Thermodynamics, Gibbs Method and Statistical Physics of Electron Gases

Thermodynamics and statistical physics study the physical properties (mecha
nical, thermal, magnetic, optical, electrical, etc.) of the macroscopic system.
The tasks and objects of study in thermodynamics and statistical physics are
identical. However, the methods of investigation into macroscopic systems are
different.
Thermodynamics is a phenomenological theory. It studies the properties
of bodies, without going into the mechanism of phenomena, i.e., not taking
into consideration the relation between the internal structure of substance
and phenomena, it generalizes experimental results. As a result of such a gen
eralization, postulates and laws of thermodynamics made their appearance.
These laws make it possible to find general relations between the different
properties of macroscopic systems and the physical events occurring in them.
Statistical physics is a microscopic theory. On the basis of the knowledge of
the type of particles a system consists of, the nature of their interaction, and
the laws of motion of these particles issuing from the construction of substance,
it explains the properties being observed on experiment, and predicts the new
properties of systems. Using the laws of classical or quantum mechanics, and
also the theory of probability, it establishes qualitatively new statistical appro
priatenesses of the physical properties of macroscopic systems, substantiates
the laws of thermodynamics, determines the limits of their applicability, gives
the statistical interpretation of thermodynamic parameters, and also works
out methods of calculations of their means. The Gibbs method is based on
statistical physics. This method is the most canonical. Therefore, in this book,
the exposition of the Gibbs method takes an important place.
Results, stemming from phenomenological thermodynamics, bear the gen
eral character and can be applied to any macroscopic systems; however, the
internal mechanism of physical phenomena and properties, being observed
in the experiments, is not disclosed. In other words, thermodynamics only
describes the phenomena and establishes the relation between them, but does
not answer the question why it happens just so.
VI
Preface
Statistical physics relates the properties of bodies to their internal con
struction, creates the microscopic theory of physical phenomena, and answers
the question why it happens just so. The disadvantage of this method resides
in the fact that results, obtained here, bear a particular character and are
right only in frames of the considered model of the structure of substance.
Thermodynamics and statistical physics study not only equilibrium sys
tems, but also systems in which specified currents and flows (the electric
current, flow of energy and substance) exist. In this case, the theory is called
thermodynamics of non-equilibrium systems or kinetics. Kinetics originates
from the Boltzmann equation (1872) and has continued developing up to the
present time.
The development of phenomenological thermodynamics started in the first
half of the nineteenth century.
The first law of thermodynamics was discovered by the German phys
iologist Julius Robert von Mayer (1842) and the English physicist James
Prescott Joule (1843). They showed the equivalence of heat and mechanical
work. The first law of thermodynamics is a law of conservation of energy for
closed processes. In 1847, the German physicist and physiologist Hermann von
Helmholtz generalized this law for any non-closed thermodynamic processes.
The second law of thermodynamics was discovered independently by
both the German physicist Rudolf Clausius (1850) and the English physi
cist William Thomson (Lord Kelvin). They introduced in the theory a new
function of state– entropy, in the statistical sense, and discovered the law of
increasing entropy.
The third law of thermodynamics was discovered in 1906 by the German
physicist–chemist Walther Nernst. According to this law, entropy of all sys
tems independently of external parameters tends to the identical value (zero)
as temperature approaches the absolute zero.
Note that the first law of thermodynamics is a law about energy, and the
second and the third ones are about entropy.
The founders of thermodynamics are J.R. von Mayer, J.P. Joule, H. von
Helmholtz, R. Clausius, W. Kelvin, and W. Nernst.
Statistical physics received its development only in the last quarter of the
nineteenth century. The founders of classical statistical physics are R. Clau
sius, J.C. Maxwell, L. Boltzmann, and J.W. Gibbs. The height of development
of classical statistical physics is the method of Josiah Willard Gibbs (1902).
The application of classical statistics to many problems provided results,
though not coinciding with the experimental facts of that time. Black radiation
(thermodynamics of a photon gas), heat capacity of metals, Pauli paramag
netism, etc. can serve as examples. These difficulties of classical statistics
were circumvented only after the rise of quantum mechanics (L. de Broglie,
W. Heisenberg, E. Schr¨odinger, and P. Dirac) and quantum statistics, created
on its basis (E. Fermi, P. Dirac, S.N. Bose, A. Einstein) during 1924–1926.
The method of thermodynamic functions and potentials, and also the
Gibbs statistical method or the methods of free energy, being the key-note
Preface
VII
of the book, occupy an important place. It is shown that of all the thermo
dynamic functions, the most important are the function of free energy and
grand thermodynamic potential, which are determined from the Gibbs canon
ical distribution. It is expalined that the basic postulate of statistical physics–themicrocanonical distribution of isolated systems– is based on the statisti
cal theory of the macroscopic properties of a system, from which all canonical
distribution stems.
Understanding free energy and grand thermodynamic potential, it is easy
to determine entropy, thermal and caloric equations of state, and also all
thermodynamic coefficients, measured by testing. To do this in the case of
classical systems, it is sufficient to know the Hamilton function– energy as
a function of coordinates and impulses of particles of the system, forming
it, and for quantum systems, it is the energy spectrum, i.˚a., the dependence
of energy on quantum numbers. It is also an essence of the Gibbs method,
which is applied to ideal and non-ideal gases, and also to a crystalline solid.
The exposition of the Fermi-Dirac and Bose–Einstein quantum statistics and
its application to different quantum gases occupy a large place. It is shown
how the difficulties of classical statistics, associated with its application to an
electron gas in metals, are circumvented. The statistics of the electron gases
are considered in detail in this book.
A separate chapter is devoted to the statistical theory of thermodynamic
properties of an electron gas in a quantizing magnetic field. Note that the
investigation of properties of an electron gas in extremal conditions, in par
ticular, at ultra-low temperatures and in strong quantizing magnetic fields, is
one of the actual tasks of contemporary physics.
In the last chapter, on the basis of the Boltzmann kinetic equation, the
electron gas in metals and semiconductors is considered in a nonequilibrium
state. Nonequilibrium processes are associated with charge carrier motion in
a crystal under external disturbances such as the electric field and the tem
perature gradient in the magnetic field. They include electric conductivity,
thermoelectric, galvanomagnetic, and thermomagnetic effects