Sep 19 Classical e/m field as collection of harmonic oscillators. Field quantization and commutation rules. Number states.
Sep 20 Non-relativistic scalar quantum field. Number states, field operators, relationships between first and second quantization.
Sep 25 Single and two-particles operators in second quantization. Examples: single-particle Hamiltonian, total number of particle, particle density, pair-potential.
Sep 26 Quantum Field Theory and Classical Field Theory, the scalar case. Mean field decouplings Hartree, Foch and BCS. Global gauge, particle number conservation and gauge symmetry breaking in the BCS decoupling scheme.
Oct 03 Interaction representation, perturbation theory, Dyson series. Euclidean time and equilibrium statistical mechanics. Partition function, theormodynamic potential, quantum-statistical average.
Oct 04 Perturbation theory for the partition function and for average values. Linear response theory, susceptibility and correlation functions. Thermodynamics: theormodynamic potentials and Legendre transform.
Oct 10 Thermodynamics, Maxwell's relations, intensive adn extensive variables. Thermodynamics of open systems, availability. Thermodynamics stability and thermodynamic fluctuations.
Oct 11 Statistical mechanics reversible time evolution vs irreversible thermodynamics. Hamiltonian description of an isolated system, time reversion, Poicar?s theorem, Loschmidt and Zermelo paradoxes. Chaos and Lyapunv exponent. Calculation of Poincar?return time.
Oct 17 Single evolotion of a classical dynamical system, observables, time average. Ensembles of initial data, Liouville's theorem and conservation of volumes of phase space. Ensemble averages. Stationary distribution. Ergodic hypotesis.
Oct 18 Mixing and ergodicity. Qualitative description of ergodic motion. Time average in QM. Density matrix. Exercises.
Oct 24 Stationary density matrix, microcanonical ensemble. Routes to equilibration, the role of the observable, the role of the interactions. Reduced denisty and denisty matrices. The Eigenstate Thermalization Hypotesis.
Oct 25 Reduced denisities and the BBGKY hierarchy for the s-particle reduced density. The s=1 case. Momentum distribution. Botzmann Equation.
Nov 02 Boltzmann equation, molecular chaos hypotesis. H-theorem equilibrium distribution function. Physical meaning of the H function.
Nov 07 Density matrix of a pure and mixed state. Populations and coherences. Reduced density matrix time evolution. Purity and reduced purity, tme evolution. Decoherence and dissipation.
Nov 08 Dissipation in a model of harmonic oscillators bath. Derivations of the equations, Ohmic spectral density and Langevin equation. Brownian motion calculation of the asymptotic velocity distribution.
Nov 14 Langevin equation, processes, correlations functions, Ornstein-Uhlembec process, velocity correlation function. Drift, Drude Formula.
Nov 15 Mobility and Einstein's relations. Overdamped Langevin equations long-time behaviour and diffusion. Fokker-Plack equation and its asymptotic limit.
Nov 21 Microcanonical ensemble, Boltzmann entropy, Boltzmann correct counting and entropy extensivity. Entropy extensivity, definition of temperature. Sakur-Tetrode formula.
Nov 22 Nerst principle and second principle in the microcanonical ensemble. Loschmidt and Zermelo paradoxes revised. Thermalized systems, canonical ensemble. Partition function and thermodynamic potential. Energy distribution. Equivalence of the ensembles and thermodynamic stability in the thermodynamic limit.
Nov 28 From micanonical to microcanonical ensemble, thermodynamic potential, equilibrium condition and stability. Second order phase transition, gas-liquid divergence of isothermal compressibility. Generalized equipartition principle.
Nov 29 Virial theorem and its application to the state equation of a real fluid. Virial expansion. Fugacity expansion in the grand canonical ensemble. Calculation of the second virial coefficient for hard spheres. A model for interacting particles with hard-cores. The Van Der Waals equation.
Dec 6 Bose-Einstein condensation.
Dec 14 Ising model, ground state, degeneracies and symmetries. Mean field solution via variational principle. Landau potential.
Dec 20 Landau theory of second order phase transitions.
Classical critical indexes and classes of universility. Non existence of phase transitions in one
dimensions. Correlation functions and responses within mean-field theory.
EXERCISES
1)
a) The energy of classical
electromagnetic field
b) Quantum e/m field commutation rules
2)
a) Two particle operators in first and second
quantization
b) gauge transformation and commutation
relations
3)
a) Perturbation theory on partition
function adn response function for perturbed harmonic oscillator,
comparison with exact results
b) Non-interacting lattice gas,
partition function and entropy
c) Specific heat of
classical and s=1/2 spin in external field
4)
a)
Thermodynamics of open systems: role of chemical potential
b)
Equilibrium of a gas/piston system
c) Dynamical evolution of N
independent harmonic oscillators, time-averages
5)
a)
Ergodic flow map: mixing and ergodicity.
b) Arnold's map:
mixing, ergodicity and recurrence.
6)
a) particle in
interaction with a bath of harmonic oscillators: statistical
properties of the noise term
b) the Kac ring
7)
a)
momentum correlation function in the Langevin equation
b)
particle in interaction with a bath of harmonic oscillators:
numerical simulation
8)
a) Two level systems, entropy
in microcanonical and canonical ensemble-
b) Momentum
distribution function for an isolated system
c) The isobaric
ensemble: the one dimensional perfect gas case
9)
a) Thermodynamic potential for hard-sphere gas
b) Calculation of the Cv for hard spheres system
c) Virial theorem for quandratic Hamiltonians
10)
a) Single particle density matrix and the meaning of the De Broglie Wave-lenght
b) Polymer lenght
c) PV,U relations for quantum gases. Equation of state of Fermi gas at T=0 and for Bose gas of massless particles
11)
a) Variational principle in quantum stat. mech., variational principle for grand-canonical and isobaric ensembles.
b) Specific heath, Entropy and Free energy in the mean field Ising model.
12)
a) Equivalence beween Ising model and Lattice Gas. Equation of state in the mean-field theory of the Lattice Gas.
b) Ising model in d=1
c) Ising model with infinte range interaction.