Statistical Mechanics

This book deals with statistical mechanics. Its goal is to give a deductive
presentation of the statistical mechanics of equilibrium systems based on a
single hypothesis– the form of the microcanonical density matrix– as well
as to treat the most important aspects of non-equilibrium phenomena. Be
yond the fundamentals, the attempt is made here to demonstrate the breadth
and variety of the applications of statistical mechanics. Modern areas such
as renormalization group theory, percolation, stochastic equations of motion
and their applications in critical dynamics are treated. A compact presenta
tion was preferred wherever possible; it however requires no additional aids
except for a knowledge of quantum mechanics. The material is made as un
derstandable as possible by the inclusion of all the mathematical steps and
a complete and detailed presentation of all intermediate calculations. At the
end of each chapter, a series of problems is provided. Subsections which can
be skipped over in a first reading are marked with an asterisk; subsidiary
calculations and remarks which are not essential for comprehension of the
material are shown in small print. Where it seems helpful, literature cita
tions are given; these are by no means complete, but should be seen as an
incentive to further reading. A list of relevant textbooks is given at the end
of each of the more advanced chapters.
In the first chapter, the fundamental concepts of probability theory and
the properties of distribution functions and density matrices are presented. In
Chapter 2, the microcanonical ensemble and, building upon it, basic quan
tities such as entropy, pressure and temperature are introduced. Following
this, the density matrices for the canonical and the grand canonical ensemble
are derived. The third chapter is devoted to thermodynamics. Here, the usual
material (thermodynamic potentials, the laws of thermodynamics, cyclic pro
cesses, etc.) are treated, with special attention given to the theory of phase
transitions, to mixtures and to border areas related to physical chemistry.
Chapter 4 deals with the statistical mechanics of ideal quantum systems, in
cluding the Bose–Einstein condensation, the radiation field, and superfluids.
In Chapter 5, real gases and liquids are treated (internal degrees of free
dom, the van der Waals equation, mixtures). Chapter 6 is devoted to the
subject of magnetism, including magnetic phase transitions. Furthermore,
related phenomena such as the elasticity of rubber are presented.
deals with the theory of phase transitions and critical phenomena; following
a general overview, the fundamentals of renormalization group theory are
given. In addition, the Ginzburg–Landau theory is introduced, and percola
tion is discussed (as a topic related to critical phenomena). The remaining
three chapters deal with non-equilibrium processes: Brownian motion, the
Langevin and Fokker–Planck equations and their applications as well as the
theory of the Boltzmann equation and from it, the H-Theorem and hydrody
namic equations. In the final chapter, dealing with the topic of irreversiblility,
fundamental considerations of how it occurs and of the transition to equilib
rium are developed. In appendices, among other topics the Third Law and a
derivation of the classical distribution function starting from quantum statis
tics are presented, along with the microscopic derivation of the hydrodynamic
equations.
The book is recommended for students of physics and related areas from
the 5th or 6th semester on. Parts of it may also be of use to teachers. It is
suggested that students at first skip over the sections marked with asterisks or
shown in small print, and thereby concentrate their attention on the essential
core material.
This book evolved out of lecture courses given numerous times by the au
thor at the Johannes Kepler Universit¨ at in Linz (Austria) and at the Technis
che Universit¨at in Munich (Germany). Many coworkers have contributed to
the production and correction of the manuscript: I. Wefers, E. J¨ org-M¨uller,
M. Hummel, A. Vilfan, J. Wilhelm, K. Schenk, S. Clar, P. Maier, B. Kauf
mann, M. Bulenda, H. Schinz, and A. Wonhas. W. Gasser read the whole
manuscript several times and made suggestions for corrections. Advice and
suggestions from my former coworkers E. Frey and U.C. T¨auber were likewise
quite valuable. I wish to thank Prof. W.D. Brewer for his faithful translation
of the text. I would like to express my sincere gratitude to all of them, along
with those of my other associates who offered valuable assistance, as well as
to Dr. H.J. K¨ olsch, representing the Springer-Verlag