Statistical Mechanics

From microstates to the renormalization group.

About this course

Eighteen lectures deriving thermodynamics from counting. It begins with microstates, Boltzmann entropy and the microcanonical ensemble, then builds the canonical and grand canonical ensembles and shows how every thermodynamic quantity follows from the partition function. Indistinguishability introduces quantum statistics — the degenerate Fermi gas, Bose-Einstein condensation, photons and phonons. The fourth module treats interacting systems: van der Waals, the Ising model, critical phenomena and the renormalization group. It closes with kinetic theory, fluctuation-dissipation and Monte Carlo methods.

What you will be able to do

  • Relate microstates to macroscopic thermodynamics through Boltzmann entropy
  • Derive thermodynamic quantities from the canonical and grand canonical partition functions
  • Distinguish Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics and apply each
  • Analyse phase transitions with van der Waals and the Ising model in mean-field theory
  • Explain scaling, universality and the renormalization-group view of critical phenomena

What this course covers

  • Limitations of the ideal gas law
  • Deriving the van der Waals equation
  • Van der Waals isotherms and the non-monotonic loop
  • Maxwell's equal area construction
  • The critical point and universal compressibility
  • Law of corresponding states
  • Clause 12 of the course spine and lecture synthesis
  • Distribution function and phase space setup
  • Boltzmann transport equation
  • H-theorem and the arrow of time
  • Relaxation time approximation (BGK model)
  • Transport coefficients from the mean free path
  • Pressure independence of gas viscosity
  • Synthesis: kinetic theory as the physics of the journey
  • The Gibbs Paradox and the N Factorial Fix
  • Quantum Indistinguishability: Symmetric and Antisymmetric Wave Functions
  • Pauli Exclusion as a Theorem from Antisymmetry
  • Three Occupation Statistics from One Orbital
  • The Degeneracy Criterion: N lambda cubed
  • Road Ahead: Fermi Sea and Bose-Einstein Condensation
  • Clause Eight of the Course Spine
  • Motivation for the Canonical Ensemble
  • Derivation of the Boltzmann Distribution
  • The Partition Function as Master Object
  • Two-Level Atom and the Schottky Anomaly
  • Quantum Harmonic Oscillator Partition Function
  • Log Z as the Universal Thermodynamic Generator
  • Helmholtz Free Energy and the Bridge to Module Two
  • The Arrow of Time as a Counting Argument
  • Microstates vs. Macrostates
  • Phase Space and Multiplicity (Ω)
  • Sharpening of the Multiplicity Peak with N
  • Counting Microstates: Combinatorics of Spins and Quanta
  • The Fundamental Postulate: Equal A Priori Probability
  • End-to-End Worked Count: 100-Spin Paramagnet
  • Entropy as logarithm of multiplicity
  • The microcanonical ensemble
  • Second law as maximum joint multiplicity
  • Statistical nature of irreversibility
  • Dual interpretation of entropy and course outlook
  • Equilibrium condition and the first glimpse of temperature
  • End-to-end Einstein solid calculation
  • Helmholtz Free Energy: Definition and Dual Identity
  • The Derivative Web: All Thermodynamics from F
  • F as a Legendre Transform of the Entropy
  • Why Fixed-Temperature Systems Minimise F
  • Energy–Entropy Competition and Phase Transitions
  • Ideal Gas Rebuilt from One Free Energy
  • The Statistical Mechanics Recipe and Course Spine
  • Four prerequisite tools for the theorem
  • Brownian motion and the Langevin equation
  • Einstein's relation: the first fluctuation-dissipation theorem
  • General fluctuation-dissipation theorem and Onsager regression
  • Johnson-Nyquist noise: the electrical fluctuation-dissipation theorem
  • Numerical verification with real parameters
  • Clause 17 and the course through-line
  • The century-old heat capacity paradox in metals
  • The Fermi sea at absolute zero
  • Counting the Fermi sea: Fermi momentum, energy, and temperature
  • Ground state energy: three-fifths of the Fermi energy per particle
  • Degeneracy pressure
  • Sommerfeld result: linear electronic heat capacity
  • Clause nine of the course spine: the degenerate Fermi gas as a unified result
