2006
DOI: 10.1002/cpa.20144
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Action minimization and sharp‐interface limits for the stochastic Allen‐Cahn equation

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 ca… Show more

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Cited by 74 publications
(110 citation statements)
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“…For problems with the complexity of micromagnetics or martensitic phase transformation this question remains open. However for the simpler case of a scalar Ginzburg-Landau model there has been some progress [42], [43].…”
Section: Ginzburg-landaumentioning
confidence: 99%
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“…For problems with the complexity of micromagnetics or martensitic phase transformation this question remains open. However for the simpler case of a scalar Ginzburg-Landau model there has been some progress [42], [43].…”
Section: Ginzburg-landaumentioning
confidence: 99%
“…In two space dimensions the problem has been studied numerically in [24] and analytically in [42], but a complete analysis is still lacking. The anticipated answer is similar to the one-dimensional case, except that (i) if the seeds are points rather than lines then their nucleation cost is negligible; and (ii) if a boundary moves via motion by mean curvature then its propagation cost is negligible.…”
Section: Ginzburg-landaumentioning
confidence: 99%
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“…• The upper bound necessary for the Gamma-convergence was formally proved [17] by the construction of good 'recovery sequences'.…”
Section: Introductionmentioning
confidence: 99%
“…• The lower bound was proved in [17] for sequences (u ε ) ε>0 such that the associated 'energy measures' have equipartitioned energy and single multiplicity as ε → 0.…”
Section: Introductionmentioning
confidence: 99%