87950 - STATISTICAL MECHANICS

Course Unit Page

Academic Year 2018/2019

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; magnetic systems;

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

[1] Sect. 1.1-1.2, 2.1-2.2

[2] Sect. 2.1-2.2

- The microscopic interpretation of thermodynamics:

i) Microcanoncal ensembles: microcanonical probability density, mean values, entropy and Boltzmann formula, thermodynamic limit; specific heat and susceptibility; 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; diamagnetism.

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

iv) Classical, and quantum statistics, from counting of states.

[1] Sect 2.1-2.3

[2] Sect. 1.1-1.6, 2.3-2.4, 3.1-3.7, 4.1-4.3

- 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.

[1] Sect. 7.1

[2] Sect. 8.2

[3] Sect. 1.1-1.2, 3.1

- 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; Maxwell-Boltzmann distribution for the velocity; A magnetic solid; Grancanonical partition function of a gas; Solid-Vapor equilibrium phase; Negative temperatures.

[4]

2. Foundations of quantum statistical mechanics (about 42 hours)

- Review of quantum mechanics and second quantization formalism

i) states and observables; the evolution operator; Dirac notation;

ii) density matrices; pure and mixed states; the probabilistic interpretation.

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

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

[1] Sect. 4.1, 5.1-5.2

- Quantum statistical mechanics:

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

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

[1] Sect. 5.3

[2] Sect. 5.1-5.2, 5.4-5.5

- Non-relativistic Fermi Gas:

i) dispersion relation and thermodynamical quantities;

ii) classical and semiclassical limit;

iii) T=0 limit and Fermi temperature.

[1] Sect. 5.3.1

[2] Sect. 6.1-6.3, 8.1

- Non-relativistic Bose Gas:

i) equations for the density and critical temperature;

ii) Bose-Einstein condensation.

[1] Sect. 5.3.2

[2] Sect. 7.1-7.2

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

[5]

Readings/Bibliography

  • The main reference books are:

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

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

[3] G. Mussardo, Statistical Field Theory, Oxford

[4] E. Ercolessi, Classical Statistical Mechanics – Exercises; available to download from the university depository

[5] E. Ercolessi, Quantum Statistical Mechanics – Exercises; available to download from the university depository

  • For a more elementary introduction to statistical mechanics, students may look at:

K. Huang, Statistical Mechanics, Wiley

  • For complements about more conceptual problems, students may read:

G. Gallavotti, Statistical Mechanics – Short Treatise (available on the web)

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 groups, under the supervision of the teacher

Assessment methods

Oral exam.

The exam consists of (at least) three questions, two of which about the different topics of Foundation of Classical and Quantum Statistical Mechanics and one about the models/applications studied in the exercise sessions.

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

They will be asked to both present an introduction to the main general topics and to prove more specific results, making connections among the different parts of the syllabus.

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

The “cum laude” honor isgranted to students who demonstrate a personal and 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.

Teaching tools

Exercises; available to download from the university depository

Office hours

See the website of Elisa Ercolessi