# Statistical Mechanics

### PHYSICS763

Canonical and grand canonical ensembles, quantum statistics, ideal Bose and Fermi systems, classical non-ideal gases, virial expansion, phase transitions, fluctuations, transport coefficients, non-equilibrium processes. 3 units.

#### Possible principal texts:

- R. K. Pathria,
*Statistical Mechanics*, 2^{nd}edition (Elsevier).

#### Other texts to consider:

- L. D. Landau and E. Lifschitz,
*Statistical Physics*(vol. 1+2). - K. Huang,
*Statistical Mechanics*(John Wiley & Sons). - L. E. Reichl,
*A Modern Course in Statistical Physics*(Univ. of Texas Press)

#### Prerequisites:

This course structure is predicated on the assumption that students have mastered an intermediate undergraduate course at the level of **Daniel Schroeder**' textbook.

In Duke Physics, there is an undergraduate "Introduction to Thermal and Statistical Physics" course (PHY 176), with the synopsis: *Thermal properties of matter treated using the basic concepts of entropy, temperature, chemical potential, partition function, and free energy. Topics include the laws of thermodynamics, ideal gases, thermal radiation and electrical noise, heat engines, Fermi-Dirac and Bose-Einstein distributions, semiconductor statistics, kinetic theory, and phase transformations.*

#### Syllabus

- Canonical ensemble
- Grand canonical ensemble
- Formulation of quantum statistics: density matrix.
- Photons, the Planck distribution, and thermal radiation.
- Lattice vibrations and Debye theory.
- Ideal Bose gas and Bose condensation.
- Ideal Fermi system: degenerate electron gas in metals.
- Magnetic behavior of an ideal Fermi gas: Pauli paramagnetism and Landau diamagnetism.
- Virial expansion; cluster expansion.
- First-order phase transitions.
- Mean field theory.
- Ising model.
- Second-order phase transitions.
- Critical phenomena, scaling.
- Brownian motion: Langevin theory, Fokker-Planck theory.
- Transport phenomena: conduction (Drude theory), diffusion, thermal transport.
- Onsager relation, fluctuation-dissipation theorem.
- Far from equilibrium systems; non-ergodicity.

Possible special topics: Density functional theory of dense liquids, Hydrodynamics, Transport equations, surfaces, evaporation and condensation.

#### Sample lecture schedule

(based on 25 lectures each of duration 75 minutes).

*Lecture 1:*__Review__of thermodynamics.*Lecture 2:*__Review__of basic statistical physics.*Lecture 3:*Canonical ensemble 1 (partition function, Helmholtz free energy.)*Lecture 4:*Canonical ensemble 2 ( examples)*Lecture 5:*Grand canonical ensemble 1 (chemical potential)*Lecture 6:*Grand canonical ensemble 2 (examples)*Lecture 7:*Formulation of quantum statistics: density matrix.*Lecture 8:*Photons, the Planck distribution, and thermal radiation.*Lecture 9:*Lattice vibrations and Debye theory.*Lecture 10:*Ideal Bose gas and Bose condensation.*Lecture 11:*Ideal Fermi system: degenerate electron gas in metals.*Lecture 12:*Magnetic behavior of an ideal Fermi gas: Pauli paramagnetism and Landau diamagnetism.*Lecture 13:*Virial expansion; cluster expansion.*Lecture 14:*First-order phase transitions.*Lecture 15:*Mean field theory.*Lecture 16:*Ising model (1d).*Lecture 17:*Ising model (2d).*Lecture 18:*Second-order phase transitions.*Lecture 19:*Critical phenomena, scaling.*Lecture 20:*Brownian motion: Langevin theory.*Lecture 21:*Brownian motion: Fokker-Planck theory.*Lecture 22:*Transport phenomena: conduction (Drude theory), diffusion, thermal transport.*Lecture 23:*Onsager relation, fluctuation-dissipation theorem.*Lecture 24:*Far from equilibrium systems; non-ergodicity.*Lecture 25:*Special topics.