Debashish Chowdhury, Dietrich Stauffer

Principles of Equilibrium Statistical Mechanics

®WILEY-VCH Weinheim • New York • Chichester • Brisbane • Singapore • Toronto

Table of Contents Part I: T H E R M O S T A T I C S 1

1 B A S I C P R I N C I P L E S OF T H E R M O S T A T I C S 5 1.1 Introduction . 5 1.2 Extensive and Intensive Variables 5 1.3 Entropy and Temperature 8 1.4 Concept of Equilibrium 9 1.5 Internal Energy and the First Law 10 1.6 "Processes" in Thermostatics; Cycles 11 1.7 Constraints and Walls 11 1.8 Fundamental Relation, Equations of State 12 1.9 Euler and Gibbs-Duhem Relations 18 1.10 Second Law of Thermostatics 20

1.11 Third Law of Thermostatics 22 1.12 Conditions for Fundamental Relations 24 1.13 Extremum Principles for Equilibrium 24

1.13.1 Entropy-maximum/Energy-minimum Principles . . . . 24 1.13.2 Concept of a Reservoir 26 1.13.3 Why T is Called Temperature 27 1.13.4 Thermostatic Potentials and Equilibrium 28

1.14 Response Functions . . . . . . . . . . . . . . . . . . . . . . . . 33 1.15 Thermostatic Relations 35 1.16 Second Law: Alternative Statements 38 1.17 Some Applications of Thermostatics 40 1.18 Chapter Summary 45 1.19 Historical Notes 46

1.19.1 First Law of Thermostatics 46 1.19.2 Second Law of Thermostatics 50 1.19.3 Third Law of Thermostatics 53 1.19.4 Gibbs and Modern Thermostatics 53 1.19.5 Duhem and Applications of Thermostatics 54

1.20 Problems 56 1.21 Supplementary Notes 57

5.1.1 Convexity and Concavity of Thermostatic Variables . 57 5.1.2 Mathematical Foundation of Thermostatics 57 5.1.3 Applications of Thermostatics 58

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2 THERMOSTATICS OF PHASE TRANSITIONS 59 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.2 Phases and Components . . . . . . . . . . . . . . . . . . . . . 59 2.3 Phase Diagrams and Critical Point . . . . . . . . . . . . . . . 61

2.3.1 Fluids 61 2.3.2 Magnets 64

2.4 Stability of Phases 66 2.4.1 Intrinsic Stability of Homogeneous State 66 2.4.2 Mutual Stability of Coexisting Phases 68

2.5 Stable, Metastable and Unstable States 74 2.5.1 Van der Waals Gas 77 2.5.2 Curie-Weiss Magnet 80

2.6 Derivatives of Thermostatic Potentials 82 2.7 Classification of Phase Transitions 83 2.8 Critical Exponents 87 2.9 Scaling Relations . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.10 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . 91 2.11 Historical Notes 92

2.11.1 Concept of Latent Heat 92 2.11.2 Critical Phenomena in Fluids 92 2.11.3 Critical Phenomena in Magnets 94

2.12 Problems 95 2.13 Supplementary Notes 96

S.2.1 Tricritical and Multicritical Phenomena 96

PART II BASIC PRINCIPLES OF STATISTICAL MECHANICS

and Rules of Calculation 99

3 RULES OF CALCULATION 109 3.1 Introduction 109 3.2 Isolated Classical Systems 109

3.2.1 Boltzmann Hypothesis 109 3.2.2 Illustration with Ideal Classical Ising Magnet I l l 3.2.3 Illustration with Ideal Classical Monatomic Gas . . .112 3.2.4 Entropy of Mixing and Gibbs'Paradox 113 3.2.5 Equipartition of Energy 116

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3.2.6 Concepts of Ensemble and Ergodicity . . . . . . . . . 117

