Statistical Mechanics
Ensembles, partition functions and the statistical basis of thermodynamics, developed as a full course that derives the laws of thermodynamics from the microscopic rules of classical and quantum mechanics: large numbers, Stirling's approximation and the 1/sqrt(N) fluctuations that make macroscopic behaviour deterministic; counting microstates for two-state paramagnets, Einstein solids and ideal gases, and the sharp division of energy between systems in contact; the microcanonical ensemble, with entropy as the logarithm of multiplicity, temperature, pressure and chemical potential as derivatives of entropy, negative temperatures, a complete derivation of the Sackur–Tetrode entropy and the Gibbs paradox; the canonical ensemble, the Boltzmann distribution derived from a reservoir, the partition function, energy fluctuations and heat capacity, the Helmholtz free energy and the equivalence of ensembles; paramagnets, Curie's law, Brillouin functions and Henry's measurements, adiabatic demagnetization, the Schottky anomaly and the equipartition theorem and its failure; the molecular ideal gas built from translational, rotational, vibrational and electronic partition functions, including the ortho–para puzzle of hydrogen; the chemical potential, the grand canonical ensemble, Poisson number fluctuations, binding sites and vapour pressure; Fermi–Dirac and Bose–Einstein statistics, the quantum concentration and the density of states; degenerate Fermi gases, the Sommerfeld heat capacity of metals and the mass–radius relation and Chandrasekhar limit of white dwarfs; photons, the Stefan–Boltzmann law and the cosmic microwave background, phonons and the Debye model, and Bose–Einstein condensation in helium and in ultracold atoms; phase transitions through the Ising model, Onsager's exact solution, mean-field and Landau theory, critical exponents, universality and the renormalization group; and a capstone on chemical equilibrium and the Saha equation, Langmuir adsorption, the entropic elasticity of polymers, maximum entropy and Landauer's principle, with runnable Python examples throughout.
PHYS-361 Year 3 3 credits 12 lessons · 4 h 18 min ▶ Video lessons coming soon
Offered by HSM University · Physics
What you'll learn
- Explain how thermodynamics emerges from microscopic statistics, estimate fluctuations in large systems, and use Stirling's approximation to handle large multiplicities.
- Count microstates for two-state paramagnets, Einstein solids and ideal gases, and show that energy shared between large systems divides into a sharply defined most probable state.
- Define entropy, temperature, pressure and chemical potential from the microcanonical entropy, derive the Sackur–Tetrode equation, and explain negative temperatures and the Gibbs paradox.
- Derive the Boltzmann distribution from a reservoir, compute partition functions, and obtain energies, fluctuations, free energies and entropies from them.
- Apply the canonical ensemble to paramagnets, Brillouin functions, adiabatic demagnetization, two-level Schottky anomalies and the equipartition theorem and its failure.
- Build the partition function of a molecular ideal gas from translational, rotational, vibrational, electronic and nuclear-spin parts, including ortho- and parahydrogen.
- Calculate chemical potentials of ideal gases, use the grand canonical ensemble for open systems, and analyse number fluctuations, binding sites and vapour pressure.
- Derive the Fermi–Dirac and Bose–Einstein distributions, decide when a gas is quantum using \(n\lambda^3\), and use the density of states to compute properties of ideal quantum gases.
- Analyse degenerate Fermi gases, including Fermi energies, the Sommerfeld heat capacity of metals, and the structure and mass limit of white dwarf stars.
- Treat photons, phonons and massive bosons, deriving the Stefan–Boltzmann law, the Debye heat capacity and the Bose–Einstein condensation temperature.
- Analyse phase transitions with the Ising model using exact results, mean-field theory and Landau theory, and explain critical exponents and universality.
- Apply statistical mechanics to chemical equilibrium, ionization, adsorption, polymer elasticity and information, including the law of mass action and Landauer's principle.
Course content
12 lessons · 4 h 18 min total · final exam
- 1 Why statistical mechanics: from microscopic rules to macroscopic laws 23 min · quiz
- 2 Microstates, multiplicity and the fundamental postulate 23 min · quiz
- 3 Entropy, temperature and the microcanonical ensemble 21 min · quiz
- 4 The canonical ensemble and the partition function 20 min · quiz
- 5 Applications of the canonical ensemble: paramagnets, two-level systems and equipartition 20 min · quiz
- 6 The ideal gas from partition functions: translational, rotational and vibrational parts 21 min · quiz
- 7 The grand canonical ensemble and chemical potential 21 min · quiz
- 8 Quantum statistics: Fermi–Dirac and Bose–Einstein distributions 21 min · quiz
- 9 Degenerate Fermi gases: electrons in metals and white dwarfs 23 min · quiz
- 10 Bose gases: photons, phonons and Bose–Einstein condensation 20 min · quiz
- 11 Phase transitions and critical phenomena: Ising model, mean-field theory and universality 21 min · quiz
- 12 Capstone: statistical mechanics at work — chemical equilibrium, adsorption, polymers and information 24 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
Ensembles, partition functions and the statistical basis of thermodynamics, developed as a full course that derives the laws of thermodynamics from the microscopic rules of classical and quantum mechanics: large numbers, Stirling's approximation and the 1/sqrt(N) fluctuations that make macroscopic behaviour deterministic; counting microstates for two-state paramagnets, Einstein solids and ideal gases, and the sharp division of energy between systems in contact; the microcanonical ensemble, with entropy as the logarithm of multiplicity, temperature, pressure and chemical potential as derivatives of entropy, negative temperatures, a complete derivation of the Sackur–Tetrode entropy and the Gibbs paradox; the canonical ensemble, the Boltzmann distribution derived from a reservoir, the partition function, energy fluctuations and heat capacity, the Helmholtz free energy and the equivalence of ensembles; paramagnets, Curie's law, Brillouin functions and Henry's measurements, adiabatic demagnetization, the Schottky anomaly and the equipartition theorem and its failure; the molecular ideal gas built from translational, rotational, vibrational and electronic partition functions, including the ortho–para puzzle of hydrogen; the chemical potential, the grand canonical ensemble, Poisson number fluctuations, binding sites and vapour pressure; Fermi–Dirac and Bose–Einstein statistics, the quantum concentration and the density of states; degenerate Fermi gases, the Sommerfeld heat capacity of metals and the mass–radius relation and Chandrasekhar limit of white dwarfs; photons, the Stefan–Boltzmann law and the cosmic microwave background, phonons and the Debye model, and Bose–Einstein condensation in helium and in ultracold atoms; phase transitions through the Ising model, Onsager's exact solution, mean-field and Landau theory, critical exponents, universality and the renormalization group; and a capstone on chemical equilibrium and the Saha equation, Langmuir adsorption, the entropic elasticity of polymers, maximum entropy and Landauer's principle, with runnable Python examples throughout.
Part of the Science stream, typically taken in year 3.