Course Overview

Statistical Mechanics is a core graduate-level course providing rigorous analytic techniques in classical and quantum statistical mechanics. Students will master partition function formalism, ensemble theory, phase transitions, and quantum gases—essential foundations for theoretical physics, condensed matter science, and quantum information applications.

Course Code & Structure

• Course Code: PH 620 (Core Course)
• Credits: 2-1-0-3 (Lecture-Tutorial-Practical-Total)
• Prerequisite: Basic knowledge of thermal physics, Lagrangian and Hamiltonian mechanics, and quantum mechanics

Course Objectives & Outcomes

Objective: This course imparts comprehensive analytic techniques in classical and quantum statistical mechanics with emphasis on partition functions and ensemble theory.

Learning Outcomes:

• Mastery of partition function formalism and related thermodynamic concepts
• Ability to differentiate between various regimes of validity for quantum and classical ensembles
• Understanding of phase transitions and critical phenomena
• Expertise in quantum statistics including Fermi and Bose gases
• Application to real physical systems and phenomena

Course Syllabus

Review of Thermodynamics

• Zeroth law and temperature concept
• First law and internal energy
• Second law, Carnot engine, and entropy definition
• Thermodynamic potentials and Maxwell relations
• Third law and approach to absolute zero
• Legendre transformations

Review of Probability Theory

• Random variables and probability distributions
• Moments, cumulants, and generating functions
• Wick’s theorem
• Central limit theorem
• Information theory and Shannon entropy
• Applications to statistical ensembles

Ensemble Theory

• Macrostates and microstates
• Concept of ensemble and phase space
• Phase space density as probability density
• Liouville’s theorem and canonical distribution
• Ergodic hypothesis and ergodicity
• Connection to thermodynamics

Classical Statistical Mechanics

• Microcanonical ensemble and entropy
• Gibbs paradox and particle indistinguishability
• Canonical ensemble and Boltzmann distribution
• Grand canonical ensemble
• Limitations of classical mechanics
• Thermal wavelength and quantum corrections

Introduction to Phase Transitions

• Ising model and ferromagnetism
• Mean-field theory and approximations
• Critical exponents
• Landau theory of phase transitions
• Order parameters and symmetry breaking

Quantum Statistical Mechanics

• Quantum macrostates and density matrices
• Liouville’s equation in quantum form
• Canonical and grand canonical ensembles for quantum systems
• Partition functions and thermodynamic relations

Ideal Quantum Gases

• Identical particles and exchange statistics
• Fermi-Dirac and Bose-Einstein statistics
• Non-relativistic quantum gases
• Degenerate Fermi gas properties
• Degenerate Bose gas and superfluidity
• Bose-Einstein condensation in ⁴He
• White dwarf and neutron star applications

Primary Textbooks

1. M. Kardar, Statistical Physics of Particles, Cambridge University Press (2007). ISBN: 978-0521873420.

2. R. K. Pathria and P. D. Beale, Statistical Mechanics, Academic Press, Elsevier (2021). ISBN: 978-9351073970.

Reference Books

1. K. Huang, Statistical Mechanics, John Wiley & Sons (2021). ISBN: 978-9354247736.

2. J. P. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Oxford University Press (2006). ISBN: 978-0198865254.

3. D. Chandler, Introduction to Modern Statistical Physics, Oxford University Press (1987). ISBN: 978-0195042771.

4. L. E. Reichl, A Modern Course in Statistical Physics, Wiley (2016).

5. F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill (1965).

Teaching Approach

This graduate course employs:

• Rigorous mathematical derivations and proofs
• Problem-solving with emphasis on applications
• Bridging classical and quantum treatments
• Connection to experimental phenomena and real materials
• Computational methods and numerical simulations

Assessment

• Problem sets and assignments (30%)
• Mid-term examination (30%)
• Final examination (40%)

Prerequisites & Preparation

Students should have solid understanding of:

• Thermodynamics fundamentals
• Quantum mechanics at least at the level of Schrödinger equation
• Lagrangian and Hamiltonian formalism
• Basic differential equations and linear algebra

A review of probability theory and classical mechanics will be provided during the course.