Metastable transitions in inertial Langevin systems: What can be different from the overdamped case?

Abstract

Metastable transitions in Langevin dynamics can exhibit rich behaviors that are markedly different from its overdamped limit. In addition to local alterations of the transition path geometry, more fundamental global changes may exist. For instance, when the dissipation is weak, heteroclinic connections that exist in the overdamped limit do not necessarily have a counterpart in the Langevin system, potentially leading to different transition rates. Furthermore, when the friction coefficient depends on the velocity, the overdamped limit no longer exists, but it is still possible to efficiently find instantons. The approach we employed for these discoveries was based on (i) a simple rewriting of the Freidlin-Wentzell action in terms of time-reversed dynamics, and (ii) an adaptation of the string method, which was originally designed for gradient systems, to this specific non-gradient system.

Parametric resonance for enhancing the rate of metastable transition

“…(2),ε h (t; t 0 ) is moving along the perturbed downhill heteroclinic orbit as we will show in subsection 2.3 (similar result originated from autonomous kinetic Langevin has been verified in [48]). From a physical point of view, the reason for such a result is that x (2),ε h (t; t 0 ) is a relaxation trajectory, i.e.…”

“…h (t − t 0 ), respectively. Different from the autonomous case described above (where the unperturbed action is invariant to the specific choice of t 0 , e.g., [50,48]), different t 0 gives different local minimum as the concatenation of x (1),ε h (t; t 0 ) and x…”

“…The uphill (resp. downhill) heteroclinic orbit will be assumed to exist in this paper (the nonexistence is considered in [48] and not our focus).…”

“…In the absence of a non-autonomous forcing (ε = 0), hopping between metastable states of kinetic Langevin equation has been studied in detail, including the characterization of MLPs and the corresponding action value. For example, Souza and Tao [47] analysed minimisers of the Freidlin-Wentzell action for the kinetic Langevin equation with respect to various types of friction coefficients for illustrating features in kinetic Langevin metastable transitions that markedly differ from the familiar overdamped picture. The idea studied in this paper is to use specific periodic perturbations (ε = 0) to facilitate metastable transitions.…”

“…An inspiring article [8] for instance extends Melnikov's method to give a condition under which the stable/unstable manifolds intersect transversely in multidimensional setting. In essence, our approach is related to Melnikov's method, but at the same time a generalization as we worked out the 1st-order perturbative expansion in higher-dimensions; (ii) Unlike in the autonomous kinetic Langevin case considered in [47], one can no longer relate the perturbed MLPs to a 2nd-order deterministic equation due to the loss of delicate statistical mechanical structure (which is, roughly speaking, a transverse decomposition [5] and consequent detailed balance after momentum reversible); instead, 4nd-order Euler-Lagrange equation is necessary. To analyze it, we adopt a perturbation analysis for approximating the rate change along perturbed MLPs directly, which is independent of specific form of perturbed MLPs.…”