ADVANCED STATISTICAL MECHANICS

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.

Contents:

• Preface
• Thermodynamics
• Foundation of Statistical Mechanics, Statistical Ensembles
• Quantum Statistical Mechanics
• Phase Transitions, van der Waals Equation
• Ising Models and Mean-Field Theories
• Ising Models: Exact Methods
• Critical Exponents, Scaling, and Renormalization Group
• Monte Carlo Methods
• Brownian Motion — Langevin and Fokker-Planck Equations
• Systems Near and Far from Equilibrium — Linear Response Theory and Jarzynski Equality
• The Boltzmann Equation
• Answers to Selected Problems
• Bibliography
• Index

Readership: Advanced undergraduate and graduate students in Physics; researchers in the field of thermal transport or nonequilibrium statistical physics.

Key Features:

• The book is a one-semester course likely covered in most of graduate schools in physics
• The materials bring the students up-to-date working in statistical physics and condensed matter physics, particularly the studies of phase transitions and transport in nonequilibrium systems
• The materials in the book have been the contents of a course with the same title for many years at National University of Singapore