Thermodynamics and Statistical Mechanics

Physics in 16 chapters — from “Entropy and the fundamental relation” to “Degenerate Fermi gases, Bose-Einstein condensation and quantum fluids”.

Chapters

  1. Entropy and the fundamental relation
    • Entropy as a state function
    • The fundamental relation S(U,V,N)
    • Entropy maximum and intensive parameters
    • Extensivity, the Euler relation and Gibbs-Duhem
    • The ideal gas as a fundamental relation
    • Problems
  2. Thermodynamic potentials, Legendre transforms and Maxwell relations
    • The Legendre transform, from mechanics to thermodynamics
    • Enthalpy, Helmholtz and Gibbs free energies
    • The grand potential
    • Natural variables and the choice of potential
    • Maxwell relations
    • Free energy as available work
    • Problems
  3. Chemical potential, open systems and chemical equilibrium
    • The chemical potential as escaping tendency
    • Particle-exchange equilibrium
    • Chemical potential of an ideal gas
    • Mixtures and the entropy of mixing
    • Phase coexistence as equality of μ
    • Chemical equilibrium and the law of mass action
    • Problems
  4. Response functions, stability and extremum principles
    • Response functions
    • The Jacobian method for thermodynamic derivatives
    • The Cp​−CV​ relation
    • The extremum principle for each potential
    • Concavity of S and thermodynamic stability
    • Le Chatelier's principle
    • Problems
  5. Thermodynamics of magnetic, elastic and surface systems
    • Generalised work terms and conjugate pairs
    • Magnetic work with a stated convention
    • Magnetic Maxwell relations and the magnetocaloric relation
    • Elastic systems and entropic elasticity
    • Surface tension and the Laplace pressure
    • Problems
  6. Probability, random walks and the statistical description of large systems
    • The variance of a sum of independent variables
    • The binomial distribution and Stirling's approximation
    • The Gaussian limit and the central limit theorem
    • Relative fluctuations and macroscopic values
    • The freely jointed chain and its entropic tension
    • Problems
  7. Microstates, phase space and the microcanonical ensemble
    • Microstate and macrostate
    • The phase-space cell of volume h3N
    • Multiplicity and the postulate of equal probabilities
    • Boltzmann entropy and statistical temperature
    • Sharpness of the most probable macrostate
    • The Gibbs entropy of a distribution
    • The Gibbs paradox and extensive entropy
    • Problems
  8. The Boltzmann factor, the canonical ensemble and the partition function
    • A small system in a heat reservoir
    • The Boltzmann factor and the canonical ensemble
    • The partition function
    • Free energy from the partition function
    • Factorisation of the partition function
    • The equipartition theorem
    • The Maxwell-Boltzmann distribution recovered
    • Problems
  9. The grand canonical ensemble, fluctuations and ensemble equivalence
    • The particle reservoir
    • The grand partition function and the grand potential
    • Fugacity and the mean particle number
    • The energy-fluctuation heat-capacity relation
    • Number fluctuations and κT​
    • Ensemble equivalence and the thermodynamic limit
    • Problems
  10. Ergodicity, Liouville's theorem and the foundations of equilibrium statistical mechanics
    • Phase-space flow and the density of copies
    • Liouville's theorem
    • Stationary distributions are functions of H
    • The ergodic hypothesis
    • Time averages, ensemble averages and mixing
    • Coarse graining and the arrow of time
    • Problems
  11. Two-level systems, paramagnetism and the Einstein solid
    • The two-level partition function
    • The ideal paramagnet and Curie's law
    • Negative temperature
    • The Schottky anomaly
    • The Einstein solid heat capacity
    • Freezing out, and where the Einstein solid fails
    • Problems
  12. The classical ideal gas from the partition function
    • The single-particle partition function
    • The N-particle gas and the Sackur-Tetrode entropy
    • The statistical chemical potential
    • Separable molecular energies and the internal partition function
    • Rotational and vibrational temperatures
    • The classical-limit condition and where it ends
    • Problems
  13. Photons and phonons: blackbody radiation and the Debye model
    • The field as a set of oscillators
    • The density of modes in k-space
    • The Planck spectrum
    • Stefan-Boltzmann law and Wien displacement
    • Radiation pressure in a photon gas
    • The Debye model of a crystal
    • The T3 law and the Debye temperature
    • Problems
  14. Indistinguishability, occupation numbers and the density matrix
    • Identical particles and the exchange of labels
    • Exchange symmetry, bosons and fermions
    • The Pauli exclusion principle
    • The occupation-number description
    • Three ways to count states
    • The density operator and mixed states
    • The canonical density operator
    • The von Neumann entropy
    • Problems
  15. Bose-Einstein and Fermi-Dirac statistics and the ideal quantum gases
    • Grand canonical factorisation over single-particle states
    • The Bose-Einstein distribution
    • The Fermi-Dirac distribution
    • Maxwell-Boltzmann as the common limit
    • The role of μ in a quantum gas
    • The density of states in three dimensions
    • Equation of state of an ideal quantum gas
    • The first quantum correction to the pressure
    • The photon gas as bosons
    • Problems
  16. Degenerate Fermi gases, Bose-Einstein condensation and quantum fluids
    • The Fermi sea and the Fermi energy
    • Degeneracy pressure
    • The electronic heat capacity of metals
    • Pauli paramagnetism
    • White dwarfs and the Chandrasekhar mass estimate
    • Bose-Einstein condensation in a box
    • Condensation in a harmonic trap
    • Liquid helium-4 and the quantum fluids
    • Fermi energy, Tc​ and the classical limit
    • Problems