# ADVANCED STATISTICAL MECHANICS

**Academic Year 2022/2023**- Teacher:

**Andrea RAPISARDA**

## Expected Learning Outcomes

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.*

## Required Prerequisites

## Attendance of Lessons

## 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)

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Thermodynamics | K. Huang, Statistical Mechanics, J. Wiley & Sons (1987) |

2 | Ensembles theory | K. Huang, Statistical Mechanics, J. Wiley & Sons (1987) |

3 | Phase transitions | K. Huang, Statistical Mechanics, J. Wiley & Sons (1987); R.K. Pathria Statistical Mechanics, Pergamon Press (1996) |

4 | Critical Phenomena | K. Huang, Statistical Mechanics, J. Wiley & Sons (1987); R.K. Pathria Statistical Mechanics, Pergamon Press (1996) |

5 | Universality and Scaling | K. Huang, Statistical Mechanics, J. Wiley & Sons (1987); R.K. Pathria Statistical Mechanics, Pergamon Press (1996) |

6 | Deterministic Chaos | E. Ott: Chaos in Dynamical systems, Cambridge University Press (1993) |

## Learning Assessment

### Learning Assessment Procedures

Intermediate tests are scheduled during the semester

For the final exam, the preparation of a written paper on one of the topics covered is expected. The student must present the thesis orally during the oral exam, starting from the thesis the main topics presented in class will be studied in depth during the exam.

The criteria adopted for the evaluation are: the relevance of the answers to the questions asked, the level of in-depth analysis of the contents presented, the ability to connect with other topics covered by the program and with topics already acquired in previous years' courses, the ability to report examples, language properties and expository clarity.

Verification of learning can also be carried out electronically, should the conditions require it.

### Examples of frequently asked questions and / or exercises

The questions / topics listed below do not constitute an exhaustive list but represent only a few examples

Thermodynamic potentials. Equivalence of ensembles. Phase transitions of 1st and 2nd order. Bose-Einstein condensation. Solutions of the Ising model. Universality and critical exponents