Statistical Mechanics I

What this course covers

• Thermodynamics as phenomenological description
• Thermal equilibrium and the zeroth law
• Empirical temperature and isotherms
• Ideal gas temperature scale
• Internal energy and the first law
• Generalized work and conjugate variables
• Heat capacity at constant volume vs. constant pressure
• Joule free expansion and energy of ideal gas
• Thermodynamic equilibrium and state coordinates
• Zeroth Law and empirical temperature
• First Law and internal energy
• Quasistatic processes
• Second Law: Kelvin and Clausius statements
• Carnot engine and its properties
• Thermodynamic temperature scale
• Clausius inequality and entropy preview
• Clausius's theorem for cyclic processes
• Entropy as a state function
• Fundamental thermodynamic relation
• Independent degrees of freedom in equilibrium
• Entropy increase for irreversible adiabatic processes
• Enthalpy and mechanical equilibrium
• Helmholtz free energy
• Gibbs free energy and grand potential
• Extensivity and the Euler relation
• Gibbs–Duhem relation
• Maxwell relations from second derivatives
• Fundamental thermodynamic relation and state variables
• Gibbs-Duhem relation and extensivity
• Maxwell relations from mixed second derivatives
• Mechanical analogy for thermodynamic stability
• General thermodynamic stability condition
• Positivity of heat capacity at constant displacement
• Positivity of isothermal compressibility and critical points
• Positive-definiteness of the response function matrix
• Third Law of Thermodynamics
• Microscopic origins of thermodynamic laws
• Definition of probability and random variables
• Cumulative and probability density functions
• Expectation values and moments
• Change of variables for random variables
• Characteristic function and moment generation
• Cumulants and the cumulant generating function
• Graphical representation of moments in terms of cumulants
• Gaussian distribution and its characteristic function
• Cumulants and the cumulant generating function
• Graphical (diagrammatic) method for moments and cumulants
• Binomial distribution and its characteristic function
• Poisson distribution derived from the binomial
• Joint probability density and joint characteristic function
• Marginal (unconditional) and conditional probabilities; Bayes' theorem
• Joint Gaussian distribution and its cumulant structure
• Wick's theorem for zero-mean Gaussian variables
• Central limit theorem via characteristic functions
• Central Limit Theorem for sums of variables
• Independent and identically distributed (IID) variables
• Scaling behavior of cumulants with N
• Intensive vs. extensive vs. exponential quantities
• Dominance of the largest term in exponential sums
• Saddle-point (Laplace) integration
• Stirling's approximation via saddle-point
• Shannon entropy and information content
• Entropy of continuous probability distributions
• Maximum-entropy principle for probability assignment
• Macro states vs. micro states
• Statistical ensemble definition
• Ensemble density (probability density in phase space)
• Ensemble average of observables
• Phase space volume conservation (Liouville's theorem)
• Liouville's equation for density evolution
• Poisson bracket formalism
• Equilibrium condition for ensemble density
• Conserved quantities and accessible phase space
• Liouville equation and phase space density
• Reduced one-particle and s-particle densities
• Gas Hamiltonian: kinetic, external, and pairwise terms
• BBGKY hierarchy of equations
• Timescales: box-crossing, collision duration, and mean free time
• Dilute-gas limit and truncation of the hierarchy
• Boltzmann equation: structure and molecular chaos
• Phase space density and Liouville equation
• Reduced single-particle distribution function F1
• BBGKY hierarchy of equations
• Collision time scales and truncation of the hierarchy
• Center-of-mass frame and impact parameter
• Molecular chaos assumption (Stosszahlansatz)
• Boltzmann equation
• Boltzmann H-theorem and irreversibility
• Liouville equation and phase space density
• BBGKY hierarchy and s-particle densities
• Dilute gas truncation of the hierarchy
• Two-body collision kinematics and impact parameter
• Molecular chaos assumption
• Boltzmann equation
• Boltzmann H-theorem
• Local equilibrium distribution
• Global equilibrium and two timescales
• Collision-conserved quantities and vanishing of collision integral
• Boltzmann equation structure and notation
• Collision invariants and conserved quantities
• Hydrodynamic moment equations
• Continuity equation for particle density
• Momentum equation and pressure tensor
• Energy equation and heat flux vector
• Local equilibrium distribution f₀
• Sound waves from linearized hydrodynamics
• Chapman-Enskog perturbative correction to f
• Boltzmann equation and collision operator
• Hydrodynamic conservation equations
• Zeroth-order local equilibrium distribution
• First-order Chapman-Enskog correction
• Viscosity and shear flow relaxation
• Thermal conductivity and heat transport
• Statistical mechanics and equal a priori probabilities
• Zeroth law from microcanonical ensemble
• First and second laws from entropy counting
• Microcanonical phase-space volume for ideal gas
• Microcanonical ensemble and equal probability postulate
• Phase space volume Ω as a hypersphere surface
• Entropy of the ideal gas from Ω
• Deriving thermodynamic relations from entropy
• Single-particle momentum distribution from microcanonical ensemble
• Gibbs paradox and entropy of mixing
• Indistinguishability correction: dividing by N! and h^{3N}
• Canonical ensemble: Boltzmann weight derivation
• Partition function and thermodynamic averages
• Energy fluctuations and heat capacity in the canonical ensemble
• Microcanonical ensemble and entropy
• Canonical ensemble and Boltzmann probability
• Energy probability distribution and saddle-point argument
• Energy fluctuations and heat capacity
• Helmholtz free energy from the partition function
• Gibbs canonical ensemble and Gibbs free energy
• Grand canonical ensemble and grand potential
• Ideal gas partition function and thermodynamics
• Ideal gas in the Gibbs and grand canonical ensembles
• Interacting systems and cumulant expansion
• Cumulant expansion for interacting systems
• Pairwise interaction potential and its divergence
• Mayer f-function and its graphical form
• Second Virial coefficient B_2
• Cluster expansion and diagrammatic representation
• Linked cluster integrals b_l
• Combinatorial multiplicity in cluster sums
• Grand canonical ensemble and the grand partition function
• Pressure as a sum over linked clusters
• Grand partition function and cluster expansion
• Cluster integrals and the virial series
• Virial coefficients from cluster diagrams
• Second virial coefficient for a realistic potential
• Validity limits of the virial expansion
• Derivation of the van der Waals equation
• Isotherms and the critical point
• Thermodynamic stability and compressibility
• Maxwell equal-area construction
• Liquid-gas phase diagram and coexistence
• Canonical partition function for interacting gas
• Mean-field (uniform density) approximation
• Derivation of the van der Waals equation
• Failure of uniform-density approximation at coexistence
• Isobaric (Gibbs) ensemble and Maxwell construction
• Phase transitions in the thermodynamic limit
• Critical point coordinates from van der Waals equation
• Law of corresponding states
• Critical exponents and universality
• Failures of classical statistical mechanics
• Partition function for a diatomic ideal gas
• Classical equipartition and the 7/2 kB prediction
• Quantization of vibrational energy levels
• Quantization of rotational energy levels
• Correspondence between quantum and classical phase space
• Einstein model of heat capacity of solids
• Phonon spectrum and the Debye T³ law
• Quantized harmonic oscillator energy levels
• Heat capacity of solids and phonon modes
• Debye T³ law derivation
• Electromagnetic modes in a cavity
• Blackbody radiation spectrum and Stefan-Boltzmann law
• Quantum microstate as a Hilbert space vector
• Quantum observables as Hermitian operators
• Density matrix for mixed quantum states
• Quantum Liouville equation and equilibrium density matrix
• Density matrix and quantum ensemble averages
• Time evolution and quantum Liouville equation
• Quantum micro-canonical ensemble
• Assumption of random phases
• Quantum canonical ensemble and partition function
• Single-particle density matrix in a box
• Identical particles and wave-function symmetry
• Symmetrized many-particle basis states
• N-particle density matrix and exchange terms
• Identical non-interacting particle Hamiltonians
• Product states and their limitations
• Symmetrization and anti-symmetrization of states
• Occupation-number representation
• Single-particle density matrix and thermal wavelength
• N-particle density matrix via permutation sum
• Statistical (exchange) interaction between identical particles
• N-particle canonical partition function and quantum corrections
• Second virial coefficient from quantum statistics
• Grand canonical ensemble for quantum ideal gases
• Fermi-Dirac and Bose-Einstein distribution functions
• Quantum gas of identical non-interacting particles
• Grand canonical ensemble and diagonal density matrix
• Grand canonical partition function and mean occupation number
• Single-particle spectrum and density of states for a non-relativistic gas
• Thermal de Broglie wavelength and the fugacity-density relation
• Fermi–Dirac and Bose–Einstein special functions f_m^η(z)
• Quantum degeneracy parameter and the classical (low-density) limit
• Degenerate Fermi gas and the Sommerfeld expansion
• Grand canonical ensemble for quantum gases
• Density of states and f_m functions
• Pressure and energy of ideal quantum gas
• Sommerfeld expansion for degenerate fermions
• Chemical potential of a Fermi gas vs. temperature
• Pressure and heat capacity of a Fermi gas
• Constraint on chemical potential for bosons
• Bose-Einstein condensation and critical temperature
• Thermodynamics below T_c: pressure and heat capacity
• Bose-Einstein condensation into ground state
• Heat capacity of a Bose-Einstein condensate
• Experimental observation of BEC momentum distribution
• Helium phase diagram and quantum melting
• Lambda transition and superfluidity in He-II
• Andronikashvili torsional oscillator experiment
• Two-fluid model of superfluidity
• Differences between BEC and superfluidity
• Superfluid helium phase diagram
• Differences between BEC and superfluid helium
• Landau excitation spectrum: phonons and rotons
• T-cubed heat capacity from linear spectrum
• Landau criterion for superfluidity
• Normal fluid density from thermal excitations (T⁴ dependence)

