Lecture Notes

Lecture Notes

PHY 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu ---- Spring 2020

Lecture Notes

My hand written class lecture notes are being scanned and uploaded for you to view. Please be warned that these are the notes I prepare for myself to lecture from - they are not in general carefully prepared for others to read. I make no guarantees about their legibility, or that they are totally free of errors. I hope, nevertheless that you will find them useful. The lectures are uploaded as pdf files, so you will need Adobe Acrobat Reader in order to read them. You can download Acrobat Reader for free here.

The lecture note files correspond roughly to the material presented in a given day's lecture. But you may on occassion find the end of one day's lecture at the start of the file for the next day's lecture, so please look there if you think there might be something missing.

Lecture 0 - Sketching the history of statistical mechanics and thermodynamics
Lecture 1 - Postulates of classical thermodynamics, entropy, temperature, pressure, chemical potential
Lecture 2 - Conditions for thermal, mechanical and chemical equilibrium, concavity of the entropy, equations of state, Euler relation, Gibbs-Duhem relation
Lecture 3 - Entropy of the ideal gas, entropy maximum vs energy minimum principle, Legendre transformation
Lecture 4 - Helmholtz and Gibbs free energies, enthalpy, the grand potential, reservoirs, extremum principles for free energies, Maxwell relations
Lecture 5 - Response functions and relations between them, stability constraints on response functions
Lecture 6 - Curvature of free energies and stability requirements for response functions, heat engines, thermodynamic efficiency, Carnot cycle
Lecture 7 - Kinetic theory of ideal gas, Maxwell velocity distribution, Ergodic hypothesis, statisitcal ensembles, density matrix
Lecture 8 - Liouville's theorem, microcanonical ensemble, density of states and connection to entropy
Lecture 9 - Entropy of the ideal gas and problem of extensivity, entropy of mixing and Gibbs paradox, indistinguishable particles
Lecture 10 - The canonical ensemble and the canonical partition function, Helholtz free energy and the canonical partition function, energy fluctuations, equivalence of canonical and microcanonical ensembles in the thermodynamic limit
Lecture 11 - Average energy vs most probably energy, Stirling's formula, factoring the canonical partition function for non-interacting objects, the ideal gas in the canonical ensemble, Maxwell velocity distribution revisited
Lecture 12 - The virial theorm, the equipartition theorem, elastic vibrations of a solid and the law of Dulong and Petit, paramagnetism of classical spins and the Curie law of paramagnetic susceptibility
Lecture 13 - A note about the proper choice of coordinates, the grand canonical ensemble and partition function
Lecture 14 - Entropy and information
Lecture 15 - The grand potential, fluctuations of the number of particles, fluctuations of energy in the grand canonical ensemble
Lecture 16 - Non-interacting particles in the grand canonical ensemble, chemical equilibrium, a gas in equilibrium with a surface of absorption sites
Lecture 17 - Quantum statistical ensembles, density matrix operator, quantum microcanonical, canonical, and grand canonical ensembles, quantum harmonic osciallator at temperature T, Bose-Einstein and Fermi-Dirac statistics, Pauli exlusion principle, occupation number representation
Lecture 18 - Particle in a box states, density matrix for two non-interacting particles, quantum correlations as an effective classical pair interaction, N-particle partition function in the position-space basis, the partition function in the occupation number representation
Lecture 19 - Partition functions for non-interacting quantum and classical ideal gas, average occupation number, the classical vs quantum limits of the ideal gas
Lecture 20 - The quantized harmonic oscillator as bosons, the Debye model for the specific heat of a solid
Lecture 21 - Black body radiation, the quantum ideal gas, standard functions, pressure, particle density, energy
Lecture 22 - Leading quantum correction to the ideal gas equation of state, the degenerate Fermi gas, Sommerfeld model for conduction electrons in a metal, Fermi energy, ground state energy, specific heat
Supplement: The Sommerfeld Expansion
Lecture 23 - Chemical potential of a degenerate Fermi gas at finite temperature, Pauli paramagnetism
Lecture 24 - The ideal Bose gas, Bose-Einstein condensation in an ideal gas of free bosons
Lecture 25 - Classical spin models, constant magnetization vs constant magnetic field ensembles, broken symmetry and the possibility for a phase transition in the thermodynamic limit
Lecture 26 - Ising phase diagram, mean-field solution of the Ising model
Lecture 27 - Mean-field solution and critical exponents of the Ising model
Lecture 28 - Maxwell construction, lattice-gas model, exact solution of the one dimensional Ising model

Supplement - Fluctuations, correlation length, Landau theory, Landau-Ginzburg approach, corelation function, upper critical dimension [these are supplemental notes; we did not go over this in lecture]