This short textbook covers roughly 13 weeks of lectures on advanced statistical mechanics at the graduate level. It starts with an elementary introduction to the theory of ensembles from classical mechanics, and then goes on to quantum statistical mechanics with density matrix. These topics are covered concisely and briefly. The advanced topics cover the mean-field theory for phase transitions, the Ising models and their exact solutions, and critical phenomena and their scaling theory. The mean-field theories are discussed thoroughly with several different perspectives — focusing on a single degree, or using Feynman–Jensen–Bogoliubov inequality, cavity method, or Landau theory. The renormalization group theory is mentioned only briefly. As examples of computational and numerical approach, there is a chapter on Monte Carlo method including the cluster algorithms. The second half of the book studies nonequilibrium statistical mechanics, which includes the Brownian motion, the Langevin and Fokker–Planck equations, Boltzmann equation, linear response theory, and the Jarzynski equality. The book ends with a brief discussion of irreversibility. The topics are supplemented by problem sets (with partial answers) and supplementary readings up to the current research, such as heat transport with a Fokker–Planck approach.
Author(s): Jian-Sheng Wang
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 224
City: Singapore
Tags: Statistical Mechanics, Phase Transitions, Ising Models, Mean-Field Theories, Monte Carlo Methods, Langevin Equation, Fokker-Plank Equation, Boltzmann Equation
Contents
Preface
1. Thermodynamics
1.1 Introduction
1.2 Callen’s postulates
1.3 Definition of temperature
1.4 Some formal thermodynamic relations
1.5 Supplementary reading — differential, total differential, and line integral
Problems
2. Foundation of Statistical Mechanics, Statistical Ensembles
2.1 Classical dynamics
2.2 Statistical description
2.2.1 Liouville’s equation
2.3 Microcanonical ensemble
2.3.1 Boltzmann’s principle
2.4 Sackur-Tetrode formula
2.4.1 Thermodynamic limit
2.4.2 Equations of states
2.5 From microcanonical ensemble to canonical ensemble
2.5.1 Connection to thermodynamics
2.5.2 Harmonic oscillators
2.6 Grand canonical ensemble
2.7 Mathematical preliminaries — Gamma function, Gaussian integral, and volume of a hypersphere
2.8 Supplementary reading — volume vs. area measure in the microcanonical ensemble
2.9 Supplementary reading — origin of Boltzmann’s principle
Problems
3. Quantum Statistical Mechanics
3.1 Density matrix and von Neumann equation
3.1.1 Wigner transform
3.2 Examples of quantum systems
3.2.1 Harmonic oscillators
3.2.2 Quantum gases
3.2.3 Density of states
3.2.4 Electrons at low temperatures
3.2.5 Bose-Einstein condensation
3.3 Supplementary reading, Boltzmann counting method for quantum particles
Problems
4. Phase Transitions, van der Waals Equation
4.1 Phase diagrams
4.2 The van der Waals equation
4.2.1 Derivation of the van der Waals equation
4.2.2 Properties near the critical point
4.2.3 Maxwell’s construction
4.3 Series expansion to equation of state
Problems
5. Ising Models and Mean-Field Theories
5.1 The Ising model
5.1.1 Paramagnetism
5.2 Mean-field theory
5.2.1 Feynman-Jensen-Bogoliubov inequality
5.2.2 The cavity method
5.3 Landau theory
5.4 Ginzburg-Landau theory of superconductivity
Problems
6. Ising Models: Exact Methods
6.1 Transfer matrix method
6.2 High-temperature expansion and pair correlation function
6.3 Duality relation
6.4 Supplementary reading: Exact solution of 2D Ising model
Problems
7. Critical Exponents, Scaling, and Renormalization Group
7.1 Critical exponents
7.2 Scaling relations
7.3 Renormalization group methods
Problems
8. Monte Carlo Methods
8.1 Random variables of specified distribution
8.1.1 Continuous distribution
8.2 Monte Carlo evaluation of finite-dimensional integrals
8.2.1 Fundamental theorems
8.2.2 Error of Monte Carlo calculations
8.3 Importance sampling (Metropolis algorithm)
8.3.1 Markov chain
8.3.2 Ergodicity
8.3.3 Detailed balance
8.3.4 Metropolis algorithm
8.3.5 Example of the Metropolis algorithm — Ising model for magnets
8.4 Analysis of the Ising model Monte Carlo data
8.4.1 Finite-size scaling
8.5 Critical slowing down
8.6 Cluster Monte Carlo algorithms
8.6.1 Swendsen-Wang algorithm
8.6.2 Wolff single-cluster algorithm
8.7 Other applications of the Metropolis algorithm
8.7.1 Simple fluid
Problems
9. Brownian Motion—Langevin and Fokker-Planck Equations
9.1 Introduction
9.2 Brownian motion
9.2.1 Random walk model
9.3 Langevin equation
9.4 Einstein relation
9.4.1 Langevin’s argument
9.4.2 Velocity correlation and diffusion constant
9.4.3 Mobility and diffusion
9.5 Fourier method, Wiener-Khinchin theorem
9.6 Fokker-Planck equation
9.6.1 Solution of the Fokker-Planck equation
9.7 Supplementary reading — A microscopic model of Brownian motion
9.8 Supplementary reading — heat transport in classical phonon systems, a Fokker-Planck equation approach
Problems
10. Systems Near and Far from Equilibrium — Linear Response Theory and Jarzynski Equality
10.1 Introduction
10.2 Linear response theory
10.2.1 Example: a quantum oscillator under external driven force
10.3 Kubo-Greenwood formula
10.4 Jarzynski equality
Problems
11. The Boltzmann Equation
11.1 Introduction
11.2 The Boltzmann equation
11.3 H-theorem
11.4 Microscopic laws and irreversibility
11.5 Supplementary reading — electron transport, Boltzmann vs. Kubo-Greenwood
Problems
Answers to Selected Problems
Bibliography
Index
