This book presents an introduction to the main concepts of statistical physics, followed by applications to specific problems and more advanced concepts, selected for their pedagogical or practical interest. Particular attention has been devoted to the presentation of the fundamental aspects, including the foundations of statistical physics, as well as to the discussion of important physical examples. Comparison of theoretical results with the relevant experimental data (with illustrative curves) is present through the entire textbook. This aspect is facilitated by the broad range of phenomena pertaining to statistical physics, providing example issues from domains as varied as the physics of classical and quantum liquids, condensed matter, liquid crystals, magnetic systems, astrophysics, atomic and molecular physics, superconductivity and many more. This textbook is intended for graduate students (MSc and PhD) and for those teaching introductory or advanced courses on statistical physics.
Key Features
A rigorous and educational approach of statistical physics illustrated with concrete examples.
A clear presentation of fundamental aspects of statistical physics.
Many exercises with detailed solutions.
Nicolas Sator is Associate Professor at Sorbonne University, Paris, France. He is a member of the Laboratory of Theoretical Physics of Condensed Matter (LPTMC) and his research focuses on the physics of liquids.
Nicolas Pavloff is Professor at Paris-Saclay University, France. He is a member of Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS) and his domain of research is quantum fluid theory.
Lénaïc Couëdel is Professor at the University of Sasktchewan, Saskatoon, Canada and researcher at CNRS, France. His research area is plasma physics with a focus on complex plasma crystals.
Author(s): Nicolas Sator, Nicolas Pavloff, Lénaïc Couëdel
Publisher: CRC Press
Year: 2023
Language: English
Pages: 450
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Contributors
Chapter 1: Microscopic Description of a Macroscopic System
1.1. Microstate of a System
1.1.1. Classical Framework
1.1.2. Quantum Framework
1.2. Deterministic Evolution of Microstates
1.2.1. Hamiltonian
1.2.2. Phase Space Dynamics
1.3. From a Microscopic Description to a Macroscopic Description
1.3.1. Macrostate and Thermodynamic Equilibrium
1.3.2. Failure of the Deterministic Approach
1.4. The Need for a Probabilistic Approach
1.4.1. An Instructive Example: the Joule Expansion
1.4.2. Rudiments of Gas Kinetic Theory
1.5. Statistical Description of a System: Statistical Ensembles
1.5.1. Statistical Ensembles
1.5.2. Probability of a Microstate and Ensemble Averages
1.6. Exercises
Chapter 2: Microcanonical Statistical Ensemble
2.1. Postulates of Equilibrium Statistical Physics
2.2. Counting Accessible Microstates
2.2.1. Uncertainty on the Energy of an Isolated System
2.2.2. Discretisation of Phase Space in the Framework of Classical Mechanics
2.2.3. Distinguishable or Indistinguishable Particles in Classical Physics
2.2.4. Ω(N, V, E) and Microcanonical Averages
2.2.5. Dependence of Ω(N, V, E) on E
2.3. Probability Distribution Function of a Macroscopic Internal Variable
2.4. Discussion on the Foundations of Statistical Physics
2.4.1. Liouville’s Theorem
2.4.2. Ergodic Problem, Hypothesis and Theory
2.4.3. Statistical Physics According to Boltzmann
2.5. Exercises
Chapter 3: Statistical Thermodynamics
