Thermodynamics and Statistical Mechanics  (Core graduate Course)

Detailed program

Week 1: Thermodynamics: Basics, Postulates 0-2, temperature, Entropy. Ideal gas, Tonks gas.

• Variables (intensive, extensive), functions, concepts.
• Reversible & irreversible processes. Postulate 0.
• Postulate 1. Postulate 2. Carnot cycle, efficiency,
• Thermodynamic temperature. Kelvin scale.
• Ideal gas.Entropy. Tonks gas.

Week 2: Van der Waals. Thermodynamic potentials. Minimax principles.

• Water phase diagram. Transitions and critical point.
• Van Der Waals (VDW) gas + law of corresponding states. 
• Legendre transforms. Thermodynamics potentials (energy U, enthalpy H, Helmholtz free energy F, Gibbs free energy G, grand potential J), Maxwell relations.
• Max S, min U, minima of F and G
• Stability conditions,  Global minimum.
• La Chatelier principle, local minimum.

Week 3: Phase diagrams. Maxwell cosntruction. Gibbs rule. Lattice models.

• VDW gas view via Helmholtz free energy. Maxwell construction.
• VDW gas view via Gibbs free energy. Maxwell construction.
• Latent heat.
• Solid-solid melt diagram (liquidus, solidus, eutectic).
• Liquid mixtures.  Definition of: Lattice binary melt vs.
• Lattice gas vs. Ising
• Gibbs phase rule.
• Black body radiation, from classical EM.
• 3rd postulate

Week 4: Postulate 3. Liouville theorem. Shannon entropy.

• 3rd postulate (coninued) C_V->0 for T=0. Meaning of "only one ground state" (examples, FD, BE, Debye phonons)
• 1D harmonic oscillator, and Bohr-Somerfeld quantization:
• demonstration that \Gamma=\int dp dq/h.
• Analytical mechanics, Lagrangian & eqns of motion,
• Canonical variables,Hamiltonian & eqns of motion.
• Poisson brackets, Poincare theorem. All this - just quotations; no proof.
• Phase space; flow in phase space is incompressible.
• Liouville's theorem (just quotation) and Liouville's equation,  in form of Poisson brackets, and in form of "mass conservation" law.
• Introduction to Shannon entropy.

Week 5: Entropy maximization in classical stat. mech. ensembles

• Extremum of entropy. Microcanonical. Gibbs paradox.
• Canonical ensemble. Example: Ideal gas, de Broglie wavelenth  
• Tonks gas. Ideal polymer (continuum). Ideal lattice polymer.
• p-T ensemble (only definition).
• Grand partition function (only definition). Example: ideal polymer

Week 6: Equipartition and virial theorems. Applications

• Ideal polymer in grand-canonical ensemble compared to canonical ensemble.
• Time dependence of entropy. Coarse-grained entropy.
• Equipartition and virial theorems. Application to quadratic forms.
• Dulong-Petit law.
• From virial theorem to equation of state using the pair correlation function

Week 7: Quantum stat. mech. Density matrix. Entropy. Lattice models

• Concepts in quantum mechanics: bra-ket, orthonormal set, operator.
• Projection operator for a pure state. Quantum average using projection operator. Mixed state. Density operator/matrix.
• Von Neuman-Liouville equation.
• Entropy and density matrix for microcanonical and canonical ensembles.
• Density matrix of single particle in a box.
• Electron spin - quantum treatment, density matrix, partition function, magnetization
• Lattice models: Ising (ferro and anti-ferro), Heisenberg (quantum and classical), XY model.
• Solution of 1D Ising w/free boundaries

Week 8: 1D, 2D, Bethe lattice phase transitions. Mean field.

• Spontaneous symmetry breaking
  • Why there is no phase transition if 1D? (free energy argument)
  • Transfer matrix for 1D Ising.
  • Why 1D argument of no-transition fails in 2D?
  • Critical point of 2D Ising: dual lattice, high/low temperature expansions.         
• Mean-field (Weiss/Bragg-Williams).
• Brief mention of Bethe-Peiels approximation.
• Description of results of Onsager.
  • Recitations: Correlation in Ising w/periodic boundaries
  • Potts model

Week 9: Quantum ideal gas. High-T for bosons and fermions.

• Quantum ideal gas, quantum corrections at high-T
  • (effective repulsion/attraction) in fermions/bosons

Week 10: Virial exapansion. Application to van der Vaals.

• Virial expansion
• Van der Waals gas from virial expansion

Week 11: Debey-Huckel. Approximate methods. Monte Carlo.

• Debye-Huckel theory
• Approximate methods: Gibbs inequality, Peierls inequality.
• Monte Carlo  method

Week 12: Second order phase transitions: Landau theory. Critical exponents.

• Phase transition types, historical remarks,liquid-magnetic analogy
• Landau function, critical exponents \alpha, \beta.
• Exponents \gamma, \delta.  Divergence of fluctuations.
• Continuous Landau functional, treatment of fluctuations, Correlation function, exponents \nu and \eta.
• Widom and Rushbrooke relations.

Week 13: Kadanoff scaling. Real space renormalization. Fixed points.

• Landau function: (1) failure due to large fluctuations; (2) 1st order transition
• Kadanoff scaling function of free energy.
• Pameter space; example 2D Ising.
• Real space renormalizxation. Fixed points (stable, unstable, mixed)
• Identifying 1/y_1=\nu, \alpha=2-d\nu. Concept of k-space RG, and epsilon-expansion.