Statistical Physics: An Advanced Approach with Applications

415 Pages·1998·12.829 MB·English

by  Professor Dr. Josef Honerkamp (auth.)

Statistical Physics
Springer-Verlag Berlin Heidelberg GmbH
Josef Honerkarnp
Statistical Physics
An Advanced Approach
with Applications
With 82 Figures, 7 Tables
and 57 Problems with Solutions
Springer
Professor Dr. Josef Honerkamp
Albert-Ludwig-Universität Freiburg
Fakultät für Physik
Hermann-Herder-Str. 3
D-79104 Freiburg, Germany
e-mail: [e-mail]
Translator:
Dr. Thomas Filk
Am oberen Kirchweg 12
D-79258 Hartheim, Germany
e-mail: [e-mail]
The cover picture was composed from Figs. 6.6,7-3, 8.7 of this book
Library of Congress Cataloging-in-Publication Data.
Honerkamp J. Statistical physics: an advanced approach with applications / joseph Honerkamp.
p. cm. Includes bibliographical references and index.
ISBN 978-3-662-03711-9 ISBN 978-3-662-03709-6 (eBook)
DOI 10.1007/978-3-662-03709-6
1. Statistical physics.l. Title. QC 174.8.H65 1998 530.13--dc21 98-14362
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from Springer-Verlag Berlin Heidelberg GmbH.
Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1998
Originally published by Springer-Verlag Berlin Heidelberg New York in 1998
Softcover reprint of the hardcover 1st edition 1998
The use of general descriptive names, registered names, trademarks, etc. in this publicatioll does not
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Typesetting: Camera-ready copy by the authors using a Springer TEX macro package
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Preface
This book emerged from a lecture I held in the 1995/96 winter term in the
Faculty of Physics at the University of Freiburg, Germany. It represents the
adaptation of the content of a one-semester course in order to meet the
present-day demands in education. After much contact with physicists in
industrial and science laboratories and in neighboring scientific disciplines,
I came to the conclusion that present-day education, with its predominant
orientation towards deterministic methods in the description of physical phe
nomena, is not weH suited to the needs of the jobs the students are likely
to enter after their education. Statistical questions become more and more
important in the various fields of research and developement, and in the
framework of statistical physics a student should learn more than statistical
mechanics alone.
Of course, the quantity of material one believes one should impart to the
students is much more than what can be taught within one course. Therefore,
I have included several extra topics but I have tried to focus on those which
in my opinion are most relevant for a student of statistical physics, and which
might also serve as a first step towards a deeper understanding.
I would like to thank everybody who assisted me in word and deed during
the developement of this book. First, I would like to mention Mrs. Fotschki,
who transformed the weekly supply of manuscripts reliably and always in
time into a 'IEX file, as weH as Dr. M. Koch, who worked the formulas and
pictures into these files with great care. The preparation of an almost com
plete manuscript during the time of one course required intensive work, which
could only have been completed successfuHy by an effective and good coHab
oration.
The weekly discussions with my coHaborators, especially Dr. H. P. Breuer,
and the discussions with participants of the lectures helped to clarify many
questions.
Since the manuscript of the weekly lecture was available to the students
at the end of each week via the Internet, I received many comments and
suggestions from listeners and readers. I had many discussions with my col
leagues, in particular with Prof. Dr. H. R. Lerche and Prof. Dr. H. Römer,
concerning the representation of certain subjects.
VI Preface
I would also like to express my gratitude to Dr. T. Filk. Not only did he do
excellent work in translating the German version of the manuscript but with
his sensible comments he also helped to eliminate a number of notational and
stylistic inconsistencies.
I hope that this book awakens in many readers an interest in methods for
handling uncertain information.
