Minimum action method for the study of rare events

Weinan E, Weiqing Ren, Eric Vanden-Eijnden

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187 Scopus citations


The least-action principle from the Wentzell-Freidlin theory of large deviations is exploited as a numerical tool for finding the optimal dynamical paths in spatially extended systems driven by a small noise. The action is discretized and a preconditioned BFGS method is used to optimize the discrete action. Applications are presented for thermally activated reversal in the Ginzburg-Landau model in one and two dimensions, and for noise-induced excursion events in the Brusselator taken as an example of a nongradient system arising in chemistry. In the Ginzburg-Landau model, the reversal proceeds via interesting nucleation events, followed by propagation of domain walls. The issue of nucleation versus propagation is discussed, and the scaling for the number of nucleation events as a function of the reversal time and other material parameters is computed. Good agreement is found with the numerical results. In the Brusselator, whose deterministic dynamics has a single stable equilibrium state, the presence of noise is shown to induce large excursions by which the system cycles out of this equilibrium state.

Original languageEnglish (US)
Pages (from-to)637-656
Number of pages20
JournalCommunications on Pure and Applied Mathematics
Issue number5
StatePublished - May 2004

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


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