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The geometric minimum action method: A least action principle on the space of curves
Author(s) -
Heymann Matthias,
VandenEijnden Eric
Publication year - 2008
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20238
Subject(s) - mathematics , action (physics) , ordinary differential equation , large deviations theory , principle of least action , context (archaeology) , partial differential equation , stochastic differential equation , statistical physics , differential equation , mathematical analysis , classical mechanics , paleontology , statistics , physics , quantum mechanics , biology
Freidlin‐Wentzell theory of large deviations for the description of the effect of small random perturbations on dynamical systems is exploited as a numerical tool. Specifically, a numerical algorithm is proposed to compute the quasi‐potential in the theory, which is the key object to quantify the dynamics on long time scales when the effect of the noise becomes ubiquitous: the equilibrium distribution of the system, the pathways of transition between metastable states and their rate, etc., can all be expressed in terms of the quasi‐potential. We propose an algorithm to compute these quantities called the geometric minimum action method (gMAM), which is a blend of the original minimum action method (MAM) and the string method. It is based on a reformulation of the large deviations action functional on the space of curves that allows one to easily perform the double minimization of the original action required to compute the quasi‐potential. The theoretical background of the gMAM in the context of large deviations theory is discussed in detail, as well as the algorithmic aspects of the method. The gMAM is then illustrated on several examples: a finite‐dimensional system displaying bistability and modeled by a nongradient stochastic ordinary differential equation, an infinite‐dimensional analogue of this system modeled by a stochastic partial differential equation, and an example of a bistable genetic switch modeled by a Markov jump process. © 2007 Wiley Periodicals, Inc.