Freiburg, January 1998 Jose! Honerkamp
Contents
1. Statistical Physics
Is More than Statistical Mechanies 1
Part I. Modeling of Statistical Systems
2. Random Variables:
Fundamentals of Probability Theory and Statistics ....... 5
2.1 Probability and Random Variables. . . . . .. . . . . . . . . . . . . . . . . . 6
2.1.1 The Space of Events . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6
2.1.2 Introduction of Probability . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.3 Random Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Multivariate Random Variables
and Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12
2.2.1 Multidimensional Random Variables. . . . . . . . . . . . . . .. 12
2.2.2 Marginal Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . .. 13
2.2.3 Conditional Probabilities and Bayes' Theorem. . . . . .. 14
2.3 Moments and Quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18
2.3.1 Moments........................................ 18
2.3.2 Quantiles........................................ 21
2.4 The Entropy. . . . . . . . . . . . . . . . . . . . . . . . .. . .. . . . . . . . . . . . . .. 23
2.4.1 Entropy for a Discrete Set of Events . . . . . . . . . . . . . . .. 23
2.4.2 Entropy for a Continuous Space of Events . . . . . . . . . .. 24
2.4.3 Relative Entropy ................................ , 25
2.4.4 Remarks........................................ 25
2.4.5 Applications..................................... 26
2.5 Computations with Random Variables. . . . . . . . . . . . . . . . . . .. 29
2.5.1 Addition and Multiplication of Random Variables. . .. 29
2.5.2 Further Important Random Variables. . . . . . . . . . . . . .. 33
2.5.3 Limit Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34
2.6 Stable Random Variables
and Renormalization Transformations. . . . . . . . . . . . . . . . . . . .. 37
2.6.1 Stable Random Variables. . . . . .. .. . . . . . . . . . . . . . . . .. 37
2.6.2 The Renormalization Transformation. . .. . . . . . . . . . .. 40
VIII Contents
2.6.3 Stability Analysis ................................ 40
2.6.4 Scaling Behavior ................................. 42
2.7 The Large Deviation Property for Sums
of Random Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 44
3. Random Variables in State Space:
Classical Statistical Mechanics of Fluids .................. 49
3.1 The Microcanonical System. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50
3.2 Systems in Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53
3.2.1 Thermal Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53
3.2.2 Systems with Exchange of Volume and Energy . . . . . .. 60
3.2.3 Systems with Exchange of Particles and Energy . . . . .. 64
3.3 Equilibrium States and Thermodynamic Potentials ......... 66
3.4 Susceptibilities......................................... 71
3.4.1 Heat Capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72
3.4.2 Isothermal Compressibility ........................ 74
3.4.3 Isobaric Expansivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 74
3.4.4 Isochoric Tension Coeflicient
and Adiabatic Compressibility . . . . . . . . . . . . . . . . . . . .. 75
3.4.5 A General Relation Between Response Functions . . . .. 75
3.5 The Equipartition Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77
3.6 The Radial Distribution Function. . . . . . . . . . . . . . . . . . . . . . . .. 79
3.7 Approximation Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85
3.7.1 The Virial Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85
3.7.2 Integral Equations
for the Radial Distribution Function . . . . . . . . . . . . . . .. 92
3.7.3 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94
3.8 The van der Waals Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97
3.8.1 The Isotherms ............................... . . .. 98
3.8.2 The Maxwell Construction ........................ 99
3.8.3 Corresponding States ............................. 101
3.8.4 Critical Exponents ............................... 103
4. Random Fields:
Textures and Classical Statistical Mechanics
of Spin Systems .......................................... 107
4.1 Discrete Stochastic Fields ............................... 108
4.1.1 Markov Fields ................................... 109
4.1.2 Gibbs Fields ..................................... 111
4.1.3 Equivalence of Gibbs and Markov Fields ............ 112
4.2 Examples of Markov Random Fields ...................... 113
4.2.1 Model with Independent Random Variables .......... 113
4.2.2 Auto-Model ..................................... 114
4.2.3 Multi-Level Logistic Model ........................ 115
4.2.4 Gauss Model. .................................... 116
Contents IX
4.3 Characteristic Quantities of Densities
for Random Fields ...................................... 116
