Statistical Mechanics in a Nutshell

tatistical Mechanics in a Nutshell Luca Peliti Translated PRINCETON by Mark Epstein UNIVERSITY PRESS . PRINCETON AND OXFORD Contents Preface the to English Edition xi Preface T] xiii Introduction 1.1 The 1.2 Subject l Matter of Statistical Mechanics Statistical Postulates 1.3 An The Ideal Gas Example: 3 1.4 Conclusions Recommended 2~| 7 Reading 8 Thermodynamics 2.1 9 Thermodynamic Systems 2.2 9 Extensive Variables 11 2.3 The Central Problem of Thermodynamics 2.4 Entropy 2.5 Simple 12 13 Problems 14 2.6 Heat and Work 2.7 The Fundamental 2.8 Energy 18 Equation Free 24 Thermodynamic Energy and Maxwell Relations and Enthalpy 2.11 Gibbs Free 2.12 The Measure of Chemical Potential 2.13 The 23 Scheme 2.9 Intensive Variables and 2.10 1 3 Energy Koenig Born Diagram Potentials 26 30 31 33 35 Contents 2.14 Other Thermodynamic Potentials 2.15 The Euler and Gibbs-Duhem 36 Equations 37 2.16 Magnetic Systems 2.17 Equations of State 40 2.18 Stability 41 39 2.19 Chemical Reactions 2.20 44 Phase Coexistence 2.21 The 2.22 The Coexistence Curve 45 Clausius-Clapeyron Equation 47 48 2.23 Coexistence of Several Phases 49 2.24 The Critical Point 50 2.25 Planar Interfaces 51 Recommended 54 Reading The Fundamental Postulate 55 3.1 Phase 55 Space 3.2 Observables 57 3.3 The Fundamental Postulate: Entropy as Phase-Space 3.4 Liouville's Theorem Volume 58 59 3.5 Quantum States 63 3.6 Systems 66 in Contact 3.7 Variational 3.8 Principle 67 The Ideal Gas 3.9 The 68 Probability Distribution 3.10 Maxwell Distribution 3.11 The 70 71 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 3.17 Variational Derivation of the Ensembles 3.18 Fluctuations Recommended of Uncorrelated Particles Reading Interaction-Free Systems 84 86 87 88 89 4.1 Harmonic Oscillators 89 4.2 Photons and Phonons 93 4.3 Boson and Fermion Gases 4.4 Einstein Condensation 4.5 Adsorption 4.6 Internal Equilibria Recommended 112 114 Degrees of Freedom 4.7 Chemical 102 Reading in Gases 116 123 124 Contents 5 | Phase Transitions 5.1 Liquid-Gas 5.3. Other Binary 5.4 125 Equation 127 Singularities 129 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 Duality 5.11 142 5.12 Mean-Field Theory 5.13 Variational Principle 144 147 5.14 Correlation Functions 5.15 The Landau 5.16 Critical 150 Theory 153 Exponents 5.17 The Einstein Theory 156 of Fluctuations 157 5.18 Ginzburg Criterion 5.19 Universality and Scaling 160 161 5.20 Partition Function of the Two-Dimensional Recommended ~6~| Ising Reading Renormalization Group 6.3 173 Ising 6.4 Relevant and Irrelevant 6.5 Finite Lattice Method Ising Model Operators Model 176 179 183 187 6.6 Renormalization in Fourier 6.7 Quadratic Anisotropy 165 173 6.2 Decimation in the One-Dimensional Two-Dimensional Model 170 6.1 Block Transformation Space and Crossover 189 202 6.8 Critical Crossover 203 6.9 208 Cubic 6.10 Limit 7] vii 125 Coexistence and Critical Point 5.2 Van der Waals | Anisotrophy n -> 209 °o 6.11 Lower and Upper Critical Recommended Reading Dimensions 214 Classical Fluids 7.1 Partition Function for 213 215 a Classical Fluid 215 7.2 Reduced Densities 219 7.3 Virial 227 Expansion 7.4 Perturbation Theory 7.5 Liquid Solutions Recommended Reading 244 246 249 viii | Contents Numerical Simulation 251 8.1 Introduction 251 8.2 Molecular 253 Dynamics 8.3 8.4 Random Monte Carlo Method 261 8.5 Umbrella 272 Sequences 259 Sampling 8.6 Discussion 274 Recommended Reading 275 Dynamics 277 9.1 Brownian Motion 277 9.2 Fractal 282 Properties of Brownian Trajectories 9.3 Smoluchowski Equation 9.4 Diffusion Processes and the Fokker-Planck 9.5 Correlation Functions 9.6 288 289 Kubo Formula and Sum Rules 292 9.7 Generalized Brownian Motion 293 9.8 296 Time Reversal 9.9 Response Functions 9.10 296 Fluctuation-Dissipation Theorem Onsager Reciprocity Relations 9.11 299 301 9.12 Affinities and Fluxes 303 9.13 Variational 306 9.14 Principle An Application Recommended 10 285 Equation 308 Reading 310 Complex Systems Polymers 10.1 Linear 10.2 Percolation 10.3 Disordered Recommended 311 in Solution 312 321 Systems 338 Reading 356 Appendices 357 Appendix A Legendre Transformation 359 A.i Legendre Transform 359 A.2 Properties of the Legendre Transform 360 A.3 Lagrange Multipliers Appendix 361 B Saddle Point Method B. i Euler Integrals and the Saddle Point Method 364 364 B.2 The Euler Gamma Function 366 B.3 Properties of"N-Dimensional Space 367 B. 4 Integral Representation Appendix C A Probability C. i Events and of the Delta Function Refresher Probability 368 369 369 Contents C.2 Random Variables C.3 Averages | ix 369 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. i Introduction D.2 375 Definitions 375 D.3 Spectral Properties 376 D.4 Ergodic Properties 377 D.5 Convergence Appendix to Equilibrium E Fundamental Physical 378 Constants 380 Bibliography 383 Index 389