STATISTICAL MECHANICS (PHYS-617)
(syllabus)
Introduction to Statistical Mechanics
and discussion of the main results of Thermodynamics
Thermodynamic potentials, the entropy. Thermodynamics versus
Statistical Mechanics.
Introduction to the formalism of classical Stat Mech and the Boltzmann
equation
Definition of $\mu$ and $\gamma$ spaces, representative points.
Derivation of the Boltzmann
equation, nature of binary collisions, atomic interaction potential.
The collision integral. Detailed balance as sufficient condition for
equilibrium. The Boltzmann distribution.
The Boltzmann distribution and the classical H-theorem
Derivation of the Boltzmann distribution for a monatomic "ideal"
gas. Relevance of the thermodynamic parameters in the determination of
the distribution.
Boltzmann's H-theorem and the necessity of detailed balance.
Discussion of the H-theorem's implications. Criticism of the H-theorem.
Ideal gas entropy and THE entropy constant.
Time and ensemble averages, the distribution function
Representative points in $\gamma$ space. Time average.
Ensembles and ensemble average.
Brief discussion of the ergodic problem. Properties of the averages,
statistical independence and the quenching of fluctuations.
Operational definition of $\rho$ (a la Landau).
The canonical ensembles from Liouville theorem and equilibrium
The Liouville theorem.
Equilibrium condition, relevance of the additive constants
of motion. Equilibrium ensembles: $\ln ( \rho c )$ in the classical
Micro Canonical Ensemble (MCE) and in the classical
Canonical Ensemble (CE).
Energy terms for dielectric and magnetic media. Geometry of spheres in M
dimensions. Introducing $W(E)$. One oscillator vs many.
The Grand Canonical Ensemble (GCE): $\ln ( P(N) \rho c )$.
Discussion of $W(E)$ and $Z$ for canonical systems. Brief discussion of
magnetism.
The entropy
Entropy as $k_B \ln \Omega (E)$. Introduction of THE $\delta_\gamma$.
Entropy in the CE ensemble. Free energy and $\rho_{CE}$. Free energy
in terms of $Z_{CE}$. Further discussion of the GCE.
Proof that $P(N) =f(N)$.
Classical results: Equipartition
Maxwellians, equipartition theorem, Einstein theory of
classical fluctuations. "Matrioska" theory of the canonical ensembles.
Gibbs' correct Boltzmann counting, Boltzmann distribution
from combinatorics, the N! story, Stirling approximation.
The black body radiation problem
Introduction to the physics of the black body (BB).
Thermodynamics and classical theory of the BB radiation.
Derivation of the generalized Wien's law.
Oscillators and radiation.
The Rayleigh-Jeans law. Planck's "crazy" solution.
The photoelectric effect.
Einstein's theory for the BB radiation, photons.
Quantum Stat Mech
Introduction to the quantum formalism and quantum ensembles.
Ideal gas of Fermi and Bose particles: Landau potentials, average
occupation numbers.
Zero temperature behavior of ideal Fermi and Bose gases.
Fermi surface, Fermi energy, Fermi etc.
Behavior of the chemical potential.
General expression for the total energy of an ideal gas.
Degeneracy corrections: Bose and Fermi function expansions in
powers of the fugacity.
Fermi gas at finite temperatures, the Sommerfeld expansion.
Heat capacity of a Fermi gas at low temperatures.
The Bose-Einstein condensation (BEC).
Non ideal gases
Introduction to non-ideal gas behavior.
Introduction to the van der Waals equation.
The classical Virial expansion. The second Virial coefficient.
Introduction to phase transitions and critical phenomena
Phase equilibria, phase diagrams. Characterization of phase
transitions. Non-analyticity. Yang and Lee theorems.
Critical exponents.
van der Waals description of the phase diagram of a one component
system. Mean field theory. Mean field exponents. Weiss theory for
the Ising model. Deviations from the mean field paradigm.
(Notice: The time actually spent in class on each of the above items
is NOT proportional to the length of their description here)
