Friedrich Schlogl Probability and Heat
Friedrich Schlogl Probability and Heat Fundamentals of Thermostatistics With 52 Figures Springer Fachmedien Wiesbaden GmbH
AJlrjghtsreserved Springer Fachmedien Wiesbaden 1989 Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft mbh, Braunschweig in 1989 Softcover reprint of the hardcover 1 st edition 1989 No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical photocopying or otherwise, without prior permission of the copyrjght holder. Set by Vieweg, Braunschwejg Bound by W. Langeltiddecke, Braunschwejg ISBN 978-3-528-06343-6 ISBN 978-3-663-13977-5 (ebook) DOI 10.1007/978-3-663-13977-5
v Contents Preface and Introduction...................................... IX How to Read this Book....................................... XII 1 General Statistics.... 1.1 Probability............................................ 1.1.1 Events.......................................... 2 1.1.2 Definitions of Probability.............................. 5 1.1.3 Random Quantities.................................. 10 1.1.4 Moments and Cumulants.............................. 12 1.1.5 The Normal Distribution.............................. 14 * 1.1.6 The Central Limit Theorem............................ 17 1.2 lnfonnation Measures..................................... 19 1.2.1 Shannon Information................................ 19 1.2.2 Information Gain................................... 23 * 1.2.3 Stochastic Matrices.................................. 25 1.3 Generalized Canonical DistnlJUtions............................ 27 1.3.1 The Unbiased Guess................................. 27 1.3.2 Properties of the Generalized Canonical distribution............ 30 1.3.3 Addition of Knowledge............................... 33 2 Thermodynamics of Equilibria................................ 36 2.1 Thennal States.......................................... 37 2.1.1 Direction of Time................................... 37 2.1.2 The Liouville Theorem............................... 39 2.1.3 Equilibrium Distributions.............................. 41 *2.1.4 Statistical Operators in Quantum Mechanics................. 45 *2.1.5 The Wigner Function... :............ 52 2.1.6 Thermal Variables................................... 55 2.2 Statistical Foundations of the Macroscopic Scheme.................. 57 2.2.1 The Second Law of Thermodynamics...................... 58 2.2.2 Work and Heat..................................... 60 2.2.3 Standard Distributions................................ 63 2.2.4 Extensities as Sharp Parameters.......................... 66 2.2.5 The Carnot Cycle................................... 70 * By an asterisc are designated parts of more special interest. They can be skipped by the reader without lost of continuity.
VI Contents 2.3 The Phenomenological Framework............................ 73 2.3.1 The Basic Laws of Thermodynamics....................... 73 2.3.2 Definition of Absolute Temperature....................... 78 2.3.3 Entropy in Phenomenology............................ 79 2.3.4 Thermodynamic Potentials............................. 83 2.3.5 Thermal and Caloric Equations of State.................... 87 2.3.6 Availability....................................... 90 2.3.7 Equilibrium Conditions............................... 95 2.3.8 Stability Relations.................................. 103 2.3.9 Specific Heat...................................... 106 2.4 The Low Temperature Regime............................... 109 2.4.1 The Nernst Theorem................................. 110 2.4.2 Characteristic Quantities at Zero Point..................... 111 2.4.3 Unattainability of Absolute Zero Point..................... 114 *2.4.4 Reactions at Low Temperature.......................... 116 3 Macroscopic Description of Special Systems..................... 118 3.1 Gases and Solutions...................................... 118 3.1.1 Ideal Gases and Dilute Solutions......................... 119 3.1.2 MixturesofldealGases... 121 3.1.3 Ideal Mixtures..................................... 124 3.1.4 Real Gases........................................ 127 3.2 Chemical Reactions....................................... 132 3.2.1 Reaction Heat..................................... 132 3.2.2 Chemical Affinity................................... 134 3.2.3 The Law of Mass Action.............................. 136 3.2.4 Electro-Chemical Potentials............................ 140 3.3 The Method of Cycle Processes............................... 142 3.3.1 An Isothermal Cycle................................. 142 3.3.2 Vapor Pressure and Radiation Cavity...................... 144 4 Microscopic Description of Special Systems...................... 147 4.1 Thermal Equations of State................................. 147 4.1.1 Ideal Gas......................................... 148 4.1.2 Grand Canonical Ensemble............................. 151 4.1.3 Real Gases........................................ 154 *4.1.4 Cell Model of a Liquid................................ 157 4.1.5 Perfect Bose and Fermi Gas............................ 159 4.1.6 The Radiation Cavity................................ 166 * By an asterisc are designated parts of more special interest. They can be skipped by the reader without lost of continuity.
