Transition path theory

The dynamical behavior of many systems arising in physics, chemistry, biology, etc. is dominated by rare but important transition events between long lived states. For over 70 years, transition state theory (TST) has provided the main theoretical framework for the description of these events [17,33,34]. Yet, while TST and evolutions thereof based on the reactive flux formalism [1, 5] (see also [30,31]) give an accurate estimate of the transition rate of a reaction, at least in principle, the theory tells very little in terms of the mechanism of this reaction. Recent advances, such as transition path sampling (TPS) of Bolhuis, Chandler, Dellago, and Geissler [3, 7] or the action method of Elber [15, 16], may seem to go beyond TST in that respect: these techniques allow indeed to sample the ensemble of reactive trajectories, i.e. the trajectories by which the reaction occurs. And yet, the reactive trajectories may again be rather uninformative about the mechanism of the reaction. This may sound paradoxical at first: what more than actual reactive trajectories could one need to understand a reaction? The problem, however, is that the reactive trajectories by themselves give only a very indirect information about the statistical properties of these trajectories. This is similar to why statistical mechanics is not simply a footnote in books about classical mechanics. What is the probability density that a trajectory be at a given location in state-space conditional on it being reactive? What is the probability current of these reactive trajectories? What is their rate of appearance? These are the questions of interest and they are not easy to answer directly from the ensemble of reactive trajectories. The right framework to tackle these questions also goes beyond standard equilibrium statistical mechanics because of the nontrivial bias that the very definition of the reactive trajectories imply -they must be involved in a reaction. The aim of this chapter is to introduce the reader to the probabilistic framework one can use to characterize the mechanism of a reaction and obtain the probability density, current, rate, etc. of the reactive trajectories.