Statistical Mechanics In a Nutshell (ebook)

Argumento de Statistical Mechanics In a Nutshell (ebook)

"Peliti's Statistical Mechanics in a Nutshell is a fantastic reference for those who know the subject, teach it, or need a quick technical reminder, especially on the topic of phase transitions, which are consistently featured in modern-day discussions. . . . Statistical Mechanics in a Nutshell provides the more general overview, with topics such as the renormalization group method. It includes a good mix of fundamental thermodynamics, phase behaviour, and other key subjects."--Physics Today

"This book provides a clear, no-nonsense approach to the basic ideas of the subject as well as an introduction to some of its modern applications. The main ideas are well illustrated by examples and exercises. There are important sections on numerical methods and dynamics, and a final chapter on complex systems gives the reader a foretaste of current research. The volume will serve as an excellent introductory graduate text for students in physics, chemistry, and biology."--John Cardy, University of Oxford

"Statistical mechanics has seen an extraordinary broadening of application in recent decades, from economics and the social sciences to computer science and biology. Statistical Mechanics in a Nutshell combines in one accessible book the main classical ideas of statistical mechanics with many recent developments. It should have a wide readership among young (and also less young) scientists seeking a clear view of modern statistical physics."--Bernard Derrida, école Normale Supérieure

"This superb text provides a balanced and thorough treatment of statistical physics. From thermodynamics and basic principles to renormalization group, dynamics, and complex systems, the presentation is a model of clarity, and the level of detail is highly appropriate for graduate students or advanced undergraduates. Each chapter concludes with a helpful list of recommended further reading. I see this becoming a standard textbook for the next generation of PhD students."--Daniel Arovas, University of California, San Diego

"This is an excellent and comprehensive introduction to statistical mechanics in all of its aspects. The exposition is stimulating and concise but always clear, avoiding pedantic details. Statistical Mechanics in a Nutshell has the potential to become a standard reference."--Giovanni Gallavotti, Sapienza University of Rome

"Unlike typical textbooks . . . [Statistical Mechanics in a Nutshell] presents statistical mechanics as a more general theory with broader applications. . . . A graduate student or researcher who wants to explore the applications of statistical mechanics would be very well served by this book."--Choice0Preface to the English Edition xi
Preface xiii

Chapter 1: Introduction 1
1.1 The Subject Matter of Statistical Mechanics 1
1.2 Statistical Postulates 3
1.3 An Example: The Ideal Gas 3
1.4 Conclusions 7
Recommended Reading 8

Chapter 2: Thermodynamics 9
2.1 Thermodynamic Systems 9
2.2 Extensive Variables 11
2.3 The Central Problem of Thermodynamics 12
2.4 Entropy 13
2.5 Simple Problems 14
2.6 Heat and Work 18
2.7 The Fundamental Equation 23
2.8 Energy Scheme 24
2.9 Intensive Variables and Thermodynamic Potentials 26
2.10 Free Energy and Maxwell Relations 30
2.11 Gibbs Free Energy and Enthalpy 31
2.12 The Measure of Chemical Potential 33
2.13 The Koenig Born Diagram 35
2.14 Other Thermodynamic Potentials 36
2.15 The Euler and Gibbs-Duhem Equations 37
2.16 Magnetic Systems 39
2.17 Equations of State 40
2.18 Stability 41
2.19 Chemical Reactions 44
2.20 Phase Coexistence 45
2.21 The Clausius-Clapeyron Equation 47
2.22 The Coexistence Curve 48
2.23 Coexistence of Several Phases 49
2.24 The Critical Point 50
2.25 Planar Interfaces 51
Recommended Reading 54

Chapter 3: The Fundamental Postulate 55
3.1 Phase Space 55
3.2 Observables 57
3.3 The Fundamental Postulate: Entropy as Phase-Space Volume 58
3.4 Liouville?s Theorem 59
3.5 Quantum States 63
3.6 Systems in Contact 66
3.7 Variational Principle 67
3.8 The Ideal Gas 68
3.9 The Probability Distribution 70
3.10 Maxwell Distribution 71
3.11 The Ising Paramagnet 71
3.12 The Canonical Ensemble 74
3.13 Generalized Ensembles 77
3.14 The p-T Ensemble 80
3.15 The Grand Canonical Ensemble 82
3.16 The Gibbs Formula for the Entropy 84
3.17 Variational Derivation of the Ensembles 86
3.18 Fluctuations of Uncorrelated Particles 87
Recommended Reading 88