3.3 Closed Classical Systems 118

3.3.1 The Part i t ion Function 119

3.3.2 Connection with Thermostatics 121

3.3.3 Illustration with Ideal Classical Ising Magnet . . . . . 125

3.3.4 Illustration with Ideal Classical Monatomic Gas . . . 125

3.3.5 Equipartit ion of Energy . . . . . . . . . . . . . . . . . 127

3.4 Open Classical Systems 128

3.4.1 Grand Parti t ion Function 128

3.4.2 Connection with Thermostatics 129

3.4.3 Illustration with Ideal Classical Monatomic Gas . . . 130

3.5 Unified Presentation of the Rules 131

3.5.1 Micro-canonical Ensemble 132

3.5.2 Canonical Ensemble 133

3.5.3 Grand-canonical Ensemble 133

3.6 Maximum Entropy Principle 133

3.6.1 Microcanonical Ensemble 134

3.6.2 Canonical Ensemble 134

3.6.3 Grand Canonical Ensemble . . . . . . . . . . . . . . . 135

3.7 Illustration with Ideal Quantum Gases 136

3.7.1 Micro-Canonical Ensemble 136

3.7.2 Canonical Ensemble 144

3.7.3 Grand-Canonical Ensemble 147

3.8 Quantum Systems; Density Operator 147

3.8.1 Micro-Canonical Ensemble . . . . . . . . . . . . . . . 148

3.8.2 Canonical Ensemble . . . . . . . . . . . . . . . . . . . 150

3.9 Chapter Summary 152

3.10 Historical Notes 154

3.10.1 James Clerk Maxwell 155

3.10.2 Ludwig Boltzmann 156

3.10.3 Josiah Willard Gibbs 157

3.10.4 Maximum Entropy Principle 158

3.11 Problems 160

3.12 Supplementary Notes 167

5.3.1 Entropy Calculation from Trajectory 167

5.3.2 Entropy and Information 167

5.3.3 Wigner Function 169

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4 F L U C T U A T I O N S , C O R R E L A T I O N S A N D R E S P O N S E 171 4.1 Introduction 171

4.2 Energy: Most Probable and Mean 172

4.3 Particle Number: Most Probable and Mean 172

4.4 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

4.4.1 Fluctuations in Energy . . . . . . . . . . . . . . . . . . 174

4.4.2 Fluctuations in Volume 175

4.4.3 Fluctuations in Particle Number 176

4.4.4 Cross Correlation Between Fluctuations 176

4.4.5 Fluctuations in Entropy . . . . . . . . . . . . . . . . . 178

4.5 Microstate Population Fluctuations . . . . . . . . . . . . . . . 179

4.5.1 Ideal Bose Gas 179

4.5.2 Ideal Fermi Gas 179

4.5.3 Ideal Boltzmann Gas 180

4.6 Classical versus Quantum Fluctuations 180

4.7 Correlation Functions 181

4.8 Fluctuation-Response Theorem 182

4.8.1 Illustration with Ising Magnets 182

4.8.2 Illustration with Simple Fluids 184

4.9 Scattering Measures Correlation 184

4.10 Foundations of Laws of Thermostatics 187

4.11 Chapter Summary 188

4.12 Historical Notes 189

4.13 Problems 190

4.14 Supplementary Notes 192

S.4.1 Radial Distribution Functions for Fluids 192

5 S T A T I S T I C A L P H Y S I C S O F I D E A L C L A S S I C A L G A S E S 196 5.1 Introduction 196

5.2 Monatomic Gas; MB Distribution 196

5.3 Full Part i t ion Function . 198

5.3.1 Monatomic Molecules 200

5.3.2 Diatomic Molecules 201

5.4 Chapter Summary 208

5.5 Historical Notes 208

5.6 Problems 210

5.7 Supplementary Notes 213

S.5.1 Kinetic Theory of Gases 213

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6 S T A T I S T I C A L P H Y S I C S OF I D E A L Q U A N T U M G A S E S 2 1 6 6.1 Introduction 216 6.2 Some General Results 216