3.1. Temperature
3.1.1. Thermal Equilibrium between Macroscopic Systems
3.1.2. Definition and Properties of Statistical Temperature
3.2. Statistical Entropy
3.2.1. Definition and Properties of Statistical Entropy
3.2.2. Principle of Maximum Entropy and Second Law of Thermodynamics
3.3. Pressure and Chemical Potential
3.3.1. Thermal, Mechanical and Chemical Equilibrium between Macroscopic Systems
3.3.2. Generalised Forces and Thermodynamic Transformations
3.4. Gibbs Entropy and Information Theory
3.5. Irreversibility and Entropy
3.5.1. Second Law of Thermodynamics
3.5.2. Statistical Interpretation of Irreversibility
3.5.3. Apparent Paradoxes of Statistical Physics
3.6. Exercises
Chapter 4: Canonical and Grand Canonical Statistical Ensembles
4.1. Canonical Ensemble
4.1.1. Canonical Probability Distribution Function and Partition Function
4.1.2. Free Energy
4.1.3. Average Values and Thermodynamic Quantities
4.1.4. Thermodynamic Equilibrium and Minimum Free Energy
4.1.5. Equipartition of Energy Theorem
4.2. Grand Canonical Ensemble
4.2.1. Grand Canonical Probability Distribution and Grand Partition Function
4.2.2. Grand Potential
4.2.3. Average Values and Thermodynamic Quantities
4.2.4. Thermodynamic Equilibrium and Minimum Grand Potential
4.3. Generalisation and Equivalence of the Statistical Ensembles
4.3.1. Statistical Ensembles
4.3.2. Gibbs Entropy and Information Theory
4.4. Exercices
Chapter 5: Simple Classical Fluids
5.1. Simple Classical Fluid Model
5.1.1. Validity of the Ideal Gas Approximation
5.1.2. Configuration Integral
5.2. Virial Expansion
5.2.1. Expressions of the Virial Coefficients
5.2.2. Second Virial Coefficient
5.3. Mean Field Approximation and van der Waals Equation of State
5.3.1. Determination of the Effective Potential
5.3.2. Van der Waals Fluid
5.3.3. Critical Behaviour and Universality
5.4. Microscopic Structure of a Fluid and Pair Correlation Function
5.4.1. Pair Correlation Function
5.4.2. Thermodynamic Quantities
5.4.3. Low Density Approximation
5.5. Exercises
Chapter 6: Quantum Statistical Physics
6.1. Statistical Distributions in Hilbert Space
6.1.1. State Counting
6.1.2. Microcanonical Ensemble
6.1.3. Canonical Ensemble
6.1.4. Grand Canonical Ensemble
6.2. Paramagnetism
6.2.1. Classical Analysis
6.2.2. Quantum Analysis
6.3. Vibrational Contribution to the Heat Capacity of Diatomic Molecules
6.4. Indistinguishable Particles in Quantum Mechanics
6.4.1. Canonical Treatment
6.4.2. Calculation of the Grand Canonical Partition Function
6.4.3. Maxwell-Boltzmann Approximation
6.5. Rotational Contribution to the Heat Capacity of Diatomic Molecules
6.5.1. Hetero-nuclear Molecules
6.5.2. Homo-nuclear Molecules
Chapter 7: Bosons
7.1. Heat Capacity of Solids: Debye Model
7.1.1. Separation of Variables
7.1.2. Equivalence of Canonical and Grand Canonical Analysis
7.1.3. Physical Discussion
7.2. Black Body Radiation
7.2.1. Cosmic Microwave Background
7.2.2. Interstellar Cyanogen
7.3. Liquid Helium
7.3.1. Superfluid Helium
7.3.2. Zero Temperature. Landau Criterion
7.3.3. Finite Temperature. Normal Fraction
7.3.4. Finite Temperature. Heat Capacity
7.4. Bose–Einstein Condensation
Chapter 8: Fermions
8.1. Free Fermion Gas
8.1.1. Classical Limit: High Temperature and Low Density
8.1.2. Degenerate Limit: Low Temperature and High Density
8.2. Relativistic Fermi Gas
8.3. White Dwarfs, Chandrasekhar Mass
8.4. Pauli Paramagnetism
8.5. Two-dimensional 3He Gas
