Principles of Equilibrium Statistical Mechanics

Transcription

1 Debashish Chowdhury, Dietrich Stauffer Principles of Equilibrium Statistical Mechanics WILEY-VCH Weinheim New York Chichester Brisbane Singapore Toronto

2 Table of Contents Part I: THERMOSTATICS 1 1 BASIC PRINCIPLES OF THERMOSTATICS Introduction Extensive and Intensive Variables Entropy and Temperature Concept of Equilibrium Internal Energy and the First Law "Processes" in Thermostatics; Cycles Constraints and Walls Fundamental Relation, Equations of State Euler and Gibbs-Duhem Relations Second Law of Thermostatics Third Law of Thermostatics Conditions for Fundamental Relations Extremum Principles for Equilibrium Entropy-maximum/Energy-minimum Principles Concept of a Reservoir Why T is Called Temperature Thermostatic Potentials and Equilibrium Response Functions Thermostatic Relations Second Law: Alternative Statements Some Applications of Thermostatics Chapter Summary Historical Notes First Law of Thermostatics Second Law of Thermostatics Third Law of Thermostatics Gibbs and Modern Thermostatics Duhem and Applications of Thermostatics Problems Supplementary Notes Convexity and Concavity of Thermostatic Variables Mathematical Foundation of Thermostatics Applications of Thermostatics 58

3 XVI 2 THERMOSTATICS OF PHASE TRANSITIONS Introduction Phases and Components Phase Diagrams and Critical Point Fluids Magnets Stability of Phases Intrinsic Stability of Homogeneous State Mutual Stability of Coexisting Phases Stable, Metastable and Unstable States Van der Waals Gas Curie-Weiss Magnet Derivatives of Thermostatic Potentials Classification of Phase Transitions Critical Exponents Scaling Relations Chapter Summary Historical Notes Concept of Latent Heat Critical Phenomena in Fluids Critical Phenomena in Magnets Problems 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 Introduction Isolated Classical Systems Boltzmann Hypothesis Illustration with Ideal Classical Ising Magnet Ill Illustration with Ideal Classical Monatomic Gas Entropy of Mixing and Gibbs'Paradox Equipartition of Energy 116

4 XVII Concepts of Ensemble and Ergodicity Closed Classical Systems The Partition Function Connection with Thermostatics Illustration with Ideal Classical Ising Magnet Illustration with Ideal Classical Monatomic Gas Equipartition of Energy Open Classical Systems Grand Partition Function Connection with Thermostatics Illustration with Ideal Classical Monatomic Gas Unified Presentation of the Rules Micro-canonical Ensemble Canonical Ensemble Grand-canonical Ensemble Maximum Entropy Principle Microcanonical Ensemble Canonical Ensemble Grand Canonical Ensemble Illustration with Ideal Quantum Gases Micro-Canonical Ensemble Canonical Ensemble Grand-Canonical Ensemble Quantum Systems; Density Operator Micro-Canonical Ensemble Canonical Ensemble Chapter Summary Historical Notes James Clerk Maxwell Ludwig Boltzmann Josiah Willard Gibbs Maximum Entropy Principle Problems Supplementary Notes Entropy Calculation from Trajectory Entropy and Information Wigner Function 169

5 XVIII 4 FLUCTUATIONS, CORRELATIONS AND RESPONSE Introduction Energy: Most Probable and Mean Particle Number: Most Probable and Mean Fluctuations Fluctuations in Energy Fluctuations in Volume Fluctuations in Particle Number Cross Correlation Between Fluctuations Fluctuations in Entropy Microstate Population Fluctuations Ideal Bose Gas Ideal Fermi Gas Ideal Boltzmann Gas Classical versus Quantum Fluctuations Correlation Functions Fluctuation-Response Theorem Illustration with Ising Magnets Illustration with Simple Fluids Scattering Measures Correlation Foundations of Laws of Thermostatics Chapter Summary Historical Notes Problems Supplementary Notes 192 S.4.1 Radial Distribution Functions for Fluids STATISTICAL PHYSICS OF IDEAL CLASSICAL GASES Introduction Monatomic Gas; MB Distribution Full Partition Function Monatomic Molecules Diatomic Molecules Chapter Summary Historical Notes Problems Supplementary Notes 213 S.5.1 Kinetic Theory of Gases 213

6 XIX 6 STATISTICAL PHYSICS OF IDEAL QUANTUM GASES Introduction Some General Results Concept of Density of Levels Ideal Fermi Gas at T = Fermi Energy Internal Energy Pressure Ideal Fermi Gas at T ф Sommerfeld Expansion Chemical Potential Specific Heat Magnetic Susceptibility Fermi-Dirac and Bose-Einstein Integrals Ideal Bose Gas: Bose-Einstein Condensation Number density and Chemical Potential Internal energy and Specific Heat Entropy Isotherms on the PV diagram Chapter Summary Historical Notes FD Statistics BE Statistics and Discovery of BE Condensation Problems Supplementary Notes Quantum Gases in Astrophysics Phonon Contribution to Specific Heat of Crystals PART III STATISTICAL MECHANICS OF INTERACTING SYSTEMS INTERACTING SYSTEMS; THERMODYNAMIC LIMIT Introduction Models of Fluids Partition Function Lattices 262

