Statistical Mechanics — Problem Bank

UNIST · Fall 2026 · Pathria & Beale, 3rd edition

Course contract. These problems define the level of the course. They are practice, not homework, and are not submitted. Examination problems will be taken from this bank or will be close variants. The idea and method will be familiar; notation, parameters, or the order of the derivation may change. Each examination contains four substantial problems, one from each block.

Work from the textbook problem statement. Do not memorize solutions. A useful test is whether you can reconstruct the calculation from the physical assumptions.

Midterm bank · Chapters 1–3

Block M-A · Entropy and thermodynamics

M11.2Why additive entropy must be logarithmic in multiplicity.
M21.9Extensivity and the Euler relation for thermodynamic variables.
M31.11Entropy of mixing for ideal gases.
M41.16Gibbs–Duhem structure and pressure derivatives.

Block M-B · Phase space and the microcanonical picture

M52.3Phase-space cells for a classical rotator and their quantum correspondence.
M62.5Semiclassical phase-space quantization in one dimension.
M72.6Pendulum trajectories and the phase-space area–period relation.
M82.8Relativistic ideal gas from phase-space volume.

Block M-C · Canonical ensemble and fluctuations

M93.5Extensivity of the Helmholtz free energy.
M103.8Ideal-gas entropy from the canonical partition function.
M113.17Energy variance from derivatives of the partition function.
M123.18Third energy fluctuation and its thermodynamic meaning.

Block M-D · Equipartition, virial theorem, finite-state statistics

M133.20Virial theorem for homogeneous interactions.
M143.22Quartic oscillator: energy partition and specific heat.
M153.32Two groups of states: partition function, energy, and entropy.
M163.44(a–b)Shannon information, equal probabilities, and correlations.

Final bank · Chapters 4–7

Block F-A · Grand canonical ensemble

F14.1Entropy and thermodynamic potentials in the grand canonical ensemble.
F24.2Dominant particle number and correspondence with the canonical ensemble.
F34.3Binomial particle-number statistics and Gaussian/Poisson limits.
F44.12Coupled energy and particle-number fluctuations.

Block F-B · Adsorption and indistinguishable particles

F54.10Adsorption on a finite set of sites in canonical and grand-canonical language.
F64.11Equilibrium between an adsorbed layer and an ideal gas.
F75.4What fails when free identical particles are treated as unsymmetrized.
F85.5Exchange as an effective statistical interaction; first quantum correction.

Block F-C · Quantum gases and occupation statistics

F96.1Entropy of Bose and Fermi gases from occupation probabilities.
F106.2Occupation-number fluctuations for MB, BE, and FD statistics.
F116.3Finite maximum occupancy and the interpolation between Fermi and Bose limits.
F126.5Molecular-energy fluctuations in a Maxwell–Boltzmann gas.

Block F-D · Bose systems and radiation

F137.2Virial expansion of the ideal Bose gas.
F147.3Behavior of the chemical potential near Bose–Einstein condensation.
F157.20Thermodynamics of blackbody radiation from oscillator partition functions.
F167.21Mean photon energy in thermal radiation.

TA problem sessions

DateProblemsFocus
Wed Sep 2M1Entropy from multiplicity
Wed Sep 9M3Mixing and extensivity
Wed Sep 16M7Phase-space geometry
Mon Sep 21M2, M6, M8Consolidation · Chapters 1–2
Wed Sep 23M9Canonical thermodynamics
Wed Sep 30M10Partition function → entropy
Wed Oct 7M12Energy fluctuations
Wed Oct 14M2, M7, M11, M14Midterm review · one problem from each block
Wed Oct 28F1Grand canonical thermodynamics
Wed Nov 4F5Adsorption and fugacity
Wed Nov 11F7Indistinguishability
Wed Nov 18F8Exchange correction
Wed Nov 25F10Occupation-number fluctuations
Wed Dec 2F14Bose condensation
Wed Dec 9F2, F6, F10, F15Final review · one problem from each block

The final examination covers Chapters 4–7. The basic language of Chapters 1–3 is assumed.

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