Chapter 4: Interaction-Free Systems 89
4.1 Harmonic Oscillators 89
4.2 Photons and Phonons 93
4.3 Boson and Fermion Gases 102
4.4 Einstein Condensation 112
4.5 Adsorption 114
4.6 Internal Degrees of Freedom 116
4.7 Chemical Equilibria in Gases 123
Recommended Reading 124

Chapter 5: Phase Transitions 125
5.1 Liquid-Gas Coexistence and Critical Point 125
5.2 Van der Waals Equation 127
5.3. Other Singularities 129
5.4 Binary Mixtures 130
5.5 Lattice Gas 131
5.6 Symmetry 133
5.7 Symmetry Breaking 134
5.8 The Order Parameter 135
5.9 Peierls Argument 137
5.10 The One-Dimensional Ising Model 140
5.11 Duality 142
5.12 Mean-Field Theory 144
5.13 Variational Principle 147
5.14 Correlation Functions 150
5.15 The Landau Theory 153
5.16 Critical Exponents 156
5.17 The Einstein Theory of Fluctuations 157
5.18 Ginzburg Criterion 160
5.19 Universality and Scaling 161
5.20 Partition Function of the Two-Dimensional Ising Model 165
Recommended Reading 170

Chapter 6: Renormalization Group 173
6.1 Block Transformation 173
6.2 Decimation in the One-Dimensional Ising Model 176
6.3 Two-Dimensional Ising Model 179
6.4 Relevant and Irrelevant Operators 183
6.5 Finite Lattice Method 187
6.6 Renormalization in Fourier Space 189
6.7 Quadratic Anisotropy and Crossover 202
6.8 Critical Crossover 203
6.9 Cubic Anisotrophy 208
6.10 Limit n 209
6.11 Lower and Upper Critical Dimensions 213
Recommended Reading 214

Chapter 7: Classical Fluids 215
7.1 Partition Function for a Classical Fluid 215
7.2 Reduced Densities 219
7.3 Virial Expansion 227
7.4 Perturbation Theory 244
7.5 Liquid Solutions 246
Recommended Reading 249

Chapter 8: Numerical Simulation 251
8.1 Introduction 251
8.2 Molecular Dynamics 253
8.3 Random Sequences 259
8.4 Monte Carlo Method 261
8.5 Umbrella Sampling 272
8.6 Discussion 274
Recommended Reading 275

Chapter 9: Dynamics 277
9.1 Brownian Motion 277
9.2 Fractal Properties of Brownian Trajectories 282
9.3 Smoluchowski Equation 285
9.4 Diffusion Processes and the Fokker-Planck Equation 288
9.5 Correlation Functions 289
9.6 Kubo Formula and Sum Rules 292
9.7 Generalized Brownian Motion 293
9.8 Time Reversal 296
9.9 Response Functions 296
9.10 Fluctuation-Dissipation Theorem 299
9.11 Onsager Reciprocity Relations 301
9.12 Affinities and Fluxes 303
9.13 Variational Principle 306
9.14 An Application 308
Recommended Reading 310

Chapter 10: Complex Systems 311
10.1 Linear Polymers in Solution 312
10.2 Percolation 321
10.3 Disordered Systems 338
Recommended Reading 356

Appendices 357
Appendix A Legendre Transformation 359
A.1 Legendre Transform 359
A.2 Properties of the Legendre Transform 360
A.3 Lagrange Multipliers 361

Appendix B Saddle Point Method 364
B.1 Euler Integrals and the Saddle Point Method 364
B.2 The Euler Gamma Function 366
B.3 Properties of N-Dimensional Space 367
B.4 Integral Representation of the Delta Function 368

Appendix C A Probability Refresher 369
C.1 Events and Probability 369
C.2 Random Variables 369
C.3 Averages and Moments 370
C.4 Conditional Probability: Independence 371
C.5 Generating Function 372
C.6 Central Limit Theorem 372
C.7 Correlations 373

Appendix D Markov Chains 375
D.1 Introduction 375
D.2 Definitions 375
D.3 Spectral Properties 376
D.4 Ergodic Properties 377
D.5 Convergence to Equilibrium 378
Appendix E Fundamental Physical Constants 380

Bibliography 383
Index 389