87950 - Statistical Mechanics

Learning outcomes

At the end of the course the student will know the statistical laws that rule the thermodynamic behaviour of macroscopic systems with a large number of particles, both in the classical and in the quantum setting, as well as the basics of phase transitions theory. The student will be able to introduce and solve statistical models to describe the physics of classical and quantum gases and of magnetic systems, by also discussing the phase diagram in some simple cases.

Course contents

1. Foundations of classical statistical mechanics (about 20 hours)

- A brief review of:

i) Thermodynamics: thermodynamical variables, the laws of thermodynamics, thermodynamical potentials, variational principles.

ii) Hamiltonian mechanics; states and observables, Hamilton equations, conservation of energy, Liouville theorem, ensembles and probabilities.

- The microscopic interpretation of thermodynamics:

i) Microcanoncal ensembles: microcanonical probability density, mean values, entropy and Boltzmann formula, thermodynamic limit; specific heat; the ideal gas in 3D;

ii) Canonical ensemble: canonical probability density, partition function, mean values, entropy and other thermodynamical potentials, (in)distinguishability; equipartition theorem; the ideal gas in 3D.

iii) Macrocanonical ensemble: macrocanonical probability density, granpartition function, mean values, entropy and other thermodynamical potentials; the virial expansion and the Van der Waals gas.

iv) Classical and quantum statistics from counting of states.

- Introduction to phase transitions:

i) The example of a classical fluid; Phase diagram and classification of phase transitions.

ii) A model system: the Ising model, formulation and mean field solution.

iii) Order parameter and correlations, scaling hypothesis and critical exponents.

- Exercises on Classical Statistical Mechanics will cover some important models and applications, such as: Density of states of a gas; The ideal non-relativistic gas; Gas of harmonic oscillators; The ideal ultra-relativistic gas; A magnetic solid; Negative temperatures.

2. Foundations of quantum statistical mechanics (about 32 hours)

- Review of quantum mechanics and second quantization formalism

i) states and observables; the evolution operator; Dirac notation; density matrices; pure and mixed states.

ii) indistinguishable particles, the permutation group and its action on wave functions; symmetric/antisymmetric functions, bosonic and fermionic particles.

iii) Single particle basis and creation/annihilation operators; Fock space; representation of single particle operators and two-particle potentials.

- Quantum statistical mechanics:

i) density operator in the microcanonical, canonical and grandcanonical ensembles; partition function and thermodynamical potentials, mean values.

ii) bosonic and fermionic quantum gases: grandcanonical partition function, distributions of Bose-Einstein and Fermi-Dirac, mean particle number and mean energy in the discrete and in the thermodynamic limit.

- Non-relativistic Fermi Gas:

i) dispersion relation and thermodynamical quantities;

ii) classical and semiclassical limit;

iii) T=0 limit and Fermi temperature.

- Non-relativistic Bose Gas:

i) equations for the density and critical temperature;

ii) Bose-Einstein condensation.

- Exercises on Quantum Statistical Mechanics will cover some important models and applications, such as: Quantum gases of dipoles and harmonic oscillators; Ultra-relativistic Fermi gas; Bose gas in 2D; Ultra-relativistic Bose gas and Planck distribution.

Readings/Bibliography

The main reference books are:

G. Morandi, F. Napoli, E. Ercolessi, Statistical Mechanics, World Scientific

R.K. Pathria, Statistical Mechanics, Butterworth (2^ndedition)

G. Mussardo, Statistical Field Theory, Oxford

Notes on the exercise sessions will be available in the Virtuale platform.

For a more elementary introduction to statistical mechanics, students may look at: K. Huang, Statistical Mechanics, Wiley

Teaching methods

The course consists of about:

- 40 hours of class lectures, given by the teacher at the blackboard and

- 12 hours of exercises, which will be solved in class by the students individually or in small groups of peers, under the supervision of the teacher.

Assessment methods

Written exam (3 hours).

The exam consists of four questions about different topics of the Foundation of Classical and Quantum Statistical Mechanics and about the models/applications studied in the exercise sessions.

Students should demonstrate to be familiar with and have a good understanding of the different subjects.

They might be asked to present an introduction to the main general topics, to prove more specific results, to make connections among the different parts of the syllabus, and to solve exercises.

The organization of the presentation and a rigorous scientific language will be also considered for the formulation of the final grade

The evaluation will be done according to the following scheme:

Grade 18-19: basic knowledge and ability to analyze only a very limited number of topics covered in the course; overall correct language.

Grade 20-25: discrete knowledge and ability to analyze only a limited number of topics covered in the course; overall correct language.

Grade: 26-28: good knowledge and ability to analyze a large number of topics covered in the course; mastery of scientific language and correct use of specific terminology.

Grade: 29-30: comprehensive preparation on the topics covered in the course, showing a very good/excellent knowledge and ability analysis; mastery of scientific language and correct use of specific terminology.

The “cum laude” honor is granted to students who demonstrate the ability to organize comparative analyses and personal/critical rethinking of the subject.

According to the general rules of the University, students will be allowed to reject the grade only once, but they can withdraw at any time during the exam.

Students with Specific Learning Disabilities (SLD) or temporary/permanent disabilities are advised to contact the University Office responsible in a timely manner (https://site.unibo.it/studenti-con-disabilita-e-dsa/en ). The office will be responsible for proposing any necessary accommodations to the students concerned. These accommodations must be submitted to the instructor for approval at least 15 days in advance, and will be evaluated in light of the learning objectives of the course.

Teaching tools

Teaching materials (slides and notes on specific topics about theory and exercises) will be available in Virtuale.

Office hours

See the website of Elisa Ercolessi