Statistical Mechanics Statistical Mechanics From Thermodynamics to the Renormalization Group James H. Luscombe First edition published 2021 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2021 Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot as- sume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright hold- ers if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. 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Title: Statistical mechanics : from thermodynamics to the renormalization group / James Luscombe. Description: First edition. | Boca Raton : CRC Press, 2021. | Includes bibliographical references and index. Identifiers: LCCN 2020048146 | ISBN 9781138542976 (paperback) | ISBN 9780367689278 (hardback) | ISBN 9781003139669 (ebook) Subjects: LCSH: Statistical mechanics. Classification: LCC QC174.8 .L87 2021 | DDC 530.13--dc23 LC record available at https://lccn.loc.gov/2020048146 ISBN: 978-0-367-68927-8 (hbk) ISBN: 978-1-138-54297-6 (pbk) ISBN: 978-1-003-13966-9 (ebk) Typeset in Computer Modern font by KnowledgeWorks Global Ltd. Access the Support Material: www.routledge.com/9781138542976 To my mother, who likes orderliness, and to my father who didn’t. They taught me without trying that the equilibrium state is one of lowest energy and maximum disorder. To my father, who taught me to love the wildness of nature, and to my mother, who taught me to seek order, which I imported into the world of ideas. To my wife, Lisa, who encouraged me, and who lends an order of her own to the written language. To my children, Jennifer and Jimmy, who represent the future and have brought me so much joy. Contents Preface xi SECTIONI StructureofStatisticalMechanics CHAPTER1(cid:4)Thermodynamics: Equilibrium, energy, entropy 3 1.1 SYSTEMS, BOUNDARIES, AND VARIABLES 3 1.2 INTERNAL ENERGY: WORK AND HEAT 5 1.3 CLAUSIUS ENTROPY: IRREVERSIBILITY, DISORGANIZATION 7 1.4 THERMODYNAMIC POTENTIALS 11 1.5 FREE ENERGY AND DISSIPATED ENERGY 13 1.6 CHEMICAL POTENTIAL AND OPEN SYSTEMS 14 1.7 MAXWELL RELATIONS 15 1.8 RESPONSE FUNCTIONS 15 1.9 HEAT CAPACITY OF MAGNETIC SYSTEMS 19 1.10 EXTENSIVITY OF ENTROPY, SACKUR-TETRODE FORMULA 20 1.11 BOLTZMANN ENTROPY: CONNECTION WITH MICROSCOPICS 22 1.12 FLUCTUATIONS AND STABILITY 24 1.13 LIMITATIONS OF THERMODYNAMICS 26 CHAPTER2(cid:4)From mechanics to statistical mechanics 31 2.1 MICROSTATES—THE MANY 31 2.2 STATE VARIABLES—THE FEW 37 2.3 ENTROPY, THE BRIDGE BETWEEN MICRO AND MACRO 38 2.4 FLUCTUATIONS: GATEWAY TO ENSEMBLES AND PROBABILITY 40 2.5 ENSEMBLE FLOWS IN PHASE SPACE, LIOUVILLE’S THEOREM 46 2.6 BIRKHOFF’S THEOREM 48 2.7 THE ROLE OF PROBABILITY 51 vii viii (cid:4) Contents CHAPTER3(cid:4)Probability theory 57 3.1 EVENTS, SAMPLE SPACE, AND PROBABILITY 57 3.2 COMBINING PROBABILITIES, CONDITIONAL PROBABILITY 60 3.3 COMBINATORICS 61 3.4 EXAMPLES INVOLVING DISCRETE PROBABILITIES 64 3.5 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 65 3.6 CENTRAL LIMIT THEOREM, LAW OF LARGE NUMBERS 73 3.7 CUMULANTS AND CHARACTERISTIC FUNCTIONS 75 CHAPTER4(cid:4)Ensemble theory 83 4.1 CLASSICAL ENSEMBLES: PROBABILITY DENSITY FUNCTIONS 83 4.2 THE THERMODYNAMIC LIMIT: EXISTENCE OF EXTENSIVITY 102 4.3 CONVEXITY AND STABILITY 105 4.4 QUANTUM ENSEMBLES: PROBABILITY DENSITY OPERATORS 107 SECTIONII ApplicationsofEquilibriumStatisticalMechanics CHAPTER5(cid:4)Ideal systems 127 5.1 THE MAXWELL SPEED DISTRIBUTION 127 5.2 PARAMAGNETS 130 5.3 HARMONIC OSCILLATORS, QUANTUM AND CLASSICAL 133 5.4 DIATOMIC GASES 135 5.5 IDENTICAL FERMIONS AND BOSONS 138 5.6 DEGENERATE FERMI GAS: T = 0 AND 0 < T (cid:28) T 146 F 5.7 DEGENERACY PRESSURE IN THE LIFE OF STARS 154 5.8 CAVITY RADIATION 159 5.9 DEGENERATE BOSE GAS, BOSE-EINSTEIN CONDENSATION 164 CHAPTER6(cid:4)Interacting systems 181 6.1 THE MAYER CLUSTER EXPANSION 181 6.2 VIRIAL EXPANSION, VAN DER WAALS EQUATION OF STATE 190 6.3 CUMULANT EXPANSION OF THE FREE ENERGY 192 6.4 THE TONKS AND TAKAHASHI GASES 198 6.5 THE ONE-DIMENSIONAL ISING MODEL 200 6.6 SCATTERING, FLUCTUATIONS, AND CORRELATIONS 211 6.7 ORNSTEIN-ZERNIKE THEORY OF CRITICAL CORRELATIONS 215 Contents (cid:4) ix CHAPTER7(cid:4)Phase transitions and critical phenomena 221 7.1 PHASE COEXISTENCE, GIBBS PHASE RULE 222 7.2 THE CLASSIFICATION OF PHASE TRANSITIONS 225 7.3 VAN DER WAALS LIQUID-GAS PHASE TRANSITION 226 7.4 A BESTIARY OF CRITICAL EXPONENTS: α,β,γ,δ 231 7.5 WEISS MOLECULAR FIELD THEORY OF FERROMAGNETISM 234 7.6 THE CRITICAL CORRELATION EXPONENTS: ν,η 238 7.7 LANDAU THEORY OF PHASE TRANSITIONS 240 7.8 MEAN-FIELD THEORY: UNCORRELATED FLUCTUATIONS 244 7.9 WHEN IS MEAN FIELD THEORY EXACT? 246 7.10 THE TWO-DIMENSIONAL ISING MODEL 249 7.11 CRITICAL EXPONENT INEQUALITIES 254 7.12 THE IMPOSSIBILITY OF PHASES IN ONE DIMENSION 256 CHAPTER8(cid:4)Scaling theories and the renormalization group 265 8.1 THE WIDOM SCALING HYPOTHESIS 265 8.2 KADANOFF SCALING THEORY: SCALING EXPONENTS 269 8.3 RENORMALIZATION: A FIRST LOOK 272 8.4 THE RENORMALIZATION GROUP 284 8.5 REAL-SPACE RENORMALIZATION 287 8.6 WILSON RENORMALIZATION GROUP IN A NUTSHELL 293 APPENDIXA(cid:4) Physical constants 307 APPENDIXB(cid:4) Useful integrals 309 APPENDIXC(cid:4) Classical mechanics 321 APPENDIXD(cid:4) Quantum mechanics of identical particles 337 APPENDIXE(cid:4) Topics in magnetism 357 APPENDIXF(cid:4) The method of the most probable distribution 363 x (cid:4) Contents Bibliography 369 Index 377