Statistical Mechanics 2019 - 2020

• Sept 23: Introduction to the course. Thermodynamics 1st and 2nd principle. Intensive and extensive variables. Thermodynamic potentials. Legendre transforms among thermodynamic potentials. Maxwell’s relations, thermodynamics of a system in a bath: availability. Thermodynamic stability.
• Sept 24: Thermodynamic fluctuations. Linear response in thermodynamics, thermodynamic fluctuations and responses. Divergence of susceptibility at the critical point.
• Sept 30: Statistical mechanics, Hamiltonian dynamics for classical systems, phase space and observables, time averages. Microscopic reversibility vs macroscopic irreversibility. Loschmidt and Zermelo’s paradoxes and their resolution.
• Oct 01: Ensemble, ensemble average, Liouville’s theorem. Invariant measure. Stationary distribution. Ergodicity. Mixing and its relation to ergodicity.
• Oct 07: Mixing and its relation to ergodicity. The recurrence time and its calculation.
• Oct 08: Ergodicity and mixing in quantum ensembles. Liapunov exponent and chaos. Some models: Kac ring, Arnold cat map, logistic map.
• Oct 14: Separable Hamiltonians and lack of asymptotic equipartition. Example: interacting hamonic oscillators, normal modes. Typical initial data and time averages. Role of observable in ergodicity, Khinchin theorem. Ergodicity of quantum systems: Eigenstate Thermalization Hypotesis.
• Oct 15: Thermalization of an isolated system, reduction of d.o.f. reduced probability density for classical systems and reduced density matrix for quantum systems. Thermalization as dissipation and decoherence in quantum systems. Reduced density matrix evolution.
• Oct 21: Purity. Purity evolution. Role of the entanglement. Notes here. Classical systems: the BBGKY hierarchy.
• Oct 22: BBGKY hierarchy for single particle evolution. Boltzmann’s equation: Molecular chaos approximation, H-theorem and the Maxwell Boltzmann distribution.
• Oct 28: Exercises correction. The significance of the function H, Entropy in the canonical ensemble. Different timescales BBGKY. Truncation of BBGKY leading to Boltzmann equation, the problem of the loss of time reversal.
• Oct 29: An example of thermalization: a classical particle interacting with N harmonic oscillators. Ohmic density and effective Langevin equation. Notes here.
• Nov 04: Langevin equation, velocity relaxation and correlation function. The Ornstein-Uhlembeck process. Brownian motion in the overdaped limit. Diffusion constant. Drift and Drude formula, mobility. Einstein’s relation. Notes here.
• Nov 05: Wiener process. Averages over histories. The Fokker-Planck equation. Stationary solution. Maxwell distribution from Fokker-Planck equation Notes here.
• Nov 11: Exercises corrections. Boltzmann’s Entropy, additivity and the second principle. Correct counting of microscopically different states, Sakur-Tetrode formula.
• Nov 12: Sakur-Tetrode and additivity.
• Nov 18: Correct Gibbs counting. Canonical distribution, partition function and thermodynamic potential.
• Nov 19: Grandcanonical ensemble. Ensembles equivalence. Energy fluctuations. Number fluctuations. Thermodynamic linear response. Generalised change of ensemble
• Nov 20: Fugacity and classical limit. Fugacity expansion for a classical gas. Generalised equipartition theorem. Virial theorem in classical mechanics and in statistical mechanics.
• Nov 25: Quantum ensembles, introduction. The second quantization. Creation and destruction operators and statistics. Field operators. Single particle operators. Examples.
• Nov 26: Two particle operators Examples. The global gauge and the total number conservation. The grand canonical partition function for interacting particles.
• Dec 02: The grand partition function for a perfect gas, Bosons Fermions Boltzmannions. Equation of state, density of states. Correct Boltzmann counting from quantum statistics. Equation of state far from the degenerate condition. Bose-Einstein condensation.
• Dec 03: 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 09: (T. Macrì) Gibbs variational princicple and Peierls inequality.
• Dec 10: (T. Macrì) Applications: 1) Ising and Heisemberg spin systems. Free energy in a Heisemberg ferromagnet vs Ising ferromagnet (Huang 15.5). Mena field theory in the Ising model (Huang 15.6). Spontaneous magnetization.
• Dec 16: Critical indexes for Ising ferromagnet. Landau functionals and classes of universality in the Landau theory of the second order phase transitions. Spontaneous symmetry breaking.
• Dec 17: Spontaneous symmetry breaking. Spontaneous symmetry breaking in the Ising model. The role of the number of degree of freedom. Correlation function and susceptibility in the Ising model. Correlation length and it critical behaviour.
• Jan 07: Linearisation of interaction Hamiltonian and mean field theory. The Ginzburg criterion and the upper critical dimension. The Mermin-Wagner theorem (no proof) and the lower critical dimension.
• Jan 09: Mean field decoupling schemes in quantum interacting systems. The superconductive transition in the BCS mean field scheme. Landau potential for the superconductive transition. Global gauge symmetry breaking in the superconductive phase.