Thermal and Statistical Physics

Units

1. 1

   Microstates and multiplicity

   2 lessons

   1. Counting microstates: two-state systemsSeparate microstates from macrostates for coins and for a two-state paramagnet, count them with Ω(N, n) = N!/(n!(N − n)!), state the fundamental assumption of statistical mechanics, and see why the most probable macrostate is the overwhelming one.
   2. The Einstein solid and very large numbersCount the microstates of an Einstein solid with Ω(N, q) = (q + N − 1)!/(q!(N − 1)!), find how energy is most probably shared between two solids, use Stirling's approximation to handle huge factorials, and show that the multiplicity peak becomes extremely sharp for large N.
2. 2

   Entropy, temperature and the second law

   3 lessons

   1. Entropy and the statistical definition of temperatureDefine entropy as S = k ln Ω, state the second law as the overwhelmingly likely approach to the largest multiplicity, define temperature by 1/T = ∂S/∂U, and derive the energy and heat capacity of an Einstein solid from its entropy.
   2. The two-state paramagnetFind the entropy of a two-state paramagnet as a function of its energy, get the temperature from 1/T = ∂S/∂U, derive the magnetization M = Nμ tanh(μB/kT) and Curie's law, and explain negative temperatures for a system whose energy has an upper bound.
   3. Pressure, chemical potential and the thermodynamic identityDefine pressure and chemical potential from the entropy, P = T(∂S/∂V) and μ = −T(∂S/∂N), show that they are equal at mechanical and diffusive equilibrium, derive the thermodynamic identity dU = T dS − P dV + μ dN, and obtain the Sackur-Tetrode entropy of a monatomic ideal gas.
3. 3

   Boltzmann statistics

   3 lessons

   1. The Boltzmann factor and the partition functionDerive the probability of a state of a small system in contact with a reservoir, P = e^(−E/kT)/Z, define the partition function Z as the sum of Boltzmann factors over states, and use it for hydrogen in the Sun, a two-level system and the density of the atmosphere against height.
   2. Average values and equipartition from ZShow that the average energy follows from the partition function as Ē = −∂ln Z/∂β, derive the equipartition theorem for quadratic energy terms, find the quantum harmonic oscillator's mean energy and heat capacity and why vibration freezes out, and write the rotational partition function of a diatomic molecule.
   3. The ideal gas from its partition functionCompute the single-particle partition function of a particle in a box, Z₁ = V/v_Q with the quantum volume v_Q, build Z for N indistinguishable particles as Z₁ᴺ/N!, recover U = (3/2)NkT, PV = NkT and the Sackur-Tetrode entropy, and derive the Maxwell speed distribution from Boltzmann factors.
4. 4

   Free energy and phase equilibria

   2 lessons

   1. Helmholtz and Gibbs free energyDefine F = U − TS and G = U − TS + PV, show F = −kT ln Z, show that F is minimised at fixed T and V and G at fixed T and P, and use −ΔF and −ΔG as the maximum work, including for fuel cells and batteries.
   2. Phase transformations and the Clausius-Clapeyron relationRead a phase diagram, explain a first-order transition as the crossing of two Gibbs free energy curves, derive the Clausius-Clapeyron equation dP/dT = L/(TΔV), and use the van der Waals equation to find the critical point and the liquid-gas coexistence.
5. 5

   Quantum statistics

   4 lessons

   1. Fermi-Dirac and Bose-Einstein distributionsDefine fermions and bosons, derive the Gibbs factor and the grand partition function for one orbital, obtain the Fermi-Dirac and Bose-Einstein mean occupations 1/(e^((ε−μ)/kT) ± 1), and show that both reduce to the Boltzmann factor when the occupation is small.
   2. The degenerate Fermi gasFill the lowest orbitals of N electrons in a box at T = 0 to find the Fermi energy E_F = (h²/8m)(3N/πV)^(2/3), derive the total energy and the degeneracy pressure P = (2/5)(N/V)E_F, balance it against gravity for a white dwarf, and explain why the electronic heat capacity at room temperature is only a small fraction of the classical value.
   3. The photon gas and blackbody radiationTreat the light in a cavity as a gas of bosons with zero chemical potential, count its modes to get the density of modes 8πf²/c³, derive Planck's spectrum, integrate it to the Stefan-Boltzmann law U/V = aT⁴, and apply both to the cosmic microwave background.
   4. Bose-Einstein condensationShow that a gas of bosons with a fixed number of particles in a box has a maximum number it can hold in excited states, 2.612 V/v_Q, find the condensation temperature T_c = (h²/2πmk)(n/2.612)^(2/3), derive the condensed fraction N₀/N = 1 − (T/T_c)^(3/2), and relate it to cold-atom condensates and superfluid helium.