8.6. Landau Diamagnetism
8.6.1. Non Degenerate Limit
8.6.2. Low Field Limit
Chapter 9: Phase Transition – Mean Field Theories
9.1. Ferromagnetism
9.1.1. Heisenberg and Ising Hamiltonians
9.1.2. Weiss Molecular Field
9.1.3. Bragg-Williams Approximation
9.2. Landau Theory of Phase Transition
9.2.1. Partial Partition Function
9.2.2. Order Parameter and Second Order Phase Transition
9.2.3. Landau Free Energy
9.2.4. Effect of an External Field. Phase Diagram
9.3. Liquid Crystals
9.3.1. Property of the Order Parameter Q
9.3.2. Landau’s Theory of the Nematic-Isotropic Transition
9.4. Tri-critical Transition: Blume-Emery-Griffiths Model
9.4.1. Tricritical Points
Chapter 10: Phase Transition – Spatial Variations of the Order Parameter
10.1. Ginzburg-Landau Theory
10.2. Functional Derivative
10.3. Several Non-homogeneous Configurations
10.3.1. Effect of a Weak Non-uniform External Field
10.3.2. Domain Wall
10.3.3. Effect of Boundary Conditions
10.4. Static Deformation in a Nematic Crystal
10.5. Ginzburg-Landau Theory of Superconductivity
10.5.1. Meissner Effect and Fluxoid Quantisation
10.5.2. Thermodynamics and Magnetism
10.5.3. Critical Magnetic Field
10.5.4. Interface Between Normal and Superconducting Phases
Chapter 11: Phase Transitions – Validity of the Mean Field Theory – Scaling Laws
11.1. Fluctuation-dissipation Theorem
11.2. Mean Field as a Saddle-point Approximation
11.2.1. Correlation Function in Landau Theory
11.3. Validity of the Mean Field Approximation: Ginzburg Criterion
11.4. Some Comparisons
11.4.1. Superconductors
11.4.2. Ising System
11.4.3. Liquid Helium
11.5. Behaviour in the Vicinity of the Critical Point
11.6. Scale Invariance
11.6.1. Power Law
11.6.2. Relationships Between Critical Exponents
11.6.3. Experimental Data
Chapter 12: Percolation
12.1. Introduction
12.2. Evaluation of the Percolation Threshold
12.2.1. Bond Percolation on a Two-dimensional Square Lattice
12.2.2. Mean Field Approximation
12.2.3. The Renormalisation Procedure
12.3. Thermodynamics and Critical Exponents
12.3.1. Cluster Size Distribution
12.3.2. Free Energy
Chapter 13: Dynamics of Phase Transitions
13.1. Time-dependent Ginzburg-Landau Equation
13.1.1. Homogeneous Out-of-equilibrium System
13.1.2. Weakly Inhomogeneous Out-of-equilibrium System
13.2. Domain Wall Motion
13.2.1. Planar Interface
13.2.2. Interface of Arbitrary Shape, Allen-Cahn Equation
13.3. Quench Dynamics
13.3.1. Short Times
13.3.2. Long Time
13.3.3. O.J.K. Theory
13.4. Kibble-Zurek Mechanism
13.5. Disappearance of a Phase
Chapter 14: Bose–Einstein Condensates of Weakly Interacting Dilute Gases
14.1. Gross-Pitaevskii Equation
14.2. Zero Temperature Ground State of a Trapped Gas
14.2.1. Variational Approach
14.2.2. Thomas-Fermi Approximation
14.3. Trapped Gas at Finite Temperature
14.3.1. Ideal Gas
14.3.2. Interacting Gas
14.4. Infinite Homogeneous Gas at Finite Temperature
14.4.1. First Approximation
14.4.2. Equation of State. Hartree-Fock Approximation
14.5. Elementary Excitations
14.5.1. Infinite and Homogeneous System
14.5.2. Trapped Gas
Appendix A: Mathematical Complements
A.1. Special Functions
A.2. Gaussian Integrals
A.3. Saddle-point Method
A.4. Volume of a Hypersphere
A.5. Method of Lagrange Multipliers
A.6. Fourier Analysis
A.6.1. Fourier Transform
A.6.2. Fourier Series
A.7. Semi-classical Calculations
A.8. Stokes and Gauss-Ostrogradski Theorems
A.9. Physical Constants
Index
