are grazing. The mathematical tool set is about Fourier multiplier representations with Oseen kernels (LiP), Laplace and Fourier analysis techniques (e.g. LeN) and scattering problem analysis techniques based on Garding type (energy norm) inequalities (like the Korn inequality). Its solutions enjoy a rather striking compactness property, which is main result of P. Lions ((LiP) (LiP1)).


The Leray-Hopf operator and the linearized Landau collision operator

In a weak H(-1/2) Hilbert space framework in the context of the Landau damping phenomenon the linerarized Landau collision operator can be interpreted as a compactly disturbed Leray-Hopf operator. 

The Leray-Hopf operator plays a key role in existence and uniqueness proofs of weak solutions of the Navier-Stokes equations, obtaining weak and strong energy inequalities.

Both operators, the Leray-Hopf (or Helmholtz-Weyl) operator and the linearized Landau collision operator are not classical Pseudo-Differential Operators, but Fourier multipliers with same continuity properties as those of the Riesz operators (LiP1).

For the related Oseen operators Fourier multiplier we refer to (LeN).

The related hypersingular integral equation theory, including the Prandtl operator, is provided in (LiI).


References

(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

(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

(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

(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

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