6.2.1 Concept of Density of Levels 217 6.3 Ideal Fermi Gas at T = 0 219

6.3.1 Fermi Energy 220 6.3.2 Internal Energy 221 6.3.3 Pressure 221

6.4 Ideal Fermi Gas at T ф 0 222 6.4.1 Sommerfeld Expansion 222 6.4.2 Chemical Potential 225 6.4.3 Specific Heat 225 6.4.4 Magnetic Susceptibility 227

6.5 Fermi-Dirac and Bose-Einstein Integrals . . . . . . . . . . . . 230 6.6 Ideal Bose Gas: Bose-Einstein Condensation . . . . . . . . . . 231

6.6.1 Number density and Chemical Potential 231 6.6.2 Internal energy and Specific Heat 236 6.6.3 Entropy 239 6.6.4 Isotherms on the PV diagram 240

6.7 Chapter Summary 243 6.8 Historical Notes 244

6.8.1 FD Statistics 244 6.8.2 BE Statistics and Discovery of BE Condensation . . . 245

6.9 Problems 247 6.10 Supplementary Notes 249

5.6.1 Quantum Gases in Astrophysics 249 5.6.2 Phonon Contribution to Specific Heat of Crystals . . . 249

P A R T III S T A T I S T I C A L M E C H A N I C S

OF I N T E R A C T I N G S Y S T E M S 253

7 I N T E R A C T I N G S Y S T E M S ; T H E R M O D Y N A M I C L I M I T 2 5 7 7.1 Introduction 257 7.2 Models of Fluids 258

7.2.1 Part i t ion Function 261 7.3 Lattices 262

7.3.1 Bravais Lattices . . . . . . . . . . . . . . . . . . . . . 262 7.3.2 Bethe Lattice . . . . . . . . . . . . . . . . . . . . . . . 264

7.4 Spin Models on Lattices . . . . . . . . . . . . . . . . . . . . . 265 7.4.1 Classical Spin-1/2 Ising Model . . . . . . . . . . . . . 266 7.4.2 Spin-1/2 Ising Model: Physical Realizations . . . . . . 270 7.4.3 Generalization: From Spin-1/2 to Spin-1 Ising Model . 276 7.4.4 Generalization: From Ising to Vector Spins . . . . . . 277 7.4.5 Generalization: Prom Ising to Potts Variables . . . . . 279 7.4.6 Quantum Spin Models 279 7.4.7 Classical Limit of Quantum Spin Models 280 7.4.8 Magnetic Physical Realizations of Spin Models . . . . 2 8 1

7.5 Restrictions on the Interactions 281 7.6 Zeroes of the Grand Parti t ion Function 285

7.6.1 Yang-Lee Theorems and Their Consequences 286 7.7 Chapter Summary 291 7.8 Historical Notes 292

7.8.1 Spin Models and Their Physical Realizations 292

7.8.2 Thermodynamic Limit 294 7.8.3 Yang-Lee Theorem 295

7.9 Problems 296 7.10 Supplementary notes 299

5.7.1 Continuum Models of Fluids 299

5.7.2 Spin Models on Discrete Lattices . 299 5.7.3 Vertex Models 300 5.7.4 Generalization: From "Hard" Spins to "Soft" Spins . . 300 5.7.5 Spin Models with Quenched Disorder 303 5.7.6 Anisotropic Hamiltonians for Spin System 303 5.7.7 Thermodynamic Limit 304 5.7.8 Complex Temperature Plane: Zeroes of Parti t ion Func­

tion 308 5.7.9 Quantum Phase Transitions and Critical Phenomena . 308

E X A C T S O L U T I O N OF S O M E I N T E R A C T I N G S Y S T E M S 3 1 0 8.1 Introduction 310 8.2 Ising Model in d = 1: Partit ion Function 311

8.2.1 Open Chain in the Absence of External Field 311 8.2.2 Closed Chain: Transfer Matrix Approach 313 8.2.3 Zeroes of the Parti t ion Function 317

