# ADVANCED STATISTICAL MECHANICS

**Academic Year 2021/2022**- 1° Year - Curriculum ASTROPHYSICS, Curriculum CONDENSED MATTER PHYSICS and Curriculum THEORETICAL PHYSICS

**Teaching Staff:**

**Andrea RAPISARDA**

**Credit Value:**6

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

**Taught classes:**35 hours

**Exercise:**15 hours

**Term / Semester:**1°

## Learning Objectives

The course aims at understanding the thermodynamic properties of macroscopic systems on the basis of the statistical-dynamic behavior of their microscopic constituents. In particular, the objectives of the course are:

**Knowledge and understanding.** Critical understanding of the most advanced developments in Modern Physics, both theoretical and experimental, and their interrelations. Adequate knowledge of advanced mathematical and numerical tools, currently used in statistical physics both in basic and applied research. Considerable knowledge of the scientific method, understanding of nature and research in physics on a statistical basis.

**Applying knowledge and understanding.** Ability to identify the essential elements of a phenomenon, in terms of orders of magnitude and level of approximation and being able to perform the required approximations. Ability to use analytical and numerical tools or science computing of statistical type, including the development of specific software.

**Making judgments.** Ability to argue personal interpretations of physical phenomena in statistical terminology, comparing themselves within working groups. Upon completion of the course, the student must be able to know the topics of the course and be able to derive the main results presented in class through the necessary analytical steps.

**Communication skills.** Ability to discuss advanced physical concepts, both in Italian and in English. Ability to present one's research activity or a review topic to both an expert and non-expert audience.

**Learning skills.** Ability to acquire adequate tools for the continuous updating of one's knowledge. Ability to access specialized literature both in the specific field of one's competence and in closely related fields.

## Course Structure

Lectures and excercises in the classroom

Note: *should the circumstances require online or blended teaching, appropriate modifications to what is hereby stated may be introduced, in order to achieve the main objectives of the course.*

## 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. Stochastic processes.

## Textbook Information

K. Huang : *Statistical Mechanics, *J. Wiley & Sons (1987)

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

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