Syllabus

1. Lectures

   1. 1. Thermodynamics Part 1
   2. 2. Thermodynamics Part 2
   3. 3. Thermodynamics Part 3
   4. 4. Thermodynamics Part 4
   5. 5. Probability Part 1
   6. 6. Probability Part 2
   7. 7. Kinetic Theory of Gases Part 1
   8. 8. Kinetic Theory of Gases Part 2
   9. 9. Kinetic Theory of Gases Part 3
   10. 10. Kinetic Theory of Gases Part 4
   11. 11. Kinetic Theory of Gases Part 5
   12. 12. Classical Statistical Mechanics Part 1
   13. 13. Classical Statistical Mechanics Part 2
   14. 14. Classical Statistical Mechanics Part 3
   15. 15. Interacting Particles Part 1
   16. 16. Interacting Particles Part 2
   17. 17. Interacting Particles Part 3
   18. 18. Interacting Particles Part 4
   19. 19. Interacting Particles Part 5
   20. 20. Quantum Statistical Mechanics Part 1
   21. 21. Quantum Statistical Mechanics Part 2
   22. 22. Ideal Quantum Gases Part 1
   23. 23. Ideal Quantum Gases Part 2
   24. 24. Ideal Quantum Gases Part 3
   25. 25. Ideal Quantum Gases Part 4
   26. 26. Ideal Quantum Gases Part 5