Statistical Mechanics 2017 - 2018

• Sept 18: Thermodynamics, first and second principle, extensive and intensive variables, thermodynamic potentials.
• Sept 19: Legendre transforms, Maxwell identities. Thermodynamics of a system in a bath, availability. Thermodynamic stability. Thermodynamic fluctuations
• Sept 25: Thermodynamic fluctuations, linear response in Thermodynamics. Exercise: correlations for gaussian variables.
• Sept 26: Statistical mechanics, macroscopic irreversibility vs microscopic reversibility. Poincare’s return time. Loschmidt’s and Zermelo paradoxes. Calculation of Poincare’s time.
• Oct 9: Ensemble of initial data. Probability density and its evolution. Liouville’s theorem. Conservation of volumes in phase space. Ergodicity. Stationary distribution.
• Oct 10: Ergodicity. Mixing and its relation to ergodicity. Role of the interaction in equalibration. Integrable Hamiltonians. Examples. Role of the observable. Reduced densities.
• Oct 16: Density matrix for pure states and ensembles, time average and ergodicity in quantum systems. Stationary density matrix for an isolated system. Eigenstate Thermalization Hypotesis.
• Oct 17: Equation of motion of the reduced density distribution: the BBGKY hierarchy. Case s=1. The single-particle velocity distribution function. The Stosszahnalsatz and the Boltzmann equation.
• Oct 18: H-theorem and equilibrium distribution of velocity. Physical meaning of the H function.
• Oct 23: Decoherence and dissipation in equilibration of a quantum system. Reduced density matrix evolution. Purity. Purity evolution. Notes here
• Oct24: An example of thermalization: a classical particle interacting with N harmonic oscillators. Ohmic density and effective Langevin equation. Notes here
• Oct 30: 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 31: General Langevin equation with additive noise. Discretization of Langevin equation. The Fokker-Planck equation and equilibrium distribution. Maxwell-Boltzmann distribution as equiliberium distribution of Fokker-Planck equation. Notes here. Exercises.
• 6 Nov: Microcanonical ensemble, Boltzmann’s entropy extensivity and second principle. The perfect gas: Sakur-Tetrode formula and Boltzmann’s Correct Counting.
• 7 Nov: Sakur-Tetrode formula and it’s meaning, Nerst theorem. From microcanonical to canonical ensemble. Equivalence of the ensembles. Exercises.
• 13 Nov: From canonical to grand-canonical ensemble, thermidynamic potential and partition function. Equivalence of the ensembles. Fluctuations of the particle number. General case. The isobaric ensemble.
• 14 Nov: The isobaric ensemble, fugacity and conditions for non-degeneration of perfect gas. Thermodynamic responses. Classical gas in interaction, fugacity expansion.
• 20 Nov: Quantum equilibrium ensembles. 2nd quantization, the Fock space, creation and destruction operators, field operators. Quantum statistics and commutation relations.
• 21 Nov: Single and two particle operators. Global gauge invariance. The Gand Canonical perfect gas for bosons and fermions. The classical limit and the fugacity expansion.
• 27 Nov: Exercise corrections. Bose and Fermi gases for a general DOS, P-V-U relation. T=0 Fermi isotherm. Bose condensation.
• 28 Nov: Bose-Einstein condesation, expression for $T_{BE}$. Interaction representation, Dyson serie and perturbation theory. Euclidiean time.
• 4 Dec: Perturbation theory in statistical mechanics, first order correctin to the free energy, linear response theory. Variational principle in Microcanonical, Canonical and Grand-Canonical ensembles.
• 5 Dec: Applications of variational principle. Ising model in mean field.
• 11 Dec: Mean field theory for the Ising model, second and first order phase transition. Order parameter. Spontaneous symmetry breaking. Landau functional and its development around the critical temperature. Phase transition and finite size effects.
• 12 Dec: Landau theory of the second order phase transitions, classical critical indexes. The lattice gas model and its equivalence with the Ising model (exercise). Response and correlation functions, the correlation lenght and its divergence at the critical point. Universality. Non-existence of phase transitions in d=1.
• 8 Jan: Universality and critical exponents for scalar order parameter. Role of dimensionality Upper and lower critical dimensions. Role of the symmetry of the order parameter. Mermin-Wagner theorem (no proof). Linearization of the Hamiltonian as alternative method to get MFT. Ginsburg criterion.
• 9 Jan: Lee-Yang theorem (no proof), first and second order “phase transitions” at finite size. Multicomponent order parameter. Introduction to BCS theory, screening and dynamical screening of the electron gas.
• 10 Jan: The BCS model and its mean field solution. Landau potential for the BCS model. Critical temperature. SSB in superconductivity.