Statistical Mechanics: Algorithms and Computations

Week 5 is the first episode of a three-weeks journey through quantum statistical mechanics. We will start by learning about density matrices and path integrals, fascinating tools to study quantum systems. In many cases, the Trotter approximation will be useful to consider non-trivial systems, and also to follow the time evolution of a system. All these topics, including the matrix-squaring technique, will be reviewed in detail in the homework session, where you will also study the anharmonic potential.

<br> Note that previous knowledge of quantum mechanics is not really necessary to go through the next three weeks. Follow us in our journey through algorithms and physics, and don't forget to ask on the forum if you have any doubt!

In Week 6, the second quantum week, we will introduce the properties of bosons, indistinguishable particles with peculiar statistics. At the same time, we will also go further by learning a powerful sampling algorithm, the Lévy construction, and in the homework session you will thoroughly compare it with standard sampling techniques.

At the end of our quantum journey, in Week 7, we discuss the Bose-Einstein condensation phenomenon, theoretically predicted in the 1920's and observed in the 1990's in experiments with ultracold atoms. In the path-integral framework, an elegant description of this phenomenon is in term of permutation cycles, which will also lead to a great sampling algorithm, to be discussed in the homework session.

In Week 8 we come back to classical physics, and in particular to the Ising model, which captures the essential physics of a set of magnetic spins. This is also a fundamental model for the development of sampling algorithms, and we will see different approaches at work: A local algorithm, the very efficient cluster algorithms, the heat-bath algorithm and its connection with coupling. All of these will be revisited in the homework session, where you will get a precise control over the transition between ordered and disordered states.

Continuing with simple models for spins, in Week 9 we start by learning about a dynamic Monte Carlo algorithm which runs faster than the clock. This is easily devised for a single-spin system, and can also be generalized to the full Ising model from Week 8. In the tutorial we move towards the simulated-annealing technique, a physics-inspired optimization method with a very broad applicability. You will also revisit this in the homework session, and apply it to the sphere-packing and traveling-salesman problems.

The lecture of Week 10 includes the alpha and the omega of our course. First we repeat the experiment of Buffon's needle, already performed in the 18th century, and then we touch the sophisticated theory of Lévy stable distributions, and their connection with the central limit theorem. In the tutorial there will be time for a review of the entire course material, and then a little party is due, to celebrate the end of the course!

<br> (There is no homework session for Week 10, but don't forget that the final exam is still there!)