Contents VII 4.2 Specific Heat........................................... 168 4.2.1 Specific Heat ofldea1 Gases............................ 169 4.2.2 Ortho- and Para-Hydrogen............................. 172 4.2.3 Specific Heat of Solids................................ 174 4.3 Magnetism............................................. 178 4.3.1 Paramagnetism..................................... 179 4.3.2 Weiss Theory of Ferromagnetism......................... 181 *4.3.3 The/singModel... 184 *4.3.4 The Long Range Order............................... 187 *4.3.5 Bethe-Peierls Approximation........................... 190 5 N onequilibria............................................. 194 5.1 Thermal Fluctuations..................................... 195 5.1.1 Fluctuations and Susceptibilities......................... 195 5.1.2 Probability of Fluctuations............................. 199 5.1.3 Applications of Einstein's Fluctuation Formula............... 202 *5.1.4 Maxwell's Demon................................... 205 5.2 Nonequilibrium Dynamics.................................. 206 * 5.2.1 The Liouville Operator............................... 207 5.2.2 TheMori Distribution... 212 *5.2.3 The Wangsness-Bloch Equation.......................... 215 *5.2.4 Thermodynamic Uncertainty Relation..................... 217 5.3 Linear Thermodynamics................................... 219 5.3.1 Fluxes and Forces................................... 220 5.3.2 Onsager Symmetry.................................. 222 5.3.3 Entropy Production................................. 223 5.3.4 Heat Conduction................................... 225 5.3.5 Thermodiffusion................................... 226 5.3.6 The Einstein Relation of Diffusion........................ 228 *5.3.7 Einstein's Deduction of Planck's Radiation Formula............ 229 *5.4 A Model of Time Scale Separation............................. 231 *5.4.1 Autocorrelations in the Harmonic Chain.................... 232 *5.4.2 Causal Functions................................... 233 *5.4.3 The Macroscopic Motion.............................. 234 Index... 242 * By an asterisc are designated parts of more special interest. They can be skipped by the reader without lost of continuity.
IX Preface and Introduction This book is based on lectures for graduate students of physics and physical chemistry. Its main aim is to represent the connections between the microdynamics of molecules and quanta with the macroscopic thermodynamics. There are many excellent textbooks on thermodynamics, and the question arises why another one should be written. In no other field of physics we can find as many different possible ways of introducing the theory as in thermodynamics. The reason is not only that here we have to distinguish between a macroscopic and a microscopic description of the same phenomena. The concept of probability in the statistical theory of thermodynamics introduces a fundamentally new element into physics. Between experience and the conventional theories, the use of probability was an intermediate new methodical element which did not occur in the fundamentals of classical physics before the development of thermodynamics. Originally, probability was not used in pure macroscopic thermodynamics, which had developed independently into a rather closed theory based on its own set of postulates. After further development, however, it became possible to connect these postulates with other fields of physics by deducing thermodynamics from statistical mechanics. As a result of this connection it is not surprising that macroscopic thermodynamics is less concrete and less transparent than other fields of classical physics, notwithstanding that, as a rule, understanding it does not require difficult calculus. Such notions as "entropy", "enthalpy", and their relatives, are more abstract and less directly connected with experience or pictures than most of the basic notions in other fields. Even the restriction that we can apply the rather elementary notion "temperature" only to certain kinds of states of a system, does not support the belief that, apart from the subtelities of time averaging, existence of equilibrium, and other conditions required for its definition, temperature is one of the simplest concepts in physics. Only because temperature is associated with a sensual perception, we accept it as familiar to us. The use of probability in physics requires certain methodological assumptions which, strictly speaking, are not based on experience, but which are necessary to organize experience. As in most cases in which such assumptions are introduced into thermodynamics without clear separation from empirical elements, they are often associated with conceptual difficulties. A central aim of this book is to overcome these difficulties in a specific way. The book strives for a clear separation of the physical fundamentals of thermodynamics from the general stochastic fundamentals, which in principle do not need to be restricted to their application in physics. This procedure, in a consistent way, is different from that of the usual procedure of textbooks on thermodynamics. It is chosen only because the author believes that it makes things simpler and more transparent in statistical thermodynamics. Indeed, many well-known basic relations of thermodynamics will turn out to be results already of the general stochastic theory. This is the reason for extensive analogies to thermodynamic concepts and connections occurring in modern theories of computer simulation of physical and nonphysical processes. We shall not deal with this field in this book. The above mentioned systematic separation of the different fundamentals, however, will also be useful for understanding the origin of these analogies.
