R. Liboff – Kinetic Theory (Third edition)

Автор: R. Liboff
Название книги: Kinetic Theory
Формат: PDF
Жанр: Физика
Страницы: 592
Качество: Изначально компьютерное, E-book

This book goes beyond the scope of other works in the field with its thorough treatment of applications in a wide variety of disciplines. The third edition features a new section on constants of motion and symmetry and a new appendix on the Lorentz-Legendre expansion.

Since the first edition of this work, kinetic theory has maintained its position as
a cornerstone of a number of disciplines in science and mathematics. In physics,
such is the case for quantum and relativistic kinetic theory. Quantum kinetic
theory finds application in the transport of particles and radiation through
material media, as well as the non-stationary quantum–many-body problem.
Relativistic kinetic theory is relevant to controlled thermonuclear fusion and to
a number of problems in astrophysics. In applied mathematics, kinetic theory
relates to the phenomena of localization, percolation, and hopping, relevant to
transport properties in porous media. Classical kinetic theory is the foundation
of fluid dynamics and thus is important to aerospace, mechanical, and chemical
engineering. Important to the study of transport in metals is the Lorentz–
Legendre expansion, which in this new edition appears in an appendix. A new
section in Chapter 1was included in this newedition that addresses constants of
motion and symmetry.Anumber of small but important revisions were likewise
made in this new edition. A more complete description of the contents of the
text follows.
The text comprises seven chapters. In Chapter 1, the transformation theory
of classical mechanics is developed for the purpose of deriving Liouville’s theorem
and the Liouville equation. Four distinct interpretations of the solution
to this equation are presented. The fourth interpretation addresses Gibbs’s notion
of a distribution function that is the connecting link between the Liouville
equation and experimental observation. The notion of a Markov process is
discussed, and the central-limit theorem is derived and applied to the random
walk problem.
In Chapter 2, the very significant BBKGY hierarchy is obtained from the
Liouville equation, and the first two equations of this sequence are applied in
the derivation of conservation of energy for a gas of interacting particles. In nondimensionalizing this sequence, parameters emerge that differentiate between
weakly and strongly coupled fluids. Correlation functions are introduced
through the Mayer expansions. Examining a weakly coupled fluid composed
of particles interacting under long-range interaction leads to the Vlasov equation
and the closely allied concept of a self-consistent solution. Prigogine’s
diagrammatic technique and related operator formalism for examining the
Liouville equation are described. The Bogoliubov ansatz concerning the equilibration
of a gas, as well as the Klimontovich formulation of kinetic theory,
are also included in this chapter.
The Boltzmann equation is derived in Chapter 3 and applied to the derivation
of fluid dynamic equations and the H theorem. Poincar´e’s recurrence theorem
is proved and is discussed relative to Boltzmann’sHtheorem. Transport coefficients
are defined, and the Chapman–Enskog expansion is developed. Results
of this technique of solution to the Boltzmann equation are compared with
experimental data and are found to be in good agreement for various molecular
samples. Grad’s method of solution of the Boltzmann equation involving
expansion in tensor Hermite polynomials is described. The chapter continues
with a derivation of the Druyvesteyn distribution relevant to a current carrying
plasma in a dc electric field. In the last section of the chapter, the topic of irreversibility
is revisited. Ergodic and mixing flows are discussed. Action-angle
variables are introduced, and the notions of classical degeneracy and resonant
domains in phase space are described in relation to the chaotic behavior of
classical systems.Astatement of the closely alliedKAMtheorem is also given.
In the first half of Chapter 4, the Vlasov equation is applied to linear
wave theory for a two-component plasma composed of electrons and heavy
ions. Landau damping and the Nyquist criterion for wave instabilities are
described. The chapter continues with derivations of other important kinetic
equations: Krook–Bhatnager–Gross (KBG), Fokker–Planck, Landau, and
Balescu–Lenard equations. A table is included describing the interrelation of
the classical kinetic equations discussed in the text. The chapter concludes with
a description of the widely used Monte Carlo numerical analysis in kinetic
theory.
Quantum kinetic theory is developed in Chapter 5. A brief review of basic
principles leads to a description of the density matrix, the Pauli equation,
and the closely related Wigner distribution. Various equivalent forms of the
Wigner–Moyal equation are derived. A quantum modified KBG equation is
applied to photon transport and electron propagation in solids. Thomas–Fermi
screening and the Mott transition are also discussed. The Uehling–Uhlenbeck
quantum modified Boltzmann equation is developed and applied to a Fermi
