First glances at the principles of Statistical Mechanics: what are Gibbs measures? Lesson 4

The theory of thermodynamics aims at understanding how two types of energy – mechanical and thermal – can be converted into each other. At the end of the XIXth century, the Austrian physicist Ludwig Boltzmann states that the laws of thermodynamics should be derived from Newton's mechanical first principles, on the basis of the atomistic theory of matter. A gas, for instance, can be represented as a collection of atoms – or point particles – moving under Newton's laws.
Mechanical systems are mainly characterized by geometric quantities, such as the positions and the velocities of its mass points. However, this description completely fails for gases, which hold properties that may interfere with their mechanical characteristics, in particular their temperature. The kinetic theory of gases has turned into what we know as statistical mechanics through the work of the American mathematical physicist J. Willard Gibbs, in the early XXth century.
In this context, finding the equilibrium value of macroscopic variables (like temperature) amounts to computing a probability distribution. Such a probability law depends on a finite number of parameters, and describes the possible states of the system composed by a huge amount of particles.
This introductory course will take mathematical looks at the first principles of statistical mechanics, and will aim at:

• defining the notion of a thermodynamical system of particles;
• understanding the natural probability measures that characterize such systems, namely the microcanonical ensemble and the Gibbs measures;
• proving (at least in simpler cases) the Theorem of Equivalence of Ensembles, which justifies the use of the Gibbs measures to compute thermodynamics quantities;
• asking some mathematical questions (some of them are still unsolved) that naturally arise in this area (like, for instance, the hard problem of proving ergodicity).

Prerequisities are: some notions of calculus (integrals, limits) and, if possible (but not absolutely necessary), some notions of probability.