7 7.3.1 Bravais Lattices Bethe Lattice Spin Models on Lattices Classical Spin-1/2 Ising Model Spin-1/2 Ising Model: Physical Realizations Generalization: From Spin-1/2 to Spin-1 Ising Model Generalization: From Ising to Vector Spins Generalization: Prom Ising to Potts Variables Quantum Spin Models Classical Limit of Quantum Spin Models Magnetic Physical Realizations of Spin Models Restrictions on the Interactions Zeroes of the Grand Partition Function Yang-Lee Theorems and Their Consequences Chapter Summary Historical Notes Spin Models and Their Physical Realizations Thermodynamic Limit Yang-Lee Theorem Problems Supplementary notes Continuum Models of Fluids Spin Models on Discrete Lattices Vertex Models Generalization: From "Hard" Spins to "Soft" Spins Spin Models with Quenched Disorder Anisotropic Hamiltonians for Spin System Thermodynamic Limit Complex Temperature Plane: Zeroes of Partition Function Quantum Phase Transitions and Critical Phenomena. 308 EXACT SOLUTION OF SOME INTERACTING SYSTEMS Introduction Ising Model in d = 1: Partition Function Open Chain in the Absence of External Field Closed Chain: Transfer Matrix Approach Zeroes of the Partition Function Ising Model in d = 1: Thermostatics 321

8 XXI Spontaneous Magnetization Magnetic Susceptibility Magnetic Specific Heat Ising Model in d = 1: Correlations Open Chain in Zero Field Closed Chain: Transfer Matrix Approach Important Concepts in Phase Transitions Order Parameter Peierls-Griffiths Argument Phase Transitions: A "Mathematical Mechanism" Lower Critical Dimension Critical Exponents Exact Solution of Fluid Models in d = Tonks Gas Takahashi Gas Chapter Summary Historical Notes From Ising to Peierls From Kramers to Onsager One-dimensional Models of Fluids Problems Supplementary Notes Peierls-Griffiths Arguments Duality and Star-Triangle Transformations; Exact T c 's Relevance of the Range of the Interaction Exact Solution of the Two-dimensional Ising Model Exact Solution of the n-vector Model Exact Solution of the Spherical Model in d-dimension COMPUTER SIMULATION METHODS Introduction Monte Carlo Simulation Random Sampling Illustrated: Random Walk Importance Sampling Illustrated: Ising Model Fluctuations Equilibration Time and Correlation Time MC Simulation in Micro-canonical Ensemble MC Simulation of Fluids Molecular Dynamics 364

9 XXII Constant-Energy Molecular Dynamics Constant Temperature/Constant Pressure MD Non-Self-Averaging Quantities Chapter Summary Historical Notes MD approach MC approach Problems Supplementary Notes Random Numbers and Random-Number Generators Multi-Spin Coding MC Simulation: Tricks of the Trade MD Simulation: Tricks of the Trade Langevin Dynamics Simulation Vector- and Parallel Processors; Special Computers Q2R update rules MEAN-FIELD THEORY I: Van der Waals-Weiss Formulation Introduction MFA for the d-dimensional Ising Model A Pedestrian's Approach An Alternative Variational Approach Ordering Temperature Thermostatic Properties Correlation Function Critical Exponents MFA for the d-dimensional Fluid The Equation of State Critical Point The Law of Corresponding States Critical Exponents Comparison of Magnets and Fluids Validity and Accuracy of MFA Chapter Summary Historical Notes Weiss and MFT of Magnets Van der Waals and MFT of Fluids Inadequacies of MFT 408

10 XXIII 10.8 Problems Supplementary notes Order-disorder Transition in Binary Alloys Bethe-Peierls-Weiss (BPW) Approximation Correlation Function of Fluids in the MFA Mean-field Theory of Polymers MEAN-FIELD THEORY II: Exact Solution in Infinite Dimension Introduction Infinite-Range Ising Model First Approach: Largest Term Method Second Approach: Method of Steepest Descent Infinite-Range Classical Gas Chapter Summary Historical Notes Problems Supplementary Notes Classical n-vector Models with Long-range Interactions Potts Model with Infinite-range Interactions Ising Model on a Bethe Lattice Classical Fluids in Infinite Dimension MEAN-FIELD THEORY III: Landau Formulation Introduction Second Order Phase Transitions Critical Exponents in Landau Theory First Order Phase Transitions Field-driven Transition T-driven Transition; Asymmetric Case T-driven Transition; Symmetric Case Landau-Ginzburg Theory Two-Point Correlation Function Fourier Transform Method Method of Solving Differential Equation Ginzburg Criterion Chapter Summary Historical Notes Problems 458

11 XXIV 12.10Supplementary notes Landau theory for Tricritical Points Gaussian Fluctuations and Ginzburg Criterion "Derivation" of Landau-Ginzburg Effective Hamiltonian BEYOND MEAN-FIELD APPROXIMATION: Scaling and Renormalization Group Introduction Scaling and Universality Scaling Hypothesis Scaling Theory: a General Formulation Concept of Universality and Universality Classes Heuristic Justification of Scaling Renormalization, Fixed Points and RG Flow Chapter Summary Historical Notes Problems Supplementary Notes 486 S.13.1 Renormalization Group Treatment of Percolation Mathematical Appendix 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 Variables 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

12 XXV A Binomial Distribution 500 A From Binomial to Gaussian Distribution 502 A.12.4 From Binomial to Poisson Distribution A.13 Problems References 508