Equilibrium Statistical Physics

Description

This textbook gradually introduces students to the statistical mechanical study of the different phases of matter and to the phase transitions between them. It uses simple yet fully detailed models of both hard and soft matter systems to demonstrate core concepts, developing the subject matter in a thorough and accessible pedagogical manner throughout. Starting from an introduction to basic thermodynamics and statistical physics, the book progresses from ideal, non-interacting systems to real systems exhibiting classical interactions and phase transitions. It concludes with a selection of more advanced topics, such as the renormalisation group approach to critical phenomena, the density functional theory of interfaces, and kinematic aspects of the phase transformation process. This updated second edition features a considerably expanded study of the topology of the phases, including applications to modern problems such as topological defects of nematic liquid crystals and the topological phase transition of a two-dimensional spin system. Along with a complete introductory overview of the theory of phase transitions, this textbook provides students with ample material for deeper study. References include suggestions for more detailed treatments and six appendices supply overviews of the mathematical tools employed in the text. Preface to the Second Edition Preface to the First Edition Contents Part I Basics 1 Mechanics 1.1 Classical Mechanics 1.2 Hamilton’s Equations 1.3 External Parameters 1.4 Dynamical Functions 1.5 Quantum Mechanics 1.6 Self-adjoint Operators 1.7 Eigenvalue Equation 1.8 Schrödinger’s Equation 1.8.1 Free Particle 1.8.2 Harmonic Oscillator 1.8.3 Particle in a Magnetic Field 1.9 System of Identical Particles Further Reading 2 Thermodynamics 2.1 Fundamental Equation 2.2 Intensive Variables 2.3 Law of Entropy Increase 2.4 Thermodynamic Potentials 2.5 Equilibrium Conditions 2.6 Stability Conditions 2.7 Coexistence Conditions 2.8 Phase Diagrams 2.8.1 Gibbs Free Energy 2.8.2 Helmholtz Free Energy 2.8.3 van der Waals Loop 2.8.4 An Example of a Phase Diagram Further Reading 3 Statistical Physics 3.1 Dynamical Functions and Fields 3.2 Liouville’s Equation 3.3 Systems in Equilibrium 3.4 Density Operator 3.5 Ergodicity 3.6 Thermodynamic Limit 3.7 Symmetry Breaking Further Reading Part II Ideal Systems 4 Microcanonical Ensemble 4.1 Classical Microcanonical Ensemble 4.2 Classical Ideal Gas 4.3 Entropy and the Gibbs Paradox 4.4 Temperature and Thermal Equilibrium 4.5 Ideal Systems 4.6 Equipartition Theorem 4.7 Equation of State 4.8 Entropy and Irreversibility 4.9 Quantum Microcanonical Ensemble 4.10 Absolute Negative Temperatures Further Reading 5 Canonical Ensemble 5.1 Classical Canonical Ensemble 5.2 Mean Values and Fluctuations 5.3 Helmholtz Free Energy 5.4 Classical Ideal Gas 5.5 Ideal Gas in an External Potential 5.6 Equipartition Theorem 5.7 Classical Theory of Radiation 5.8 Classical Theory of Solids 5.9 Quantum Canonical Ensemble 5.10 Ideal Quantum Systems 5.11 Maxwell–Boltzmann Statistics 5.12 Maxwell–Boltzmann’s Ideal Gas 5.13 Brillouin’s Paramagnetism 5.14 Photon Gas 5.15 Phonon Gas Further Reading 6 Grand Canonical Ensemble 6.1 Classical Grand Canonical Ensemble 6.2 Mean Values and Fluctuations 6.3 Grand Potential 6.4 Classical Ideal Gas 6.5 Classical Ideal Gas in an External Potential 6.6 Two-Particle Distribution Function 6.7 Density Fluctuations 6.8 Correlations at the Critical Point 6.9 Quantum Grand Canonical Ensemble 6.10 Bose–Einstein and Fermi–Dirac Statistics 6.11 Virial Expansions in the Classical Limit 6.12 Boson Gas: Bose–Einstein Condensation 6.12.1 Specific Heat 6.13 Fermion Gas Further Reading Part III Non-ideal Systems 7 