2006-2007, Term 1 (Fall)

MA4G3 - Introduction to Statistical Mechanics

Here is the official site from the department.

Time: Mo 10-11, We 12-1, Fr 12-1. Place: MS.05. Assessment: 3-hour examination.

Lecture notes:

Chapter 1 - Thermostatics 1.1, 1.2, 1.3, 1.4

Chapter 2 - Entropy of the classical gas
2.1, 2.2, 2.4, 2.6, 2.7

Chapter 3 - Equivalence of ensembles
3.1, 3.2, 3.3

Chapter 4 - Cluster expansions

Chapter 5 - The Ising model
5.1, 5.2, 5.4, 5.5, 5.6

Chapter 6 - Shannon entropy
6.1, 6.2, 6.3, 6.4

Index (until Chapter 6)

Exam with solution

Short description:

Statistical mechanics describes physical systems with a huge number of particles. The microscopic description of a system involves the position and momentum of each individual particle. The macroscopic description of the system is given by Thermodynamics, and involves only a few parameters such as temperature, pressure, etc... Statistical Mechanics provides the theoretical setting that allows to derive thermodynamic laws starting from the microscopic description.

This lecture will start with a crash course of Thermodynamics, explaining in particular the notion of entropy. A formula due to Boltzmann allows to define the entropy in terms of microscopic variables - this is the gate from the micro- to the macroscopic world. We will discuss alternate formulae for the ``free energy'' or for the pressure (in local jargon, this is called ``equivalence of ensembles''). We will define equilibrium states and discuss phase transitions. These notions will be illustrated in systems of continuous classical particles. Quantum systems will be discussed if time allows.

Recently, much progress has been achieved on certain statistical physics models such as Ising, Potts, etc... Many interesting aspects will be taught in MA4G0 Mathematics of Phase Transitions in Term 2. The present course will help understand the context.

Knowledge of Year 2 Mathematics, and Year 1 Physics, are enough. More advanced mathematical notions will be explained when needed. But several proofs will be challenging and at the Year 4 level.


Notes are provided. Relevant books include:
David Ruelle, Statistical Mechanics: Rigorous Results, World Scientific, 1999.
Teunis C. Dorlas, Statistical Mechanics, IOP, 1999.
Kerson Huang, Statistical Mechanics, John Wiley, 1987.
H. B. Callen, Thermodynamics, John Wiley, 1960.
David MacKay, Information Theory, Inference, and Learning Algorithms, Cambridge, 2003.