Statistical Mechanics 2020 - 2021

• Oct 05: 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.
  notes video

• Oct 06: Thermodynamic fluctuations. Linear response in thermodynamics, thermodynamic fluctuations and responses. Divergence of susceptibility at the critical point.
  notes video

• Oct 12: 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.
  notes video

• Oct 13: Ensemble of initial data, ensemble average. Ensembles evolution, Liouville’s theorem and phase space volumes conservation. Stationary distribution. Ergodicity. Mixing (definition).
  notes video

• Oct 19: Again on ergodicity and mixing. An example of ergodic but not mixing system. The recurrence time and its calculation. Chaos. Lyapunov exponent. Again of Loschmidt and Zermelo’s paradoxes. Metric transitivity. The perfect gas.
  notes video

• Oct 20: 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 in quantum systems, time averages in QM. Density matrix for a pure state.
  notes video

• Oct 26: Density matrix for a pure and mixtures of states, ensemble averages in QM. Liouville theorem in QM and stationary density matrix. Thermalization of an isolated system ,Eigenstate Thermalization Hypotesis. Thermalization as dissipation and decoherence in quantum systems.
  video

• Oct 27: Thermalization of an isolated system, ergodicity breaking, reduction of d.o.f. reduced probability density for classical systems and reduced density matrix for quantum systems. Reduced density matrix evolution.
  notes video

• Nov 02: BBGKY hierarchy for reduced density for pair interactions. $s=1$ and $s=2$ examples. Time/length scales in the $s=1$ and $s=2$ equation of the hierarchy. Truncation for low density.
  notes video

• Nov 03: Different timescales in BBGKY. Truncation of BBGKY leading to Boltzmann equation. Boltzmann’s equation: Molecular chaos approximation, H-theorem and the Maxwell Boltzmann distribution, the problem of the loss of time reversal.
  notes video

• Nov 09: An example of thermalization: a classical particle interacting with N harmonic oscillators. Ohmic density and effective Langevin equation.
  notes video

• Nov 10: 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 video

• Nov 16: Again on drift and diffusion. A general form for Langevin equation. Averages over histories and it significance. The Fokker-Planck equation. Stationary solution. Maxwell distribution from Fokker-Planck equation. The Boltzmann equation as a Msater Equation and it’s relation with Fokker-Planck equation.
  notes video

• Nov 17: Boltzmann’s Entropy, additivity and the second principle. Correct counting of microscopically different states, Sackur-Tetrode formula.
  video

• Nov 23: Gibbs paradox. Canonical distribution, partition function and thermodynamic potential. Grandcanonical ensemble. Ensembles equivalence. Energy fluctuations.
  notes video

• Nov 24: Number fluctuations. Thermodynamic linear response. Generalised change of ensemble. Generalised equipartition theorem. Virial theorem in classical mechanics and in statistical mechanics.
  notes video

• Nov 30: Fugacity and classical limit. Fugacity expansion for a classical gas. Quantum ensembles, introduction. The second quantization. Creation and destruction operators and statistics.
  video

• Dec 01: Again on second quantisation Field operators. Single particle operators. Examples.
  video

• Dec 07: Two particle operators Examples. The global gauge and the total number conservation. The grand canonical partition function for interacting particles. Bosons Fermions Boltzmannions. Equation of state, density of states. Fugacity expansion for quantum gases, the role of exchanges.
  video 1 video 2

• Dec 14: The Bose-Einstein condensation. The Fermi gas at $T=0$. Interaction picture, Dyson series and perturbation expansion.
  video

• Dec 15: 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 homogeneous and non-homogeneous cases. Examples.
  notes video

• Dec 21: Gibbs variational principle. Effective Hamiltonians. The Ising model: introduction.
  notes video

• Dec 22: Mean field theory in the Ising model from variational principle. The mean field theory as linearisation in fluctuations. Self-consistency and its solution, the second order phase transition and the spontaneous symmetry breaking for scalar order parameter.
  notes video

• Jan 11: Again on second order phase transition in the Ising model, susceptibility. The Ising model and the lattice gas. The gas-liquid transition. The order parameter. The Landau theory of phase transitions. Classical critical exponent.
  notes video

• Jan 12: Susceptibility in mean field theory. Correlation length and its classical critical exponent. Long range correlations and the failure of the mean-field theory. The Landau functional for non-homogeneous fluctuations.
  notes 1 notes 2 video

• Jan 18: The Ginzburg criterion and the upper critical dimension. The Mermin-Wagner theorem (no proof) and the lower critical dimension. Ising model in d=1 and in infinite dimensions.
  notes 1 notes 2 notes 3 notes 4 notes 5 notes 6 video

• Jan 19: Mean field decoupling schemes in quantum interacting systems. Overscreening and effective attraction of electrons in metals. The superconductive transition in the BCS mean field scheme. Landau potential for the superconductive transition. Global gauge symmetry breaking in the superconductive phase.
  notes 1 notes 2 notes 3 notes 4 video