Action minimization and sharp-interface limits for the stochastic allen-cahn equation

Robert Kohn, Felix Otto, Maria G. Reznikoff, Eric Vanden Eijnden

Research output: Contribution to journalArticle

Abstract

We study the action minimization problem that is formally associated to phase transformation in the stochastically perturbed Allen-Cahn equation. The sharpinterface limit is related to (but different from) the sharp-interface limits of the related energy functional and deterministic gradient flows. In the sharp-interface limit of the action minimization problem, we find distinct "most likely switching pathways," depending on the relative costs of nucleation and propagation of interfaces. This competition is captured by the limit of the action functional, which we derive formally and propose as the natural candidate for the Γ-limit. Guided by the reduced functional, we prove upper and lower bounds for the minimal action that agree on the level of scaling.

Original languageEnglish (US)
Pages (from-to)393-438
Number of pages46
JournalCommunications on Pure and Applied Mathematics
Volume60
Issue number3
DOIs
StatePublished - Mar 2007

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Allen-Cahn Equation
Stochastic Equations
Nucleation
Phase transitions
Minimization Problem
Gradient Flow
Phase Transformation
Costs
Energy Functional
Upper and Lower Bounds
Pathway
Likely
Scaling
Propagation
Distinct

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Action minimization and sharp-interface limits for the stochastic allen-cahn equation. / Kohn, Robert; Otto, Felix; Reznikoff, Maria G.; Vanden Eijnden, Eric.

In: Communications on Pure and Applied Mathematics, Vol. 60, No. 3, 03.2007, p. 393-438.

Research output: Contribution to journalArticle

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