Statistical mechanics
Undergraduate · Physics
Syllabus focus
Standard syllabus · Theoretical / proof-based
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Topics typically covered
Standard syllabus
Ensembles and counting
- Microstates, macrostates, and multiplicity
- Boltzmann, Gibbs, and Shannon entropy links
- Canonical, microcanonical, and grand canonical ensembles
- Partition functions and thermodynamic potentials
- Classical vs quantum statistics
Ideal systems
- Classical ideal gas and equipartition
- Quantum gases: Fermi–Dirac and Bose–Einstein
- Blackbody radiation as photon gas
- Phonon heat capacity (Debye model)
- Two-level systems and paramagnetism
Interacting systems
- Ising model and mean-field theory
- Phase transitions and critical exponents (intro)
- Classical fluids: virial expansion
- Chemical equilibrium from partition functions
- Fluctuation–dissipation preview
Theoretical / proof-based
Rigorous statistical foundations
- Liouville's theorem and ergodic hypothesis
- Density matrices for statistical ensembles
- Grand partition function as generating function
- Cumulant expansion and linked cluster theorem (sketch)
- Lee–Yang circle theorem context (optional)
Quantum statistics formalism
- Fock space and occupation number representation
- Bose–Einstein condensation mathematics
- Fermi surface and Sommerfeld expansion
- Second quantization for ideal quantum gases
- Kubo formulas for linear response (intro)
Critical phenomena theory
- Landau theory of phase transitions
- Correlation functions and susceptibilities
- Scaling hypotheses and universality
- Renormalization group fixed points (overview)
- Exact solutions: 1D Ising transfer matrix
Notes
Topics reflect common upper-division physics syllabi at US colleges and universities. Prerequisites, lab pairing, and mathematical depth vary by department.