Lectures on Statistical Physics

Felix Berezin Lectures on Statistical Physics Translated from the Russian and edited by D. Leites Abdus Salam School of Mathematical Sciences Lahore, Pakistan Summary. The book is an edited and updated transcript of the lectures on Sta- tistical Physics by the founder of supersymmetry theory | Felix Berezin. The lectures read at the Department of Mechanics and Mathematics of Moscow State University in 1966{67 contain inimitable Berezin's uniform approach to the Bose and Fermi particles, and his unique approach to quantization. These approaches were used with incredible success in the study of solid state physics, disorder and chaos (K. Efetov), and in a recent (and not yet generally accepted claim of) solution of one of Clay's Millennium problems (A. Dynin). The book is addressed to mathematicians (from sophomore level up) and physicists of any level. ISBN 978-969-9236-02-0 °c Dimitry Leites, translation into English and editing, 2009. °c Felix Berezin; his heirs, 2009. Contents Author's Preface . v Editor's preface . vi Part I The Classical Statistical Physics 1 Background from Classical Mechanics . 3 1 The Ensemble of Microscopic Subsystems ::::::::::::::::: 9 2 Physical assumptions. Ergodic Hypothesis. Gibbs's distribution. 9 3 An heuristic deduction of the Gibbs distribution . 14 4 A complete deduction of the Gibbs distribution . 17 5 Relation to thermodynamics . 24 6 Properties of the entropy . 33 7 Analytical appendix to Chapter 1 . 39 2 Real gases ::::::::::::::::::::::::::::::::::::::::::::::::: 43 8 Physical assumptions . 43 9 The Gibbs Distribution in the Small Canonical Ensemble . 46 10 The correlation functions in the small ensemble . 50 11 The Bogolyubov equations . 55 12 The Gibbs distribution in the grand ensemble . 59 13 The Kirkwood{Salzburg equations . 64 14 A Relation between correlation functions . 71 15 The existence of potential in the grand ensemble . 73 16 Existence of the potential in the grand ensemble (cont.) . 75 17 Properties of the grand and small statistical sums . 80 18 The existence of the thermodynamic potential. 84 19 The mean over the distribution of the number of particles . 86 20 Estimates of the small statistical sum . 89 iv Contents Part II THE QUANTUM STATISTICAL PHYSICS 21 Background from Quantum Mechanics . 95 3 Ensemble of microscopic subsystems ::::::::::::::::::::::: 103 22 The mean with respect to time. The ergodic hypothesis. 103 23 The Gibbs distribution . 106 24 A relation to thermodynamics. Entropy . 110 4 Quantum gases :::::::::::::::::::::::::::::::::::::::::::: 117 25 The Method of second quantization . 117 26 Macroscopic subsystems . 125 27 The ideal Bose gas . 131 28 The ideal Fermi gas . 136 29 The BCS model of superconductivity . 140 30 A relation between the quantum and classical statistical physics150 Appendix :::::::::::::::::::::::::::::::::::::::::::::::::::::: 156 A Semi-invariants in the classical statistical physics . 157 B Continual integrals and the Green function . 161 C Review of rigorous results (R. A. Milnos) . 172 References ::::::::::::::::::::::::::::::::::::::::::::::::::::: 175 Index :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 181 Author's Preface I read this lecture course in 1966{67 at the Department of Mechanics and Mathematics of the Moscow State University. Currently, statistical physics is more than 90% heuristic science. This means that the facts established in this science are not proved, as a rule, in the mathematical meaning of this word although the arguments leading to them are quite convincing. A few rigorous results we have at the moment usually justify heuristic arguments. There is no doubt that this situation will continue until rigorous mathematical methods will embrace the only branch of statistical physics physicists did not represent with su±cient clarity, namely, the theory of phase transitions. I think that at the moment statistical physics has not yet found an ade- quate mathematical description. I believe, therefore, that the classical heuristic arguments developed, starting with Maxwell and Gibbs, did not lose their value. The rigorous results obtained nowadays are only worthy, I think, only if they can compete in the simplicity and naturalness with the heuristic arguments they are called to replace. The classical statistical physics has such results: First of all, these are theorems due to Van Hove, and Lee and Yang on the existence of thermodynamic potential and the Bogolyubov{Khatset{ Ruelle theorem on existence of correlation functions. The quantum statistical physics has no similar results yet. The above described point of view was the cornerstone of this course. I tried to present the material based on the classical heuristic arguments pay- ing particular attention to the logical sequence. The rigorous results are only given where they are obtained by su±ciently natural methods. Everywhere the rigorous theorems are given under assumptions that maximally simplify the proofs. The course is by no