8.3 Ising Model in d = 1: Thermostatics 321

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8.3.1 Spontaneous Magnetization . . . . . . . . . . . . . . . 321 8.3.2 Magnetic Susceptibility . . . . . . . . . . . . . . . . . 321 8.3.3 Magnetic Specific Heat . . . . . . . . . . . . . . . . . . 322

8.4 Ising Model in d = 1: Correlations 322 8.4.1 Open Chain in Zero Field 322 8.4.2 Closed Chain: Transfer Matrix Approach . . . . . . . 324

8.5 Important Concepts in Phase Transitions . . . . . . . . . . . 326 8.5.1 Order Parameter 326 8.5.2 Peierls-Griffiths Argument 327

8.5.3 Phase Transitions: A "Mathematical Mechanism" . . 328 8.5.4 Lower Critical Dimension 329

8.6 Critical Exponents . 329 8.7 Exact Solution of Fluid Models in d = 1 . . . . . . . . . . . . 330

8.7.1 Tonks Gas 330 8.7.2 Takahashi Gas 333

8.8 Chapter Summary 334 8.9 Historical Notes 335

8.9.1 From Ising to Peierls 335 8.9.2 From Kramers to Onsager 335 8.9.3 One-dimensional Models of Fluids 337

8.10 Problems 338 8.11 Supplementary Notes 340

5.8.1 Peierls-Griffiths Arguments 340 5.8.2 Duality and Star-Triangle Transformations; Exact Tc 's 340 5.8.3 Relevance of the Range of the Interaction 341 5.8.4 Exact Solution of the Two-dimensional Ising Model . 3 4 1 5.8.5 Exact Solution of the n-vector Model . . . . . . . . . 343 5.8.6 Exact Solution of the Spherical Model in d-dimension 343

9 C O M P U T E R S I M U L A T I O N M E T H O D S 347 9.1 Introduction 347

9.2 Monte Carlo Simulation 350 9.2.1 Random Sampling Illustrated: Random Walk 351 9.2.2 Importance Sampling Illustrated: Ising Model 353 9.2.3 Fluctuations 361

9.2.4 Equilibration Time and Correlation Time 361 9.2.5 MC Simulation in Micro-canonical Ensemble 362 9.2.6 MC Simulation of Fluids 363

9.3 Molecular Dynamics 364

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9.3.1 Constant-Energy Molecular Dynamics 364 9.3.2 Constant Temperature/Constant Pressure MD . . . . 368

9.4 Non-Self-Averaging Quantities . . . . . . . . . . . . . . . . . 369 9.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . 370 9.6 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 370

9.6.1 MD approach . . . . . . . . . . . . . . . . . . . . . . . 370 9.6.2 MC approach 371

9.7 Problems 373

9.8 Supplementary Notes 374 5.9.1 Random Numbers and Random-Number Generators . 374 5.9.2 Multi-Spin Coding . . . . . . . . . . . . . . . . . . . . 374 5.9.3 MC Simulation: Tricks of the Trade 375 5.9.4 MD Simulation: Tricks of the Trade 376 5.9.5 Langevin Dynamics Simulation 376 5.9.6 Vector- and Parallel Processors; Special Computers . . 376 5.9.7 Q2R update rules . 377

10 M E A N - F I E L D T H E O R Y I: Van der Waals -Weiss Formulat ion 379 10.1 Introduction 379 10.2 MFA for the d-dimensional Ising Model 379

10.2.1 A Pedestrian's Approach 380 10.2.2 An Alternative Variational Approach 381 10.2.3 Ordering Temperature 383 10.2.4 Thermostatic Properties 385 10.2.5 Correlation Function 388 10.2.6 Critical Exponents 391

10.3 MFA for the d-dimensional Fluid 392 10.3.1 The Equation of State 392 10.3.2 Critical Point 394 10.3.3 The Law of Corresponding States 396 10.3.4 Critical Exponents 396

10.4 Comparison of Magnets and Fluids 398 10.5 Validity and Accuracy of MFA 400 10.6 Chapter Summary . 403 10.7 Historical Notes 405

