Matter, and more specifically molecules, are constantly moving... However, a mathematical model can explain this movement, which is both ordered and chaotic: the Langevin equation.
So let Ω ⊂ R^d be a bounded smooth domain and b : Ω → R^d be a smooth vector field. We focus on the associated overdamped Langevin equation:

dXt = b(Xt)dt + \sqrt{h}dBt

in the low temperature regime h → 0 and in the case where b admits the decomposition b = −∇ f − ℓ with ∇ f · ℓ = 0 on Ω.
To study this equation, we analyse the spectrum of the infinitesimal generator of the dynamics:

L_h = − h/2 ∆ + ∇ f · ∇ + ℓ · ∇

with Neumann boundary conditions.
In this case, moving particles remain trapped inside the domain. More precisely the process remains trapped, for some time, in a certain region of the domain before going to another area. These regions are called metastable and correspond to neighborhoods of minima of f .
Finally, thanks to the spectral theory of L_h , we can describe the return to equilibrium of this metastable dynamics.