Università Roma Tre

6 CFU compulsory

Thermodynamic potentials and availability function. Thermodynamic stability.
Gibbs phases rule. Probability of small deviations and central limit theorem.
Clapeyron equation. Classification of phase transitions. Van der Waals equation.
Divergence of the compressibility at the critical point.

The Boltzmann equation. H theorem and its connection with the entropy.
Derivation of the Maxwell-Boltzmann distribution and extension to the quantum case.
Evaluation of the thermal conductivity of a Maxwell gas.
Evaluation of the electrical conductivity of a Fermi gas.

The problem of the thermodynamic limit. Lee-Yang theorems and zeros of the grand partition function.
Proof of the first theorem of Lee-Yang. Proof of the second theorem of Lee-Yang.

Microscopic derivation of the van der Waals equation with the mean-field approximation. Expansion of the van der Waals equation near the critical point. Critical indices.

Ferromagnetic transition. Weiss field. Ising model. Mean-field theory for the Ising model. The Curie-Weiss equation.
Critical indices. Ising model with long range interaction.

Fluctuation-dissipation theorem. Landau theory of phase transitions.
Fluctuations within the Landau theory. Ginzburg criterion. Functional formulation of the Ising model.
Fourier representation and continuous limit of the functional formulation.
Gaussian fluctuations and multi-component order parameter.
The Mermin-Wagner theorem.

The scaling hypothesis. Scaling laws.
The properties of the renormalization group.
The Kadanoff-Wilson transformation. The Gaussian model.
The non-Gaussian fixed point and the ε-expansion.

2 CFU optional

The application of the renormalization group method to the theory of the metal-insulator transition.
Normal Fermi liquids. Luttinger liquid.

1) Carlo Di Castro, Roberto Raimondi
Statistical Mechanics and Applications in Condensed Matter
Cambridge University Press, Cambridge, UK, 2015

2) Kerson Huang
Statistical Mechanics
John Wiley & Sons, Inc., 1987