Michael V. Sadovskii Statistical Physics Texts and Monographs in Theoretical Physics | Edited by Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA Michael V. Sadovskii Statistical Physics | 2nd edition Physics and Astronomy Classification 2010 05.20.-y, 05.20.Dd, 05.20.Gg, 05.30.-d, 05.30.Ch, 05.30.Fk, 05.30.Pr, 05.70.Ph, 68.18.Jk, 68.18.Ph Author Prof. Dr. Michael V. Sadovskii Russian Academy of Sciences Institute for Electrophysics Amundsenstreet 106 Ekaterinburg 620016 Russia [email protected] ISBN 978-3-11-064510-1 e-ISBN (PDF) 978-3-11-064848-5 e-ISBN (EPUB) 978-3-11-064521-7 ISSN 2627-3934 Library of Congress Control Number: 2019930858 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Cover image: EQUINOX GRAPHICS / SCIENCE PHOTO LIBRARY Typesetting: VTeX UAB, Lithuania Printing and binding: CPI Books GmbH, Leck www.degruyter.com Preface This book is essentially based on the lecture course on “Statistical Physics”, that was taught by the author at the physical faculty of the Ural State University in Ekaterinburg since 1992. This course was intended for all physics students, not especially for those specializing in theoretical physics. In this sense the material presented here contains the necessary minimum of knowledge of statistical physics (also often called statisti- cal mechanics), which is in the author’s opinion necessary for every person wishing to obtain a general education in the field of physics. This posed the rather difficult prob- lem of the choice of material and appropriately compact presentation. At the same time, it necessarily should contain all the basic principles of statistical physics, as well as its main applications to various physical problems, mainly from the field of the theory of condensed matter. Extended version of these lectures were published in Russian in 2003. For the present English edition, some of the material was rewritten, and several new sections and paragraphs were added, bringing the contents more up to date and adding more discussion on some more difficult cases. Of course, the au- thor was much influenced by several classical books on statistical physics [19, 20,37], and this influence is obvious in many parts of the text. However, the choice of material and the form of presentation is essentially his own. Still, most attention is devoted to rather traditional problems and models of statistical physics. One of the few excep- tions is an attempt to present an elementary and short introduction to the modern quantum theoretical methods of statistical physics at the end of the book. Also, a lit- tle bit more attention than usual is given to the problems of nonequilibrium statistical mechanics. Some of the more special paragraphs, of more interest to future theorists, are denoted by asterisks or moved to the appendices. Of course, this book is too short to give a complete presentation of modern statistical physics. Those interested in further developments should address more fundamental monographs and modern physical literature. The second edition of this book has been expanded with boxes presenting brief summaries of the lives and achievements of the major founders and contributors to the field of “Statistical Physics”. The biographical details complement the scientific content of the book and contextualize the discoveries within the framework of global research in Theoretical Physics. In my personal opinion, this information can be useful for readers and lecturers alike. Ekaterinburg, 2018 M. V. Sadovskii https://doi.org/10.1515/9783110648485-201 Contents Preface | V 1 Basic principles of statistics | 1 1.1 Introduction | 1 1.2 Distribution functions | 2 1.3 Statistical independence | 7 1.4 Liouville theorem | 9 1.5 Role of energy, microcanonical distribution | 13 ∗ 1.6 Partial distribution functions | 17 1.7 Density matrix | 21 1.7.1 Pure ensemble | 22 1.7.2 Mixed ensemble | 24 1.8 Quantum Liouville equation | 27 1.9 Microcanonical distribution in quantum statistics | 28 ∗ 1.10 Partial density matrices | 30 1.11 Entropy | 33 1.11.1 Gibbs entropy. Entropy and probability | 33 1.11.2 The law of entropy growth | 36 2 Gibbs distribution | 45 2.1 Canonical distribution | 45 2.2 Maxwell distribution | 50 2.3 Free energy from Gibbs distribution | 53 2.4 Gibbs distribution for systems with varying number of particles | 54 2.5 Thermodynamic relations from Gibbs distribution | 57 3 Classical ideal gas | 63 3.1 Boltzmann distribution | 63 3.2 Boltzmann distribution and classical statistics | 64 3.3 Nonequilibrium ideal gas | 66 3.4 Free energy of Boltzmann gas | 69 3.5 Equation of state of Boltzmann gas | 70 3.6 Ideal gas with constant specific heat | 72 3.7 Equipartition theorem | 74 3.8 One-atom ideal gas | 75 4 Quantum ideal gases | 79 4.1 Fermi distribution | 79 4.2 Bose distribution | 81 4.3 Nonequilibrium Fermi and Bose gases | 82 VIII | Contents 4.4 General properties of Fermi and Bose gases | 84 4.5 