Statistical Mechanics 2025 - 2026
- Sept 22: Introduction to the course. Thermodynamics 1st and
2nd principle. State functions, work and heat. Intensive and extensive
variables. Thermodynamic potentials. Legendre transforms among
thermodynamic potentials. Maxwell’s relations. Open thermodynamics
systems, availability.
notes
- Sept 23: Thermodynamic stability. Thermodynamic responses
and thermodynamic stability. Thermodynamic fluctuations. Linear response
in thermodynamics, thermodynamic fluctuations and responses.
notes
- Sept 29: Again on linear response in thermodynamics,
thermodynamic fluctuations and responses. Mechanical foundations of
statistical mechanics: phase space and observables.
notes
- Sept 30: Time averages. Ensemble of initial data, ensemble
average.Liouville’s theorem. Stationary distribution. Ergodic hypotesis.
notes
- Oct 06: (S. Paganelli) The microcanonical ensemble. The
Boltzmann’s Entropy. Correct counting of microscopically different
states, Sackur-Tetrode formula. The mixing entropy and the correct
Boltzmann’s counting.
notes
notes
- Oct 07: (S. Paganelli) The Boltzmann’s Entropy, additivity
and the second principle. Internal energy of a subsystemas and its
fluctuations. The canonical distribution.
notes
- Oct 13: (S. Paganelli) Canonical distribution, partition
function and thermodynamic potential. Energy fluctuations. Equivalence
between Canonical and Microcanonical ensemble. The perfect gas.
notes
- Oct 14: (S. Paganelli) Grandcanonical ensemble. Ensembles
equivalence. Number fluctuations. Generalised change of ensemble.
Grandcanonical partition function for a perfect gas. Fugacity and
classical limit. Grandcanonical partition function for finite system and
Lee-Yang theorem (hints).
notes
- Oct 20: (S. Paganelli) Inperfect gases virial theorem.
Equation of state of classical interacting gas in the low-density limit.
Comparison of virial expansion to the van der Waals equation. Quantum
statistical mechanics: the density matrix, the Liouville’s theorem in
the quantum realm. Density matrix of a pure state and of a mixture.
Coherences.
- Oct 21: (S. Paganelli) Reduced density matrix. Decoherence
and dissipation. Entangled states. Von Neumann entropy. Equilibrium
Ensembles in Quantum Statistical Mechanics.
notes
- Oct 27: (S. Paganelli) Quantum noninteracting gases.
Identical particles.
- Oct 28: (S. Paganelli) The second quantization. Creation
and destruction operators and statistics. Field operators. One-body and
two-body particle operators. Examples. The global gauge and the total
number conservation.
notes
- Nov 03: Bosons Fermions Boltzmannions. Equation of state,
density of states. Fugacity expansion for quantum gases (exercise), the
role of exchanges. The cell volume and the correct Boltzmann counting as
output of quantum statistics in the classical limit. Quantum perfect
gases. Bosons Fermions Boltzmannions. Degenerate quantum gases: the
Fermi gas at T = 0.
- Nov 04: Degenerate quantum gases: the Fermi gas at T = 0. The Bose-Einstein
condensation.
notes
reference
- Nov 10: Single particle density matrix and off-diagonal
long-range order.
notes
- Nov 11: Symmetries and thermal averages. Translation and
rotation symmetry in a generic correlation function. Gibbs variational
principle. The entropy functional: microcanonical canonical and
grand-canonical distribution.
notes
- Nov 17: Again on variational theorem. Effective
Hamiltonians. Mean field theory of Ising model. The mean field
equations. The spontaneous symmetry breaking.
notes
- Nov 18: Again on Ising model and lattice gas models,
introduction to exercise session #4. Ising model and lattice-gas model
the gas-liquid phase transition and the para-ferromagnet. Universality.
Adiabatic
expansion
- Nov 19: Ising model the free energy. Back to the principles
of statistical mechanics: microscopic reversibility vs macroscopic
irreversibility. Loschmidt and Zermelo’s paradoxes and their resolution.
Ergodic trajectory in the Phase Space. The
recurrence time and its calculation.
notes
- Nov 24: Ergodicity and mixing. Chaos. The logistic map. The
Kac ring and equilibrium in time-reversal systems. Separable
Hamiltonians and lack of asymptotic equipartition. Fermi Pasta Uhlam
Hamiltonian.
- Nov 25: Reduced probability density for classical systems
and reduced density matrix for quantum systems. Reduced density matrix
evolution. The BBGKY hierarchy. BBGKY hierarchy for s = 1 and s = 2.
notes
- Dec 01: The master equation. The Boltzmann’s equation as a
master equation. Molecular chaos approximation. Proof of the H-theorem,
the problem of the loss of time reversal. Molecular chaos approximation
form BBGKY estimate.
notes
notes
- Dec 02: Ergodicity in quantum systems, time averages in QM.
Thermalization of an isolated system ,Eigenstate Thermalization
Hypotesis. Purity and decoherence in open quantum systems.
notes.
notes.
- Dec 09: An example of thermalization: a classical particle
interacting with N harmonic oscillators. Ohmic density and effective
Langevin equation. Momentum relaxation Ornstein-Uhlembeck process.
notes
reference
- Dec 12 : Brownian motion in the overdaped limit. Diffusion
constant. Drift and Drude formula, mobility. Einstein’s relation. Again
on drift and diffusion. A general form for Langevin equation and how to
numerically integrate it. The Fokker-Planck equation. The Krames
equation and the Maxwell-Boltzmann distribution as asymptotic limit.
notes:
drift and diffusion
notes:
stochastic eqs.
notes:
Fokker Plank
- Dec 15: The Boltzmann equation as a Master Equation and
it’s relation with Fokker-Planck equation, the Kramers-Moyal expansion
(hints). Again on second order phase transition in the Ising model,
susceptibility. The mean field solution. The order parameter. The
spontaneous symmetry breaking and the thermodynamic limit. The Landau
theory of phase transitions. Critical indexes.
notes:
Kramers eq.
notes:
Landau theory of 2nd order phase transitions
- Dec 16: Critical
indexes and classes of universality. Mean field linearization
schemes in in the Ising model and in quantum interacting systems. On the
validity of the Mean Field Theory. Correlations and Responses in the
Ising model. The Ornstein-Zernike formula for magnetic susceptibility.
notes:
Correlations in MFT
notes:
Susceptibility in MFT
- Dec 22: Long range correlations and the failure of the
mean-field theory. The
Ginzburg criterion and the upper critical dimension. The Ising model
in d=1 (zero magnatic field) and the lower critical dimensions.
Spontaneous symmetry breaking and the Ising model in infinite
dimensions. The Mermin-Wagner theorem (no proof).
notes:
1d Ising model
notes:
Infinite range Ising model
notes:
Spontaneous Symmetry Breaking and thermodynamic limit
- Jan 08: 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:
The BCS model
notes:
steps to BCS solution
notes:
The Bogoliubov diagonalization
reference:
The BCS theory