4.4 Simple Random Fields .......................... : ....... 118
4.4.1 The White Random Field
or the Ideal Paramagnetic System .................. 118
4.4.2 The One-Dimensional Ising Model .................. 120
4.5 Random Fields with Phase 'Iransitions .................... 124
4.5.1 The Curie-Weiss Model ........................... 124
4.5.2 The Mean Field Approximation .................... 126
4.5.3 The Two-Dimensional Ising Model .................. 131
4.6 The Landau Free Energy ................................ 137
4.7 The Renormalization Group Method
for Random Fields and Scaling Laws ...................... 140
4.7.1 The Renormalization 'Iransformation ............... 140
4.7.2 Scaling Laws ..................................... 141
5. Time-Dependent Random Variables:
Classical Stochastic Processes ............................. 145
5.1 Markov Processes ...................................... 146
5.2 The Master Equation ................................... 149
5.3 Examples of Master Equations ........................... 155
5.4 Analytic Solutions of Master Equations ................... 160
5.4.1 Equations for the Moments ........................ 160
5.4.2 The Equation for the Characteristic Function ........ 161
5.4.3 Examples ....................................... 162
5.5 Simulation of Stochastic Processes and Fields .............. 164
5.6 The Fokker-Planck Equation ............................ 172
5.7 The Linear Response Function
and the Fluctuation-Dissipation Theorem ................. 177
5.8 Approximation Methods ................................. 181
5.8.1 The [l Expansion ................................ 181
5.8.2 The One-Particle Picture .......................... 187
5.9 More General Stochastic Processes ........................ 189
5.9.1 Self-Similar Processes ............................. 189
5.9.2 Fractal Brownian Motion .......................... 190
5.9.3 Stable Levy Processes ............................. 191
5.9.4 Autoregressive Processes .......................... 191
6. Quantum Random Systems ............................... 199
6.1 Quantum-Mechanical Description
of Statistical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
6.2 Ideal Quantum Systems: General Considerations ........... 205
6.2.1 Relations Between the Thermodynamic Potential
and Other System Variables ....................... 208
6.2.2 Expansion in the Classical Regime .................. 209
X Contents
6.2.3 First Quantum-Meehanical Correetion Term ......... 210
6.3 The Ideal Fermi Gas .................................... 211
6.3.1 The Fermi-Dirac Distribution ...................... 211
6.3.2 Determination of the System Variables .............. 214
6.3.3 Applieations of the Fermi-Dirac Distribution ........ 216
6.4 The Ideal Bose Gas ..................................... 219
6.4.1 Particle Number and the Bose-Einstein Distribution .. 219
6.4.2 Bose-Einstein Condensation ....................... 221
6.4.3 Pressure ......................................... 224
6.4.4 Energy and Specifie Heat .......................... 226
6.4.5 Entropy ......................................... 227
6.4.6 Applieations of Bose Statistics ..................... 227
6.5 The Photon Gas and Blaek Body Radiation ................ 228
6.5.1 The Kirehhoff Law ............................... 234
6.5.2 The Stefan-Boltzmann Law ....................... 235
6.5.3 The Pressure of Light ............................. 236
6.5.4 The Total Radiative Power of the Sun .............. 238
6.5.5 The Cosmie Background Radiation ................. 240
6.6 Lattiee Vibrations in Solids: The Phonon Gas .............. 241
6.7 Ideal Moleeules with Internal Degrees of Freedom ........... 247
6.8 Magnetie Properties of Fermi Systems ..................... 254
6.8.1 Diamagnetism ................................... 254
6.8.2 Paramagnetism .................................. 258
6.9 Quasiparticles .......................................... 259
6.9.1 Models for the Magnetie Properties of So lids ......... 261
6.9.2 Superfluidity ..................................... 265
7. Changes of External Conditions .......................... 269
7.1 Reversible State Transformations, Heat, and Work .......... 269
7.2 Cyclie Processes ........................................ 274
7.3 Exergy and Relative Entropy ............................ 278
7.4 Time Dependenee of Statistical Systems . . . . . . . . . . . . . . . . . . . 280
Part 11. Analysis of Statistical Systems
8. Estimation of Parameters .. ............................... 289
8.1 SampIes and Estimators ................................. 290
8.2 Confidenee Intervals .................................... 297
8.3 Propagation of Errors ................................... 300
8.4 The Maximum Likelihood Estimator ...................... 302
8.5 The Least Squares Estimator ............................ 306