Degenerate gas of electrons | 87 ∗ 4.6 Relativistic degenerate electron gas | 90 4.7 Specific heat of a degenerate electron gas | 91 4.8 Magnetism of an electron gas in weak fields | 93 ∗ 4.9 Magnetism of an electron gas in high fields | 97 4.10 Degenerate Bose gas | 99 4.11 Statistics of photons | 102 5 Condensed matter | 107 5.1 Solid state at low temperature | 107 5.2 Solid state at high temperature | 110 5.3 Debye theory | 111 5.4 Quantum Bose liquid | 115 5.5 Superfluidity | 119 ∗ 5.6 Phonons in a Bose liquid | 124 5.7 Degenerate interacting Bose gas | 127 5.8 Fermi liquids | 132 ∗ 5.9 Electron liquid in metals | 137 6 Superconductivity | 141 6.1 Cooper instability | 141 6.2 Energy spectrum of superconductors | 145 6.3 Thermodynamics of superconductors | 154 ∗ 6.4 Coulomb repulsion | 157 6.5 Ginzburg–Landau theory | 161 7 Fluctuations | 171 7.1 Gaussian distribution | 171 7.2 Fluctuations in basic physical properties | 175 7.3 Fluctuations in ideal gases | 179 8 Phase transitions and critical phenomena | 183 8.1 Mean-field theory of magnetism | 183 ∗ 8.2 Quasi-averages | 190 8.3 Fluctuations in the order parameter | 194 8.4 Scaling | 200 9 Linear response | 211 9.1 Linear response to mechanical perturbation | 211 9.2 Electrical conductivity and magnetic susceptibility | 217 9.3 Dispersion relations | 221 Contents | IX 10 Kinetic equations | 227 10.1 Boltzmann equation | 227 10.2 H-theorem | 233 ∗ 10.3 Quantum kinetic equations | 235 10.3.1 Electron–phonon interaction | 236 10.3.2 Electron–electron interaction | 241 11 Basics of the modern theory of many-particle systems | 243 11.1 Quasi-particles and Green’s functions | 243 11.2 Feynman diagrams for many-particle systems | 253 11.3 Dyson equation | 256 11.4 Effective interaction and dielectric screening | 260 11.5 Green’s functions at finite temperatures | 263 A Motion in phase space, ergodicity and mixing | 267 A.1 Ergodicity | 267 A.2 Poincare recurrence theorem | 273 A.3 Instability of trajectories and mixing | 276 B Statistical mechanics and information theory | 279 B.1 Relation between Gibbs distributions and the principle of maximal information entropy | 279 B.2 Purging Maxwell’s “demon” | 284 C Nonequilibrium statistical operators | 291 C.1 Quasi-equilibrium statistical operators | 291 C.2 Nonequilibrium statistical operators and quasi-averages | 295 Bibliography | 299 Index | 301 1 Basic principles of statistics We may imagine a great number of systems of the same nature, but differing in the configurations and velocities which they have at a given instant, and differing not merely infinitesimally, butit may be so as to embrace every conceivable combination of configuration and velocities. And here we may set the problem not to follow a particular system through its succession of configurations, but to determine how the whole number of systems will be distributed among the various conceivable configurations and velocities at any required time, when the distribution hasbeen given at some specific time. The fundamental equation for this inquiry is that which gives therate of change of the number of systems which fall within any infinitesimal limits of configuration and velocity. Such inquiries have been called by Maxwell statistical. They belong to a branch of mechanics which owes its origin to the desire to explain the laws of thermodynamics on mechanical principles, and of which Clausius, Maxwell and Boltzmann are to be regarded as principal founders. J. Willard Gibbs, 1902 [11] 1.1 Introduction Traditionally, statistical physics (statistical mechanics) deals with systems consisting of large numbers of particles, moving according to the laws of classical or quantum mechanics. Historically it evolved, by the end of 19th century, from attempts to pro- vide mechanistic derivation of the laws of thermodynamics in the works by J. Maxwell and L. Boltzmann. The formalism of statistical mechanics was practically finalized in the fundamental treatise by J. W. Gibbs [11], which appeared at the beginning of the 20th century. The remarkable advantage of Gibbs method, which was created long before the appearance of modern quantum theory, is its full applicability to the stud- ies of quantum (many-particle) systems. Nowadays, statistical physics has outgrown the initial task of justification of thermodynamics, its methods and ideology actually penetrating all the basic parts of modern theoretical physics. Still being understood mainly as the theory of many (interacting) particle systems, it has deep connections with modern quantum field theory, which is at present the most fundamental theory of matter. At the same time, it is now also clear that even the description of mechanical motion of relatively few particles moving according to the laws of classical mechan- ics often requires the use of statistical methods, as this motion, in general (nontrivial) cases, is usually extremely complicated (unstable).