Springer-Verlag Berlin Heidelberg GmbH

Transcription

1 Statistical Physics

2 Springer-Verlag Berlin Heidelberg GmbH

3 Josef Honerkarnp Statistical Physics An Advanced Approach with Applications With 82 Figures, 7 Tables and 57 Problems with Solutions Springer

4 Professor Dr. Josef Honerkamp Albert-Ludwig-Universität Freiburg Fakultät für Physik Hermann-Herder-Str. 3 D Freiburg, Germany hon@physik.uni-freiburg.de Translator: Dr. Thomas Filk Am oberen Kirchweg 12 D Hartheim, Germany thomas.filk@t-online.de 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 ISBN (ebook) DOI / Statistical physics.l. Title. QC H dc This work is subject to copyright. All rights are reserved, wh ether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained 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 imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready copy by the authors using a Springer TEX macro package Cover design: design <& production GmbH, Heidelberg SPIN / Printed on acid-free paper

5 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 phenomena, 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 complete manuscript during the time of one course required intensive work, which could only have been completed successfuhy by an effective and good cohaboration. 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 colleagues, in particular with Prof. Dr. H. R. Lerche and Prof. Dr. H. Römer, concerning the representation of certain subjects.

6 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

7 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 Probability and Random Variables The Space of Events Introduction of Probability Random Variables Multivariate Random Variables and Conditional Probabilities Multidimensional Random Variables Marginal Distributions Conditional Probabilities and Bayes' Theorem Moments and Quantiles Moments Quantiles The Entropy Entropy for a Discrete Set of Events Entropy for a Continuous Space of Events Relative Entropy..., Remarks Applications Computations with Random Variables Addition and Multiplication of Random Variables Further Important Random Variables Limit Theorems Stable Random Variables and Renormalization Transformations Stable Random Variables The Renormalization Transformation

8 VIII Contents Stability Analysis Scaling Behavior The Large Deviation Property for Sums of Random Variables Random Variables in State Space: Classical Statistical Mechanics of Fluids The Microcanonical System Systems in Contact Thermal Contact Systems with Exchange of Volume and Energy Systems with Exchange of Particles and Energy Equilibrium States and Thermodynamic Potentials Susceptibilities Heat Capacities Isothermal Compressibility Isobaric Expansivity Isochoric Tension Coeflicient and Adiabatic Compressibility A General Relation Between Response Functions The Equipartition Theorem The Radial Distribution Function Approximation Methods The Virial Expansion Integral Equations for the Radial Distribution Function Perturbation Theory The van der Waals Equation The Isotherms The Maxwell Construction Corresponding States Critical Exponents Random Fields: Textures and Classical Statistical Mechanics of Spin Systems Discrete Stochastic Fields Markov Fields Gibbs Fields Equivalence of Gibbs and Markov Fields Examples of Markov Random Fields Model with Independent Random Variables Auto-Model Multi-Level Logistic Model Gauss Model

9 Contents IX 4.3 Characteristic Quantities of Densities for Random Fields Simple Random Fields... : The White Random Field or the Ideal Paramagnetic System The One-Dimensional Ising Model Random Fields with Phase 'Iransitions The Curie-Weiss Model The Mean Field Approximation The Two-Dimensional Ising Model The Landau Free Energy The Renormalization Group Method for Random Fields and Scaling Laws The Renormalization 'Iransformation Scaling Laws Time-Dependent Random Variables: Classical Stochastic Processes Markov Processes The Master Equation Examples of Master Equations Analytic Solutions of Master Equations Equations for the Moments The Equation for the Characteristic Function Examples Simulation of Stochastic Processes and Fields The Fokker-Planck Equation The Linear Response Function and the Fluctuation-Dissipation Theorem Approximation Methods The [l Expansion The One-Particle Picture More General Stochastic Processes Self-Similar Processes Fractal Brownian Motion Stable Levy Processes Autoregressive Processes Quantum Random Systems Quantum-Mechanical Description of Statistical Systems Ideal Quantum Systems: General Considerations Relations Between the Thermodynamic Potential and Other System Variables Expansion in the Classical Regime

10 X Contents First Quantum-Meehanical Correetion Term The Ideal Fermi Gas The Fermi-Dirac Distribution Determination of the System Variables Applieations of the Fermi-Dirac Distribution The Ideal Bose Gas Particle Number and the Bose-Einstein Distribution Bose-Einstein Condensation Pressure Energy and Specifie Heat Entropy Applieations of Bose Statistics The Photon Gas and Blaek Body Radiation The Kirehhoff Law The Stefan-Boltzmann Law The Pressure of Light The Total Radiative Power of the Sun The Cosmie Background Radiation Lattiee Vibrations in Solids: The Phonon Gas Ideal Moleeules with Internal Degrees of Freedom Magnetie Properties of Fermi Systems Diamagnetism Paramagnetism Quasiparticles Models for the Magnetie Properties of So lids Superfluidity Changes of External Conditions Reversible State Transformations, Heat, and Work Cyclie Processes Exergy and Relative Entropy Time Dependenee of Statistical Systems Part 11. Analysis of Statistical Systems 8. Estimation of Parameters SampIes and Estimators Confidenee Intervals Propagation of Errors The Maximum Likelihood Estimator The Least Squares Estimator

11 Contents XI 9. Signal Analysis: Estimation of Spectra The Discrete Fourier Transform and the Periodogram Filters Filters and Transfer Functions Filter Design Consistent Estimation of Spectra Frequency Distributions for Nonstationary Time Series Filter Banks and Discrete Wavelet Transformations Wavelets Wavelets as Base Functions in Function Spaces Wavelets and Filter Banks Solutions of the Dilation Equation Inverse Problems: Estimation of Distributions Identification of Stochastic Models from Observations Hidden Systems The Maximum aposteriori (MAP) Estimator The MAP Estimator for the Deconvolution Problem, Inverse Problems The Least Squares Estimator as a Special MAP Estimator Strategies for Choosing the Regularization Parameter The Regularization Method Examples of the Estimation of a Distribution Function by a Regularization Method Appendix: Random Number Generation for Simulating Realizations of Random Variables Problems Hints and Solutions References Index