Classical Systems with Interactions 7.1 Thermodynamic Integration 7.2 Thermodynamic Perturbation Theory 7.3 Virial Expansions 7.4 Direct Correlation Function 7.5 Density Functional Theory 7.6 Mean Field Theory 7.7 Numerical Simulations 7.7.1 Molecular Dynamics 7.7.2 Monte Carlo Method Further Reading 8 Phases of Matter 8.1 Crystals 8.1.1 Crystal Structure 8.1.2 Cell Theory 8.1.3 van der Waals Theory 8.1.4 Variational Theory 8.2 Fluids 8.2.1 Dense Fluids 8.2.2 Fluid Structure 8.2.3 Fluids and Glasses 8.2.4 Vapor and Liquid 8.3 Mixtures 8.3.1 Binary Mixtures 8.3.2 Colloidal Suspensions 8.3.3 Asakura–Oosawa Potential 8.3.4 DLVO Potential 8.4 Liquid Crystals 8.4.1 Maier–Saupe Theory 8.4.2 Onsager Theory 8.5 Polymers 8.5.1 Radius of Gyration 8.5.2 Flory–Huggins Theory Further Reading 9 Phase Transitions 9.1 Structural Transitions 9.1.1 Fluid–Solid Transition 9.1.2 Isotropic–Nematic Transition 9.2 Isostructural Transitions 9.2.1 Liquid–Vapor Transition 9.2.2 Solid–Solid Transition 9.3 Symmetry Breaking and Order Parameters 9.4 Landau Theory 9.4.1 Continuous Transitions 9.4.2 Discontinuous Transitions 9.5 Bifurcation Theory 9.6 Critical Points 9.6.1 Isolated Critical Points 9.6.2 Liquid–Vapor Critical Point 9.6.3 Solid–Solid Critical Point 9.6.4 Consolute Critical Point 9.6.5 Critical Lines 9.7 Summary 9.8 Triple Points 9.8.1 Ordinary Triple Point 9.8.2 Critical Endpoint 9.8.3 Bicritical Point 9.8.4 Tricritical Point Further Reading Part IV More Advanced Topics 10 Critical Phenomena 10.1 Classical Theory 10.2 Critical Exponents 10.3 Scaling Hypothesis 10.4 Correlation Length and Universality 10.5 Renormalization Group (RG) Idea 10.6 Critical Exponents and the Flory Exponent 10.7 RG Calculation of the Flory Exponent 10.7.1 RG Transformations 10.7.2 Fixed Points of the RG 10.7.3 Non-classical Flory Exponent 10.7.4 Critical Dimension 10.7.5 ε-Expansion 10.7.6 Differential RG Equations 10.7.7 Stability of the Fixed Points 10.7.8 Numerical Value of the Flory Exponent 10.8 Numerical Values of the Critical Exponents Further Reading 11 Interfaces 11.1 Non-uniform Systems 11.2 Density Profile 11.3 Pressure Profile 11.4 Surface Tension Further Reading 12 Topological Defects and Topological Phase Transitions 12.1 Topological Defects in a 3D Nematic Liquid Crystal 12.1.1 Anchoring 12.1.2 Frustration 12.1.3 Topological Defects 12.1.4 Elastic Deformations 12.1.5 Frank–Oseen Elastic Constants 12.1.6 Disclinations 12.1.7 Schlieren Texture 12.2 Topological Phase Transition in a 2D Spin System 12.2.1 A Classical Spin System 12.2.2 Absence of Long-Range Order in 2D Systems 12.2.3 Topological Phases 12.2.4 Relation to the Vorticity 12.2.5 Relation to the 2D Coulomb Plasma 12.2.6 A Topological Phase Transition 12.2.7 Absence of Screening in 2D Coulomb Systems 12.2.8 RG Equations of the Kosterlitz–Thouless Transition Further Reading 13 Phase Transformation Kinetics 13.1 Homogeneous Nucleation 13.1.1 Becker–Döring Theory 13.1.2 Zeldovich–Frenkel Theory 13.1.3 Avrami Theory 13.2 Heterogeneous Nucleation 13.3 Spinodal Decomposition 13.4 Glass Transition Further Reading Appendix A Legendre Transformations A.1 Case of One Variable A.2 Case of n Variables A.2.1 Some Examples Appendix B Random Variables B.1 One-Dimensional Random Variable B.1.1 Some Examples B.2 Approximation Methods B.3 n-Dimensional Random Variable B.4 n-Dimensional Gaussian Fluctuations Appendix C Functional and Functional Derivative C.1 Definition of a Functional C.2 The Functional Derivative C.3 Some Examples Appendix D A Quasicrystalline Lattice D.1 Forbidden Symmetries of Periodic 2D Lattices D.2 A Quasiperiodic 2D Lattice Appendix E The Bi-axial Nematic Appendix F The Helmholtz Decomposition of a Vector Field Appendix G The d-Dimensional Coulomb Potential Index