X Preface and Introduction The general stochastic fundamentals comprise more than the mathematical aspects of probability theory. They also include methods of applying probability theory to experience in an appropriate way. Therefore, in the first section of this book, which is devoted to the stochastic fundamentals, we shall also be concerned with elements of information theory. The main objective of this book is to analyse the connection between thermodynamics and the dynamics on the level of the most detailed description, in the "phase space". Statistics as the foundation of thermodynamics is here understood to be statistics over microstates in phase space. This topical restriction implies that generally we shall not use statistics on a "mesoscopic" level as an ad hoc assumption and as a starting point in its own rights. Due to this restriction of the book we shall not be concerned with the Boltzmann collision equation, with the master equation, with Langevin forces, or with stochastic processes, albeit these methods have proved to be very successful in nonequilibrium thermodynamics. There we refer to expositions existing in the rich literature on the statistical theory of nonequilibrium processes. Mesoscopic statistics will occur only as result of the basic statistics over the microstates. Another restriction is that we consider nonequilibrium processes only if they occur in the regime of "linear thermodynamics". Hence we shall not include the steadily expanding field of modern research of "nonlinear thermodynamics" because this would lead too far. Although the major concern of the book is the foundation of thermodynamics on the statistical theory, the macroscopic theory will not be represented only as result of the statistics. The macroscopic theory will be developed once again and independently within the pure macroscopic, phenomenological framework. This is done because this framework is an impressive, logically closed system which can be understood without complicated mathematics. The macroscopic theory is moreover a very useful tool for applications which can also be handled without knowledge of the statistical theory. It should be stressed that the simplicity aspired to in the representation of the basic postulates of the phenomenological thermodynamics is not primarily that of logical economy with respect to the independence of basic axioms. It is rather the simplicity of an utmost transparent connection of the basic assumptions with experience on the one hand, and with their consequences on the other hand. Thus the book does not compete with representations of the mathematical axiomatics of phenomenological thermodynamics, which developed into a separate field of research. The book may differ from the conventional representations in textbooks not only in the previously mentioned striving for a clear separation of methodical from empirical elements but also in other points. This is the case, for instance, with the interpretation of specific heat as statistical measure in chapter 2.3.10, together with the chapters 5.1.2, and 5.2.4. Another point may be the accent placed on the significance of "information gain" and its correlates "availability" and "produced entropy" in thermodynamics. These concepts elucidate some connections more clearly and at times allow for shorter deductions. In the last subsection 5.4, which has rather the character of an appendix, a model system is discussed which at first sight may look a bit far afield from the primary theme of this book. Nevertheless, this subsection deals with two central questions in thermodynamics. The first one is how a mechanical system with reversible dynamics on the microscopic level can exhibit irreversible macroscopic dynamics. The second question is how the motion of such a system can develop a distinctive time scale separation between microscopic and macroscopic processes. These questions cannot be answered analytically in a general way. For the considered model, however, a rigorous solution of these problems is possible.
Preface and Introduction XI The book is meant to be primarily a textbook. Therefore most of the examples of special thermodynamic systems are standards in different fields. The number of the examples, however, is restricted. For more applications we refer the reader to the rich literature on thermodynamics. The author hopes that the book, which is concerned above all with basic connections, will be interesting not only for students but also for academic teachers and other scientists who like the structural analysis of fundamentals in physics. According to the character of a textbook, this book is not intended to demonstrate new results. Nevertheless, the way of the logical deductions, and of the presentations used in this book, as well as the choice of illustrating examples are not only influenced by literature but also by discussions with colleagues and friends. In this respect I should like to mention the Professors A. Stahl, J. Meixner, R. Bausch, H.-K. Janssen, R. Bessenrodt, Dr. E. Scholl, and Dr. C. Escher in Germany, as well as Professor C. A. Mead in Minneapolis, and Professor R. St. Berry in Chicago. Particular thank is directed to Professor V. Dohm for critically reading certain parts of the manuscript and making valuable proposals for improvements. Above all I should like to mention my unforgettable late academic teacher Professor Richard Becker in Gottingen, who, now half a century ago, first raised my love for this field. I gratefully acknowledge the help of my niece Dr. Aenne Hannon, Houston, Texas, in reducing the linguistic shortcomings of the manuscript I had written in a language which is not my native tongue. Last but not least, I want to thank the publisher, whose suggestion and interest made this book possible. Aachen, December 1988 Friedrich Schlag!
XII How to Read this Book The book is divided into main sections designated by only one serial number, into subsections designated by two numbers, and into chapters designated by three numbers. The first number always marks the main section, the second the subsection, and the third the chapter. Within a subsection, the equations are designated by two numbers, the first corresponding to the chapter, the second to the equation itself. Footnotes and figures are designated by serial numbers running continuously through the whole book. The reader who is familiar with probability theory may skip subsection 1.1 and use this part only for occasional reference. The same is true with respect to information theory for subsection 1.2. Subsection 1.3, however, is essential for the whole structure of the book because, even at an early stage, it presents important basic relations of thermodynamics which are results of a general statistical theory, independent of underlying physics. Several chapters and parts of chapters, in particular those which require more advanced knowledge, for instance of quantum mechanics, may be skipped without loss of the logical continuity, both by beginners and by the reader who is mainly interested in the practical application of thermodynamics. These parts are marked by an asterisk in the title. The already mentioned subsection 5.4 is actually an appendix for specially interested readers. Subsection 2.3 does not rely on the statistical theory, with the exception of some additional excursions which are made and are obvious as such. This part of the book should be readable without knowledge of all the other parts and the author hopes that it can be helpful also for a reader who is only interested in macroscopic thermodynamics. References with respect to special literature are made in footnotes. They are made only if the subject is not considered to be of textbook standard.