2019 - 2020

0321-4110   Thermodynamics & Statistical Mechanics                                                             
FACULTY OF EXACT SCIENCES
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Course description

Thermodynamics and Statistical Mechanics  (Core graduate Course)
תיבת טקסט: Weekly division
week	subject
1	Thermodynamics: Basics, Postulates 0-2, temperature, Entropy.
Ideal gas, Tonks gas
2	Van der Waals. Thermodynamic potentials. Minimax principles.
3	Phase diagrams. Maxwell cosntruction. Gibbs rule. Lattice models.
4	Postulate 3. Liouville theorem. Shannon entropy.
5	Entropy maximization in classical stat. mech. ensembles
6	Equipartition and virial theorems. Applications
7	Quantum stat. mech. Density matrix. Entropy. Lattice models
8	1D, 2D, Bethe lattice phase transitions. Mean field.
9	Quantum ideal gas. High-T for bosons and fermions.
10	Virial exapansion. Application to van der Vaals.
11	Debey-Huckel. Approximate methods. Monte Carlo.
12	Second order phase transitions: Landau theory. Critical exponents.
13	Kadanoff scaling. Real space renormalization. Fixed points.

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 (IF TIME PERMITS)

  • 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.

     

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