Statistical Mechanics 2018 - 2019

• Sept 17: Introduction to the course. Thermodynamics 1st and 2nd principle. Intensive and extensive variables. Thermodynamic potentials. Legendre transforms among thermodynamic potentials.
• Sept 18: Maxwell’s relations, themrodynamics of a system in a bath: availability. Thermodynamic stability. Thermodynamic fluctuations.
• Sept 24: Linear response in thermodynamics. Statistical mechanics, Hamiltonian dynamics for classical systems, phase space and observables, time averages. Microscopic reversibility vs macroscopic irreversibility.
• Sept 25: Loschmidt and Zermelo’s paradoxes and their resolution. Ensemble, ensemble average, Liouville’s theorem. Invariant measure.
• Oct 01: Stationary distribution. Ergodicity. Mixing and its relation to ergodicity. Ergodic flow and Arnold’s maps.
• Oct 02: Separable Hamiltonians, role of the observable in ergodicity, ergodic characteristics of harmonic oscillators, equipartition and lack of thereof. Quantum ensembles, density matrix for pure and mixed states. Decoherence and dissipation. Time evolution of density matrix in the energy representation.
• Oct 08: Ergodicity and mixing in quantum ensembles. The Eigenstate Thermalization Hypotesis. Reduced densities in classical and quantum mechanics. Evolution of reduced density matrix.
• Oct 09: Evolution of reduced density matrix and classical reduced density. The BBGKY hierarchy. Case s=1. The single-particle velocity distribution function. The Stosszahnalsatz and the Boltzmann equation.
• Oct 15: Boltzmann equation and H-theorem. Equilibrium distribution function. Physical significance of the H function.
• Oct 16: role of the interaction on relaxation time. The Boltzmann equation in the non-homogeneous case. Decoherence and dissipation in equilibration of a quantum system. Reduced density matrix evolution. Purity. Purity evolution. Role of the entanglement. Notes here.
• Oct 22: An example of thermalization: a classical particle interacting with N harmonic oscillators. Ohmic density and effective Langevin equation. Notes here.
• Oct 23: Langevin equation, velocity relaxation and correlation function. The Ornstein-Uhlembeck process. Brownian motion in the overdaped limit, the Wiener process. Diffusion constant. Drift and Drude formula, mobility. Einstein’s relation. Notes here.
• Oct 29: Stochastic equations. Simulation of stochastic equations with additive noise term. Averages over histories. The Fokker-Planck equation.
• Oct 30: The Fokker-Planck equation, stationary solution. Maxwell distribution from Fokker-Planck equation. The microcanonical density, thermodynamic potential, extensivity.
• Nov 5: Sakur-Tetrode formula. Correct Gibbs counting. Exercise.
• Nov 6: Canonical distribution, partition function and thermodynamic potential, ensemble equivalence. Energy fluctuations.
• Nov 12: Grandcanonical ensemble and its equivalence with canonical ensemble. Number fluctuations. Thermodynamic linear response.
• Nov 13: Generalised change of ensemble. Fugacity and classical limit. Fugacity expansion for a classical gas. Virial coefficients. Generalised equipartition theorem.
• Nov 19: Virial theorem in mechanics and in statistical mechanics. Equation of state role of repulsive and attractive interaction, isothermal compressibility and virial coefficients. Quantum ensembles, introduction.
• Nov 20: The second quantization. Creation and destruction operators and statistics. Field operators. Single particle operators. Examples.
• Nov 22: Exercises corrections. The prefect gas of bosons and fermions in the grand-canonical ensemble.
• Nov 26: Exercises corrections. The grand partition function for a perfect gas, Bosons Fermions Boltzmannions. Equation of state, density of states.
• Nov 27: Fugacity expansion for quantum gases, correct Boltzmann counting from quantum statistics. Equation of state far from the degenerate condition. Bose-Einstein condensation.
• Nov 28: Exercises corrections. Isobaric ensemble.
• Dec 3: Interaction picture, Dyson series and perturbation expansion. Imaginary time and perturbation expansion in statistical mechanics. Expansion of the thermodynamic potential. Calculation of a generic thermal average in a perturbative framework. Linear response theory at equilibrium.
• Dec 4: Functionals and functional derivatives. Variational principle in isolated, closed and open systems. Application of variational principle.
• Dec 9: The Ising model. The mean field theory derived from variational principle in the Ising model. The Curie-Weiss self-consistency equation. Second order and first order phase transitions. Similarities between liquid-gas transition and the magnetic transition in the Ising model.
• Dec 10: The order parameter in the Ising model and in the Lattice gas model. Universality. Spontaneous symmetry breaking in the second order phase transitions. Behaviour around the critical point. The landau theory of the second order phase transiones. The critical indexes.
• Dec 17: Susceptibility in mean field theory. Correlation length and its classical critical exponent. Long range correlations and the failure of the mean-field theory. Ising model in one dimensions.
• Dec 18: Linearization of the Hamiltonian as alternative method to get MFT. Ginsburg criterion. for the upper critical dimension. Role of dimensionality Upper and lower critical dimensions. Role of the symmetry of the order parameter. Mermin-Wagner theorem (no proof).
• Jan 7: Mean field theory for quantum Hamiltonians, Hartree-Foch decoupling schemes. BCS decoupling schemes. Origin of attractive interactions between electrons in solids. BCS model and mean field decoupling.
• Jan 8: The BCS model derivation of the Landau potential. Gap equation and evaluation of the critical temperature.