Finally, using this spectral decomposition, we can also compute the time evolution of observables, such as the occupation numbers of metastable states. This is shown here for the alanine tetrapeptide molecule, for an initial distribution centered in the least stable state of the molecule.

Each eigenpair gives information about the transition mechanism, shown here for the alanine dipeptide molecule. We even show that good information can be obtained from a simulation biased with a poor collective variable (here the psi angle)

To do so, we only need to compute averages with respect to the Boltzmann distribution, which means that biased simulations can be used by simply reweighting the averages.

In this paper, we model the relaxation from an initial probabiluty distribution towards the Boltzmann one with a Langevin equation. Using this assumption, we learn the associated infinitesimal generator and its spectral decomposition.