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.
Author(s): Marc Baus, Carlos F. Tejero
Edition: 2
Publisher: Springer Nature Switzerland AG
Year: 2021
Language: English
Pages: 437
Tags: Thermodynamics, Statistical Physics, Ensemble, Interactions, Phases, Transitions, Interfaces
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
