• General introduction to the "rare events" problem. Introduction to the concepts of free energy, rate constant, mean-first-passage time, separation of time scales, committor distribution, etc.


• Computing the free energy in complex systems:


• Thermodynamic integration and umbrella sampling techniques.


• Ferrenberg and Swendsen method for computing the density of states and weighted-histogram analysis.


• History-dependent reconstruction of the free energy: metadynamics and Wand-Landau sampling.


• Techniques for computing the rate constants:


• Classical transition state theory. Kramers theory and Bennett-Chandler method for computing the recrossing corrections.


• Methods for finding the saddle point in complex potential energy surfaces: nudged elastic band, eigenvalue following and the dimer method.


• Path integral formulation of the rare event problem: transition path sampling. Methods for computing the committor distribution (finite-temperature string, etc.).

The successful and proficient attendance of the course will be evaluated by assigning to each student a different problem, that can be solved by numerical simulation implementing one of the approaches discussed in the course. The outcome of the exercise will be discussed in a oral exam.