We analyze the low temperature asymptotics of the quasistationary distribution
associated with the overdamped Langevin dynamics (also known as the
Einstein–Smoluchowski diffusion equation) in a bounded domain. This analysis is
useful to rigorously prove the consistency of an algorithm used in molecular dynamics
(the hyperdynamics) in the small temperature regime. More precisely, we show that
the algorithm is exact in terms of state-to-state dynamics up to exponentially small
factors in the limit of small temperature. The proof is based on the asymptotic
spectral analysis of associated Dirichlet and Neumann realizations of Witten
Laplacians. In order to widen the range of applicability, the usual assumption
that the energy landscape is a Morse function has been relaxed as much as
possible.
Keywords
quasistationary distributions, Witten Laplacian, low
temperature asymptotics and semiclassical asymptotics