31 research outputs found
Non-reversible Metastable Diffusions with Gibbs Invariant Measure II: Markov Chain Convergence
No full textThis article considers a class of metastable non-reversible diffusion
processes whose invariant measure is a Gibbs measure associated with a Morse
potential. In a companion paper [32], we proved the Eyring-Kramers formula for
the corresponding class of metastable diffusion processes. In this article, we
further develop this result by proving that a suitably time-rescaled metastable
diffusion process converges to a Markov chain on the deepest metastable
valleys. This article is also an extension of [45], which considered the same
problem for metastable reversible diffusion processes. Our proof is based on
the recently developed resolvent approach to metastability.Comment: 39 pages, 4 figures (the article is significantly revised at
2022-07-20; the resolvent approach is used to simplify the argument
Analysis of Metastable Behavior via Solutions of Poisson Equations (Stochastic Analysis on Large Scale Interacting Systems)
No full text"Stochastic Analysis on Large Scale Interacting Systems". November 5-8, 2018. edited by Ryoki Fukushima, Tadahisa Funaki, Yukio Nagahata, Hirofumi Osada and Kenkichi Tsunoda. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We herein review the recent progress on the study of metastability based on the analysis of solutions of Poisson equations related to the generators of the underlying metastable dynamics. This review paper is based on the joint work with Claudio Landim [24] and Fraydoun Rezakhanlou [26]
Analysis of Metastable Behavior via Solutions of Poisson Equations (Stochastic Analysis on Large Scale Interacting Systems)
No full textMetastability and time scales for parabolic equations with drift 1: the first time scale
No full textConsider the elliptic operator given by for some smooth vector field and a small parameter . Consider the initial-valued problem for some bounded continuous function . Denote by the set of critical points of which are stable stationary points for the ODE . Under the hypothesis that is finite and , where is a divergence-free field orthogonal to , the main result of this article states that there exist a time-scale , as , and a Markov semigroup defined on such that for all and in the domain of attraction of for the ODE . The time scale is critical in the sense that, for all time scale such that , , for all . Namely, is the first scale at which the solution to the initial-valued problem starts to change. In a companion paper [Landim, Lee, Seo, forthcoming] we extend this result finding all critical time-scales at which the solution evolves smoothly in time and we show that the solution is expressed in terms of the semigroup of some Markov chain taking values in sets formed by unions of critical points of .60 pages, 3 figure
Metastability of Nonreversible Random Walks in a Potential Field and the Eyring-Kramers Transition Rate Formula
No full textInternational audienc
Metastability of Ising and Potts models without external fields in large volumes at low temperatures
No full textIn this article, we investigate the energy landscape and metastable behavior
of the Ising and Potts models on two-dimensional square or hexagonal lattices
in the low temperature regime, especially in the absence of an external
magnetic field. The energy landscape of these models without an external field
is known to have a huge and complex saddle structure between ground states. In
the small volume regime where the lattice is finite and fixed, the
aforementioned complicated saddle structure has been successfully analyzed in
[20] for two or three dimensional square lattices when the inverse temperature
tends to infinity. In this article, we consider the large volume regime where
the size of the lattice grows to infinity. We first establish an asymptotically
sharp threshold such that the ground states are metastable if and only if the
inverse temperature is larger than the threshold in a suitable sense. Then, we
carry out a detailed analysis of the energy landscape and rigorously establish
the Eyring-Kramers formula when the inverse temperature is sufficiently larger
than the previously mentioned sharp threshold. The proof relies on detailed
characterization of dead-ends appearing in the vicinity of optimal transitions
between ground states and on combinatorial estimation of the number of
configurations lying on a certain energy level.Comment: 67 pages, 21 figure
