The motivation of this presentation comes from the analysis of metastable stochastic process in statistical physics.
One way to bridge the scale between full atomistic models and more coarse-grained descriptions is to use Markov State
models parameterized by the Eyring Kramers formulas. These formulas give the hopping rates between local minima of the
potential energy function. They require to identify the local minima and saddle points of the potential energy function.
This approach is for example used in materials science (kinetic Monte Carlo models).
In this talk, I will first present a recent result obtained in collaboration with D. Le Peutrec (Université d'Orléans)
and B. Nectoux (Université Clermont Auvergne) about the mathematical foundations of this approach, by deriving these
Eyring-Kramers exit rates starting from the overdamped Langevin dynamics [1]. I will then introduce a recent algorithm
we proposed together with P. Parpas (Imperial College London) in order to locate saddle points [2]. I will explain why
these two works both rely on concentration properties of the eigenvectors of Witten Laplacians, in the small temperature
regime.
References:
[1] TL, D. Le Peutrec and B. Nectoux, Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with
saddle points on the boundary, https://arxiv.org/abs/2207.09284.\\
[2] TL, P. Parpas, Using Witten Laplacians to locate index-1 saddle points, to appear in SIAM Journal on Scientific
Computing https://arxiv.org/abs/2212.10135.
