# ADVANCED STATISTICAL MECHANICS

**Academic Year 2016/2017**- 1° Year - Curriculum ASTROFISICA, Curriculum FISICA DELLA MATERIA and Curriculum FISICA TEORICA

**Teaching Staff:**

**Andrea RAPISARDA**

**Credit Value:**6

**Scientific field:**FIS/02 - Theoretical physics, mathematical models and methods

**Taught classes:**42 hours

**Term / Semester:**1°

## Learning Objectives

The course aims at the understanding of the thermodynamic properties of macroscopic systems on the basis of statistical-dynamical behavior of their microscopic constituents.

## Detailed Course Content

Principles of Thermodynamics. Thermodynamic equilibrium. Thermodynamic Potentials. Kinetic Theory. H theorem of Boltzmann. Maxwell-Boltzmann distribution. Ensemble theory of Gibbs. Classical Statistical Mechanics: Phase space. Liouville's theorem. Principle of a priori equiprobability. Microcanonical ensemble. Virial theorem. Equipartition of energy. Classical ideal gas. Derivation of thermodynamics for almost isolated systems. Gibbs paradox. System in contact with a thermostat. Statistical concept of temperature. Canonical ensemble. Energy fluctuations. Systems with variable number of particles. Chemical potential. Grand-canonical ensemble. Fluctuations in density in open systems. Gibbs paradox and correct counting of microscopic states. Postulates of quantum statistical mechanics. Density matrix. Quantum Liouville equation. Formulation of the quantum theory of Gibbs ensemble. Third Law of Thermodynamics. Ideal gas of Fermi and Bose. Bose-Einstein condensation and superfluid systems. Electromagnetic excitations in a cavity. Thermal excitations in solids. Statistical equilibrium in white dwarf stars. Electron gas in metals. Low-temperature behavior of Bose and Fermi of a weakly imperfect gas. Elementary excitations in helium liquid. Classical interacting systems. Development cluster for a classic real gas. Development of the virial equation of state of a perfect gas. Derivation of Van der Waals forces. Phase transitions and critical phenomena. Critical indices and scale invariance. The Ising model for ferromagnetism and model of the lattice gas. The mean field theory. Renormalization group theory and its applications. Numerical Methods: The Monte Carlo method and molecular dynamics - Some algorithms and applications. Deterministic chaos and the foundations of statistical mechanics - Lyapunov Exponents - Kolmogorov-Sinai entropy.

## Textbook Information

K. Huang : *Meccanica Statistica, *Zanichelli (1997)

R.K. Pathria : *Statistical Mechanics, *Pergamon Press (1996)

E. Ott: *Chaos in Dynamical systems*, Cambridge University Press (1993)