  • Four prerequisite tools for the Ising model
  • The Ising model: spins, energy, and cooperative order
  • Order parameter, spontaneous magnetization, and symmetry breaking
  • Mean-field theory and the Weiss effective field
  • Self-consistency equation and the critical temperature
  • Critical exponents predicted by mean-field theory
  • Mean-field failure and the two-dimensional Ising benchmark
  • Temperature as entropy's energy slope
  • Pressure and chemical potential as entropy slopes
  • The thermodynamic identity from three slopes
  • Ideal gas law derived from entropy slopes
  • Negative absolute temperature
  • Three equilibria unified under one principle
  • Module one capstone and course spine clause three
  • Diverging Correlation Length at Criticality
  • Critical Exponents and Log-Log Reading
  • Universality and Universality Classes
  • Scaling Hypothesis and Homogeneous Free Energy
  • Scaling Relations Among Exponents
  • Mean Field's Place: Exact Above Four Dimensions
  • Preview: Renormalization Group as the Explanation
  • Bose-Einstein occupation and the ceiling on mu
  • Saturation of excited states and convergence of the integral
  • Macroscopic condensation as a surplus forced into the ground state
  • Critical temperature from the degeneracy criterion
  • Condensate fraction as an order parameter
  • Real numbers: rubidium-87 and helium-4
  • Clause ten of the course spine and lecture synthesis
  • Partition function toolkit recap
  • Equipartition theorem derivation
  • Counting squares: ideal gas and solids
  • Energy fluctuations and the fluctuation-response relation
  • Why thermodynamics is exact: the 1/√N argument
  • Heat capacity staircase and quantum freeze-out
  • Validity of equipartition: classical limit of quantum mechanics
  • The enumeration wall: why direct summation fails
  • Importance sampling
  • The Metropolis algorithm
  • Metropolis applied to the 2-D Ising model
  • Equilibration and autocorrelation
  • Critical slowing down and cluster algorithms
  • Monte Carlo as the capstone of statistical mechanics
  • Grand canonical ensemble and its weight
  • Chemical potential as thermodynamic water level
  • Grand partition function and grand potential
  • Bose-Einstein and Fermi-Dirac distributions from one orbital
  • Bose-Einstein condensation as a constraint on μ
  • Classical limit and the ideal gas law from Ξ
  • Langmuir adsorption isotherm
  • Chemical potential pinned to zero for non-conserved bosons
  • Photon density of states and the Planck radiation law
  • Rayleigh-Jeans and Wien limits of the Planck spectrum
  • Stefan-Boltzmann law from integrating the Planck spectrum
  • Phonons as a mu = 0 boson gas with a Debye cutoff
  • Debye heat capacity: Dulong-Petit plateau and T³ low-temperature law
  • Unified architecture: light and sound from one distribution
  • The multi-scale crisis at criticality
  • The renormalization group idea: coarse grain, rescale, repeat
  • RG fixed points and the nature of the critical temperature
  • Universality from relevant and irrelevant directions
  • Exact decimation of the 1-D Ising chain
  • No finite fixed point means no phase transition in 1-D
  • Exponents, universality classes, and the RG as the capstone explanation

Syllabus

  1. Foundations

    1. Microstates, Macrostates & Phase Space
    2. Boltzmann Entropy & the Microcanonical Ensemble
    3. Temperature, Pressure & Chemical Potential

  2. Canonical & Grand Canonical Ensembles

    1. The Boltzmann Distribution & the Partition Function
    2. Free Energy & Thermodynamics from Z
    3. Equipartition, the Ideal Gas & Fluctuations
    4. The Grand Canonical Ensemble & Chemical Potential

  3. Quantum Statistics

    1. Indistinguishability — MB, BE & FD Statistics
    2. The Degenerate Fermi Gas
    3. Bose–Einstein Condensation
    4. Photons & Phonons — The Quantum Gases of Light and Sound

  4. Interacting Systems & Phase Transitions

    1. Van der Waals & the Liquid–Gas Transition
    2. The Ising Model & Mean-Field Theory
    3. Critical Phenomena — Scaling & Universality
    4. The Renormalization Group

  5. Non-Equilibrium & Computational

    1. Kinetic Theory & Transport
    2. The Fluctuation–Dissipation Theorem
    3. Monte Carlo Methods