                    The Boltzmann entropy

weakness: The continuous (Boltzmann) entropy cannot be derived from a Shannon (discrete) entropy in the limit of n.

overcoming: replace current continuous entropy definition, which is the inner product of f and log(f) with respect to the H(0)=L(2) Hilbert space, by the corresponding inner product of the distributional Hilbert space H(-1/2).

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 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.

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 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.

Nernst’s theorem (the third law of thermodynamics) states that “to the thermodynamic state of a system at absolute zero there corresponds only one dynamical state, namely, the dynamical state of lowest energy compatible with the given crystalline structure or state of aggregation of the system”.

The Gibbs principle states that the maximum entropy of a thermodynamic system is achieved, when all considered macroscopic parameters of the described systems yield stationary values. The entropy concept is applied in (non-linear) partial differential evolution equations to analyze well-posed ness of those equations providing qualitative descriptions of the behavior of its solutions, especially in the long-term regime.

In order to analyze the convergence of the entropy generation resp. its dissipation, the term is tried to be estimated to the below by the relative entropy itself. Applying the Gronwall lemma then it one can show the exponential convergence to zero. The lemma of Gronwall is also sometime applied in finite element approximation analysis of parabolic or hyperbolic PDE enabled by the Ritz/Galerkin approximation theory. In all those cases there might be quasi-optimal error estimates derived with respect to the expected convergence factor. However, applying the lemma of Gronwall to derive those estimates always require purely additional regularity assumptions, just for technical reasons. This is already be the case for the most simple linear parabolic PDE, the heat equation. From a mathematical point of view the underlying handicap is just about the not appropriately used norms.

The continuous (Boltzmann) entropy cannot be derived from Shannon (discrete) entropy in the limit of n, which is the number of symbols in distribution P(x) of a discrete random variable X. Marsh C., Introduction to Continuous Entropy

Alternatively, in line with the proposed distributional H(-1/2)-Hilbert space concept of this homepage, we suggest to define “continuous entropy” in a weak H(-1/2)- frame to overcome the weaknesses of the Boltzmann (concept) entropy. This enables also corresponding Garding-type inequalities (e.g. the Korn inequalities) and related Hilbert space approximation theory with appropriate norms, including norms with "exponential decay".

(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)." Lions P. L., On Boltzmann and Landau equations Lions P. L., Compactness in Boltzmann s Fourier integral operators and applications

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 "Landau" collision operator and the "Leray-Hopf (or Helmholtz-Weyl) operator are not classical pseudo-differential operators, but Fourier multipliers with same continuity properties as those of the Riesz operators (LiP1). As a consequence, in a weak H(-1/2) Hilbert space frame, the "Landau" collision operator can be interpreted as a compactly disturbed Leray-Hopf operator. This links back to other sections of the homepage. For a related integral representation of the NSE solution we refer to Peralta-Fabi R., An integral representation of the Navier-Stokes equation-1

For the related Oseen operators Fourier multiplier we refer to Lerner N., A note on the Oseen kernels

This concept corresponds also to the proposed distributional H(1/2)- energy Hilbert space with an alternative Schrödinger (Calderón) momentum operator and related domain, enabling a model for Schrödinger 's heat bath concept (see section YME). In (JaE), p. 628, Schrödinger's heat bath concept is applied to build two systems I and II, the system I under investigation and the system II representing the heat bath. This Schrödinger (Calderón) momentum operator P corresponds to the Calderón-Zygmund integrodifferential operator in (EsG), example 3.5. The related hypersingular integral equation theory, including the Prandtl operator, is provided in (LiI):

the inner product ((u,v)) of the alternatively proposed energy Hilbert space H(1/2) is equivalent to the induced energy norm (Pu,v) of the alternatively proposed Schrödinger operator P with respect to the inner product (u,v) of the L(2)=H(0) (measurement) Hilbert space. For alternative (Fourier multiplier) representations of the operator P, e.g. linking to the Riesz operators, we refer to (EsG), examples 3.3, 3.4, (StE). For the Fourier transform analysis of the log-function, the theory of "hyper-/sub-ellipticity" for properly supported pseudodofferential operators and related Garding inequality techniques we refer to (PeB).

Regarding the log(f) term of the Boltzmann entropy definition we note

1. that the explicit formula for the Hilbert transform of k:=log(h) given in (MaJ), where k denotes the absolute value of the function f and f is a function in the Cartwright class

2. the H(-1/2) inner product identity (((f,g))) = (((u,v)))) = (((w,z))), where u resp. v denote the Fourier transform of f and g, resp. where w and z denote the Hilbert transforms of f and g.

We also note, that if it is assumed that k is Lipschitz, then its corresponding Hilbert transform is at least continuous (MaJ).

Regarding the alternatively proposed energy Hilbert space H(1/2) with its related H(-1/2) quantum state Hilbert space we note, that this concept also supports Schrödinger's concept of a "Bose quantum oscillator with half-odd integer quantum numbers, starting with -1/2", as discussed and supported, but finally not adopted ("as (infinite) zero-point energy seems to change by infinite amounts!") in (ScE), THE n-PARTICLE PROBLEM:

“According to modern view, a gas must not be regarded as consisting of n identical systems in loose-energy contact, since the energy levels of the gas are not the sums of the energy levels of its n constituents in all combinations. They are numerically equal to them, certainly. But any two gas levels which differ only by an exchange of roles between two (or more) of the n identical atoms or molecules, have to be regarded as one and the same level of the gas. Briefly reflection will show that this produces an entirely different sum-over-states for the gas as a whole.

