An Improved Adaptive Minimum Action Method for the Calculation of Transition Path in Non-Gradient Systems

摘要

The minimum action method (MAM) is to calculate the most probable transitionpath in randomly perturbed stochastic dynamics, based on the idea of actionminimization in the path space. The accuracy of the numerical path betweendifferent metastable states usually suffers from the "clustering problem" nearfixed points. The adaptive minimum action method (aMAM) solves this problem byrelocating image points equally along arc-length with the help of moving meshstrategy. However, when the time interval is large, the images on the path maystill be locally trapped around the transition state in a tangle, due to thesingularity of the relationship between arc-length and time at the transitionstate. Additionally, in most non-gradient dynamics, the tangent direction ofthe path is not continuous at the transition state so that a geometric cornerforms, which brings extra challenges for the aMAM. In this note, we improve theaMAM by proposing a better monitor function that does not contain the numericalapproximation of derivatives, and taking use of a generalized scheme of theEuler-Lagrange equation to solve the minimization problem, so that both thepath-tangling problem and the non-smoothness in parametrizing the curve do notexist. To further improve the accuracy, we apply the Weighted Essentiallynon-oscillatory (WENO) method for the interpolation to achieve betterperformance. Numerical examples are presented to demonstrate the advantages ofour new method.