Statistical Mechanics

Syllabus

• Foundation of Statistical Physics:
  The second and third law of thermodynamics, thermodynamic ensembles, ergodicity, Liouville theorem, The virial theorem.


• Ideal gases:
  Classical, Fermi and Bose gases, black body radiation, Bose Einstein condensation, Landau diamagnetism, ideal gases with internal degrees of freedom.


• Classical non-ideal gases:
  The virial expansion, van der Waals fluid, liquid-gas phase transitions.


• Phase transitions:
  General phenomenology, first and second order phase transitions, models of magnetic systems, lattice gas models, spontaneous symmetry breaking, breaking of ergodicity, Landau theory, mean field approximation, upper and lower critical dimensions, transfer matrix, critical exponents, universality classes, scaling theory.


• Fluctuations:
  Thermodynamic fluctuations and correlation functions, spatial and temporal correlations, fluctuation-dissipation theorem, Onsager relations.

Requirements

• Assignment 1: submit by September 28
  Problems 1.4, 1.8, 1.12, 2.7, 3.6, 3.8, 3.9, 3.13.


• Midterm exam: October 19 16:00-18:30 (Room 409) - the exam will include exercises from Pathria Ch. 1-3.
  Midterm exam


• Assignment 2: cancelled


• Final exam: Dec 7 16:00-19:00

Sources

1. Statistical Mechanics by Pathria and Beale


2. Statistical Mechanics: Theory and Molecular Simulation by Tuckerman

Lectures

• Lecture Notes 1-3 (Warning: handwritten, numerous typos and errors)


• Lecture Notes 4-5 (Warning: handwritten, numerous typos and errors)

1. The Statistical Basis of Thermodynamics
   Macroscopic and microscopic states; The link between Thermodynamics and Statistical Mechanics; Classical Ideal Gases; The Gibbs paradox


2. Ensemble Theory
   Phase space of a classical system; Liouville's theorem; The microcanonical ensemble; Quantum states and the phase space.


3. The Canonical Ensemble
   The partition function; The classical systems; Energy fluctuations in the canonical ensemble; Equipartition and virial theorems; Harmonic oscillators; Paramagnetism; Negative temperatures


4. The Grand Canonical Ensemble
   The grand canonical ensemble; Density and energy fluctuations in the grand canonical ensemble; Thermodynamic phase diagrams; Phase equilibrium and the Clausius-Clapeyron equation.


5. Quantum Statistics
   Quantum-mechanical ensembles; The density matrix; Indistinguishable particles; free particles.


6. The Theory of Simple Gases
   An ideal gas in a quantum-mechanical quantum-mechanical ensembles; Statistics of the occupation numbers; Internal degrees of freedom; Chemical equilibrium.


7. Ideal Bose Systems
   Bose-Einstein condensation; Blackbody radiation; Sound waves; Liquid helium II.


8. Ideal Fermi Systems
   Thermodynamic behavior of an ideal Fermi gas; Magnetic behavior of an ideal Fermi gas; The electron gas in metals; Ultra-cold atomic Fermi gases; White dwarf stars; Statistical model of the atom.


9. Statistical Mechanics of Interacting Systems: Cluster Expansions
   Virial expansion of the equation of state; The second virial coefficient; Cluster expansion for a quantum-mechanical system; Correlations and scattering.


10. Statistical Mechanics of Interacting Systems: The Method of Quantized Fields
    Second quantization; Low-temperature behavior of an imperfect Bose gas; Energy spectrum of a Bose liquid; States with quantized circulation; Quantized vortex rings and the breakdown of superfluidity; Energy spectrum of a Fermi liquid; Condensation in Fermi systems.


11. Phase Transitions: Criticality, Universality, and Scaling
    Condensation of a van der Waals gas; A dynamical model of phase transitions; The lattice gas and the binary alloy; Ising model; The critical exponents; Landau's phenomenological theory; Scaling hypothesis for thermodynamic functions; The role of correlations and fluctuations; The critical exponents and the problem with mean field theory.


12. Phase Transitions: Exact (or Almost Exact) Results
    The 1D Ising model; The 1D n-vector models; The 2D Ising model; The spherical model in arbitrary dimensions; The ideal Bose gas in arbitrary dimensions.


13. Phase Transitions: The Renormalization Group Approach
    The concept of scaling; The renormalization group; Applications of the renormalization group Finite-size scaling.


14. Fluctuations and Nonequilibrium Statistical Mechanics
    Equilibrium thermodynamic fluctuations; Brownian motion; The Langevin theory; The Fokker-Planck equation; Spectral analysis of fluctuations; The fluctuation-dissipation theorem; The Onsager relations.