The sum-over-state Z formula expression embraces several different physical cases: the theory of black-body radiation, the theory of ordinary Bose-Einstein gases, and thereby the theory of the so-called chemical constant; the theory of a Fermi-Dirac gas, of which the most important application is to the electrons of metals.

There are two distributions to be considered:

1.     The distribution of single-particle states “on the momentum line” (p)
2.     The distribution of single-particle states “on the energy line” (a)

In order to calculate the first distribution the number of states (of a single particle) pertaining to a “physically infinitesimal “ element of the phase-space in integrated with respect to the first three variables over the volume V and also over the “4 directions” of the momentum, where p is the absolute value of this momentum. If the particles are endowed with spin, this number has still to be multiplied by a small integer 2 or 3, according to the different orientations of the spin that are possible (2 for spin ½ and also for spin 1, when the rest-mass vanishes (photon); 3 for spin 1, when the rest-mass does not vanishes (meson)). What is needed for evaluating the sum-over-state Z, is the distribution of single-particle states “on the energy line”. The general relation between a and p for a free particle would embrace all cases. But the square root makes it inconvenient. It can be avoided, since in point of fact only the two limiting cases actually arise, either m=0 (photons), or p small compared to m*c for all occupied levels. (This holds for all particles other than photons at the temperatures that actually have to be considered.) There are two equivalent ways of looking at the general relation equation between a and p, either as counting the number of quantum states of a particle, or as counting the number of wave-mechanical proper vibrations of the enclosure. The second attitude makes us think of the “n(s) particles present in state a(s)" as of a proper vibration (or a “hohlraum” oscillator to use a customary expression) in its n(s)th quantum state. This attitude really corresponds to so-called second quantization or field quantization.

The wave point of view in both cases, or least in all Bose cases, raises another interesting question. Since in the Bose case we seem to be faced, mathematically, with simple oscillator of the Planck type, of which the n(s) is the quantum number, we may ask whether we ought not to adopt for n(s) half-odd integers ( ½, 3/2, 5/2, …(n+1)/2, …) rather than integers. One must, I think, call that an open dilemma. From the point of analogy one would very much prefer to do so. For, the “zero-point energy” of a Planck oscillator is not only borne out by direct observation in the case of crystal lattices, it is also so intimately linked up with the Heisenberg uncertainty relation that one hates to dispense with it. On the other hand, if we adopt it straightaway, we get into serious trouble, especially on contemplating changes of the volume (e.g. adiabatic compression of a given volume of black-body radiation), because in this process the (infinite) zero-point energy seems to change by infinite amounts! So we do not adopt it, and we continue to take for the n(s) the integers, beginning with 0.”

The Landau damping for the non-linear Vlassov equation

The physics of gases is about increasing entropy, information loss (e.g. regarding the initial state) and stable macroscopic parameters in case of maximum entropy with "maximum" chaos. 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.

weakness
: the regularity assumptions of the famous Villani proof of exponential Laudau damping for the non-linear Vlassov equation.

overcoming: apply the alternative distributional (energy) Hilbert space H(1/2), which is in line with the non-stationary, non-linear NSE problem and the YME problem solutions of this homepage. This is about two complementary subspaces building H(1/2), whereby one those subspace is compactly embedded into H(1/2), which is identical to the standard energy Hilbert space H(1) with its inner product, the Dirichlet integral.

It is based on analytical regularity assumptions and corresponding analytical norms having up to 5 parameters (which is far away from any physical meaning). One central tool is the Gronwall lemma, which also requires certain mathematical assumptions w/o any physical meaning. The lemma of Gronwall is also sometime applied in finite element approximation analysis of parabolic or hyperbolic PDE enabled by the Ritz/Galerkin approximation theory. In all those cases there might be quasi-optimal error estimates derived with respect to the expected convergence factor. However, applying the lemma of Gronwall to derive those estimates always require purely additional regularity assumptions, just for technical reasons. This is already the case for the simplest linear parabolic PDE, the heat equation. From a mathematical point of view the underlying handicap is just about the not appropriately applied norms. Mouhot C., Villani C., On Landau Damping

The note below is about the proposed (distributional) H(-1/2)- Hilbert space as the proposed appropriate Hilbert space frame for a variation representation of the nonlinear transport equation. The corresponding Galerkin-Ritz method is proposed to calculate corresponding (quasi-optimal) approximation solutions, e.g. with underlying boundary elements approximation spaces enabled by the wavelet analysis tool.

latest update: Feb 7, 2018
changes to previous versions since Jan 31, 2018: section 1 Braun K., Unusual Hilbert or Hoelder space frames for the elementary particles transport (Vlasov) equation

References

(EsG) Eskin G., Boundary Value Problems for Elliptic Pseudodifferential Equations, American Mathematical Society, Providence, Rhode Island, 1981

(JaE) Jaynes E. T., Information Theory and Statistical Mechanics, The Physical Review, Vol. 106, No 4, (1957) 620-630 Jaynes E. T., Information Theory and Statistical Mechanics

(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

(MaJ) Mashreghi J., Hilbert transform of logf

(PeB) Petersen B. E., Introduction to the Fourier Transform and Pseudo-Differential Operators, Pitman Advanced Publishing Program, Boston, London, Melbourne, 1983

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

(StE) Stein E. M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970        