10.7.1 Weiss and M F T of Magnets 405 10.7.2 Van der Waals and MFT of Fluids 406 10.7.3 Inadequacies of M F T 408

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10.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 10.9 Supplementary notes 413

5.10.1 Order-disorder Transition in Binary Alloys 413 5.10.2 Bethe-Peierls-Weiss (BPW) Approximation 413 5.10.3 Correlation Function of Fluids in the MFA 417 5.10.4 Mean-field Theory of Polymers 417

11 M E A N - F I E L D T H E O R Y II: Exac t So lut ion in Infinite D i m e n s i o n 418 11.1 Introduction 418 11.2 Infinite-Range Ising Model 418

11.2.1 First Approach: Largest Term Method 419 11.2.2 Second Approach: Method of Steepest Descent . . . . 422

11.3 Infinite-Range Classical Gas 425 11.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . 426 11.5 Historical Notes 427 11.6 Problems 428 11.7 Supplementary Notes 429

5.11.1 Classical n-vector Models with Long-range Interactions429 5.11.2 Pot ts Model with Infinite-range Interactions 429 5.11.3 Ising Model on a Bethe Lattice 430 5.11.4 Classical Fluids in Infinite Dimension 431

12 M E A N - F I E L D T H E O R Y III: Landau Formulat ion 432 12.1 Introduction 432 12.2 Second Order Phase Transitions 433

12.2.1 Critical Exponents in Landau Theory 435 12.3 First Order Phase Transitions 437

12.3.1 Field-driven Transition 438 12.3.2 T-driven Transition; Asymmetric Case 439 12.3.3 T-driven Transition; Symmetric Case 441

12.4 Landau-Ginzburg Theory 443 12.5 Two-Point Correlation Function 448

12.5.1 Fourier Transform Method 449 12.5.2 Method of Solving Differential Equation 451

12.6 Ginzburg Criterion 453 12.7 Chapter Summary 455 12.8 Historical Notes 456 12.9 Problems 458

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12.10Supplementary notes 459 5.12.1 Landau theory for Tricritical Points 459 5.12.2 Gaussian Fluctuations and Ginzburg Criterion . . . . 464 5.12.3 "Derivation" of Landau-Ginzburg Effective Hamiltonian466

13 B E Y O N D M E A N - F I E L D A P P R O X I M A T I O N : Scal ing and Renormal i za t ion Group 470 13.1 Introduction 470 13.2 Scaling and Universality 470

13.2.1 Scaling Hypothesis 472 13.2.2 Scaling Theory: a General Formulation . . . . . . . . 474 13.2.3 Concept of Universality and Universality Classes . . . 477 13.2.4 Heuristic Justification of Scaling . . . . . . . . . . . . 478

13.3 Renormalization, Fixed Points and RG Flow 479 13.4 Chapter Summary 483 13.5 Historical Notes 484 13.6 Problems 485 13.7 Supplementary Notes 486

S.13.1 Renormalization Group Treatment of Percolation . . . 486

M a t h e m a t i c a l A p p e n d i x 489 A.l Introduction 489 A.2 Some Useful Integrals 489 A.3 Exact Differentials 489 A.4 Homogeneous Functions 489

A.4.1 Homogeneous Function of a Single Variable 489 A.4.2 Homogeneous Function of Arbitrary Number of Vari­

ables 490 A.4.3 Generalized Homogeneous Function 490

A.5 Convex and Concave Functions 490 A.5.1 Definitions 490 A.5.2 Geometrical Interpretation 491

A.6 Legendre Transformation 492 A.7 Volume of A d-Dimensional Sphere 494 A.8 Saddle point or Steepest-descent method 496 A.9 Functional derivatives 496 A.10 Stirling Formula 498 A. 11 Perron-Frobenius Theorem 498 A. 12 Probability and Statistics 499

A.12.1 Uniform Distribution 500

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A. 12.2 Binomial Distribution 500 A. 12.3 From Binomial to Gaussian Distribution 502 A.12.4 From Binomial to Poisson Distribution . . . . . . . . . 503

A.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

References 508