means an exhaustive textbook on statistical physics. I hope, however, that it will be useful to help the reader to compre- hend the journal and monographic literature. The initial transcription of the lectures was performed by V. Roitburd to whom I am sincerely thankful. Thanks are also due to R. Minlos, who wrote at my request Appendix C with a review of rigorous results. F. Berezin, 1972 Editor's preface It was for a long time that I wanted to see these lectures published: They are more lucid than other book on the same topic not only to me but to professional physicists, the ones who understand the importance of supersym- metries and use it in their everyday work, as well, see [KM]; they are also much shorter than any other text-book on the topic. The recent lectures by Minlos [M1] is the only exception and, what is unexpected and nice for the reader, these two books practically do not intersect and both are up to date. In a recent book by V. V. Kozlov1) [Koz], where in a lively and user-friendly way there are discussed the ideas of Gibbs and Poincar¶enot much understood (or not understood at all) during the past 100 years since they had been published, these notes of Berezin's lectures are given their due. In particular, Kozlov gives a reference (reproduced below) to an answer to one of Berezin's questions obtained 30 years after it was posed. About the same time that F. A. Berezin read this lecture course, he was close with Ya. G. Sinai; they even wrote a joint paper [BS]. As one can discern even from the above preface of usually very outwardly reserved and discreet author, Berezin depreciatingly looked at the mathematicians' activity aimed at construction of rigorous statistical physics. He always insisted on proving theorems that might have direct applications to physics. This attitude of Berezin to the general desire to move from problems of acute interest to physics towards abstract purely mathematical ones of doubt- ful (at least, in Berezin's view) physical value (we witness a similar trend of \searching the lost keys under the lamp-post, rather than where they were lost, because it it brighter there"in mathematical approaches to economics), the attitude that he did not hide behind politically correct words, was, perhaps, the reason of certain coldness of Berezin's colleagues towards these lectures. To my astonishment, publication of this book in Russian was not universally approved. One of the experts in the ¯eld wrote to me that this was unrealistic then and still is. The mathematical part of science took the road indicated by Dobrushin and Gri±ths who introduced the key notion | the limit Gibbs distribution | and then studying the phase transitions based on this notion, see [DLS]. It seems to this expert that this activity came past Berezin, although still during his lifetime (see [DRL, PS]). Berezin found an elegant approach to the deducing Onsager's expression for the 2D Ising model. He spent a lot of time and e®ort trying to generalize this approach to other models but without any (known) success. The 3D case remained unattainable, cf. [DP], where presentation is based on Berezin's ideas. 1 A remarkable mathematician, currently the director of Steklov Mathematical in- stitute of the Russian Academy of Sciences, Moscow. Editor's preface vii Some experts told me that the content of the lectures was rather standard and some remarks were not quite correct; Minlos's appendix is outdated and far too short. Sure. It remains here to make comparison with Minlos's vivid lectures [M1] more graphic. Concerning the standard nature of material, I'd like to draw attention of the reader to Berezin's inimitable uniform approach to the description of Bose and Fermi particles and its quantization, later developed in [B4, B5, B6]. Berezin's quantization is so un-like any of the mathematicians's descriptions of this physical term that for a decade remained unclaimed; but is in great de- mand nowadays on occasion of various applications. This part of the book | of considerable volume | is not covered (according to Math Reviews) in any of the textbooks on Statistical Physics published during the 40 years that had passed since these lectures were delivered. Close ideas were recently used with huge success in the study of the solid body [Ef]. Possible delusions of Berezin, if any are indeed contained in this book, are very instructive to study. As an illustration, see [Dy]. Finally, if these lectures were indeed \of only historical interest", then the pirate edition (not listed in the Bibliography) which reproduced the preprint (with new typos inserted) would had hardly appear (pirates are neither Sal- vation Army nor idiots, usually) and, having appeared, would not have im- mediately sell out. On notation. Although some of Berezin's notation is not conventional now (e.g., the English tr for trace pushed out the German sp for Spur) I always kept in mind, while editing, an anecdote on a discussion of F.

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