liquid. The chapter continues with an overview of classical and quantum hierarchies
of equations connecting reduced distributions. A table of hierarchies
is included where the reader is easily able to view distinctions among these
sets of equations. The Kubo formula, described previously in Chapter 3, is
revisited and applied to the derivation of a quantum expression for electrical conductivity. The chapter concludes with an introduction to Green’s function
analysis and related diagrammatic representations.
Chapter 6 addresses relativistic kinetic theory. The discussion begins with elementary
concepts, including a statement of Hamilton’s equations in covariant
form. Stemming from a covariant distribution function in four space, together
with Maxwell’s equations in covariant form, a relativistic Vlasov equation
is derived for a plasma in an electromagnetic field. An important component
of this chapter is the derivation and compilation of a table of Lorentz
invariants in kinetic theory. The chapter continues with a derivation of the
relativistic Maxwellian and concludes with a brief description of relativity in
non-Cartesian coordinates.
Chapter 7 has been added in this new edition, and addresses kinetic and
thermal properties of metals and amorphous media. The first component
of the chapter begins with a review of the notion of thermopower, and the
Wiedermann–Franz law is derived. The discussion continues with a formulation
for electrical and thermal conductivity in metals (encountered previously
in Chapter 5), stemming from the quantum Boltzmann equation, in which
Bloch’s classic low-temperature T 5 dependence of metallic resistivity and
canonical high-temperature linear T dependence are derived. In addition, the
formalism yields a residual resistivity at 0 K. The chapter continues with a
discussion of properties of amorphous media and related processes of localization,
hopping, and percolation. Bloch waves and the notion of extended
states are reviewed. Anderson’s parameter of the ratio of the spread-of-states
to the band width is introduced. Localization occurs at some critical value of
this parameter. At smaller values of the parameter, energies of localized and
extended states are separated at the “mobility edge.” Transition of the Fermi
energy from the domain of extended states to the domain of localized states
represents the Mott metal–insulator transition. Mechanisms of electrical conduction
are discussed in three temperature intervals in which the notions of
thermally assisted and variable-range hopping emerge. The chapter continues
with the concepts of bond and site percolation. A number of percolation scaling
laws are discussed. The chapter concludes with a review of localization
in second quantization. Throughout the chapter, many discussions related to
material science are included.
Each chapter is preceded by a brief introductory statement of the subject
matter contained in the chapter. Problems appear at the end of each chapter,
many of which carry solutions. A number of problems include self-contained
descriptions of closely allied topics. In such cases, these are listed in the chapter
table of contents under the heading, Topical Problems. In addition to references
cited in the text, a comprehensive list of references is included in Appendix E.
Assorted mathematical formulas are included in Appendices A and B, including
a list of properties of Laguerre and Hermite polynomials (B4). Appendix
D, addressing the Lorentz–Legendre expansion in kinetic theory, is new to this
edition. Stemming from the observation that science and society are inextricably
entwined, a time chart is included (Appendix E) listing early contributors to
science and technology of the classical Greek and Roman eras. The reader will
note that a central figure in this display is the Greek philosopher, Democritus,
who, at about 400 BCE, was the first to propose an atomic theory of matter.
Readers of my earlier work [Introduction to the Theory of Kinetic Equations,
Wiley, NewYork (1969)] will recall that it, too, included a time chart describing
contributions to dynamics from the fifteenth to the nineteenth centuries. The
appendix on Mathematical Formulas has been expanded in this new edition to
include a list of properties of Laguene and Hermite polynomials (Appendix
B4).
Many individuals have contributed to the development of thiswork. I remain
indebted to these kind colleagues and would like here to express my sincere
gratitude for their encouragement, support, and constructive criticism: Sidney
Leibovich, Terrence Fine, Robert Pay, Christof Litwin, Kenneth Gardner, Neal
Maresca, K. C. Liu, Danny Heffernan, Edwin Dorchek, Philip Bagwell, Ronald
Kline, Steve Seidman, S. Ramakrishna, G. George, Timir Datta,William Morrell,
Wayne Scales, Daniel Koury, Erich Kunhardt, Marvin Silver, Hercules
Neves, James Hartle,Kenneth Andrews, Clifford Pollock,Veit Elser, Chuntong
Ying, Michael Parker, Jack Freed, Richard Zallen, Abner Shimony, Philip
Holmes, Lloyd Hillman, Arthur Ruoff, L. PearceWilliams, Lloyd Motz, John
Guckenheimer, Isaac Rabinowitz, Gregory Schenter, and Ilya Prigogine.
Some of these individuals are former students. It is due to my association
with these gifted and talented colleagues that the talmudic inscription for this
work is motivated.