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"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, ?cole Normale Sup?rieure

"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

Preface 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

Statistical mechanics is one of the most exciting areas of physics today, and it also has applications to subjects as diverse as economics, social behavior, algorithmic theory, and evolutionary biology. Statistical Mechanics in a Nutshell offers the most concise, self-contained introduction to this rapidly developing field. Requiring only a background in elementary calculus and elementary mechanics, this book starts with the basics, introduces the most important developments in classical statistical mechanics over the last thirty years, and guides readers to the very threshold of today's cutting-edge research.

Statistical Mechanics in a Nutshell zeroes in on the most relevant and promising advances in the field, including the theory of phase transitions, generalized Brownian motion and stochastic dynamics, the methods underlying Monte Carlo simulations, complex systems--and much, much more. The essential resource on the subject, this book is the most up-to-date and accessible introduction available for graduate students and advanced undergraduates seeking a succinct primer on the core ideas of statistical mechanics.


-Provides the most concise, self-contained introduction to statistical mechanics
-Focuses on the most promising advances, not complicated calculations
-Requires only elementary calculus and elementary mechanics
-Guides readers from the basics to the threshold of modern research
-Highlights the broad scope of applications of statistical mechanics