- 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
- The third (Nernst distribution) law governs the
distribution of a solute between two non miscible solvents.
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.
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).
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
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.
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.
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.
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“.
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 equationis
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
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
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).
(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
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
J. A., Approximation Theory in Hilbert Scales
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