Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation
We study the action minimization problem that is formally associated to phase transformation in the stochastically perturbed Allen‐Cahn equation. The sharp‐interface limit is related to (but different from) the sharp‐interface limits of the related energy functional and deterministic gradient flows....
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| Veröffentlicht in: | Communications on pure and applied mathematics 2007-03, Vol.60 (3), p.393-438 |
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| container_title | Communications on pure and applied mathematics |
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| creator | Kohn, Robert V. Otto, Felix Reznikoff, Maria G. Vanden-Eijnden, Eric |
| description | We study the action minimization problem that is formally associated to phase transformation in the stochastically perturbed Allen‐Cahn equation. The sharp‐interface 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. © 2006 Wiley Periodicals, Inc. |
| doi_str_mv | 10.1002/cpa.20144 |
| format | Article |
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The sharp‐interface 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. © 2006 Wiley Periodicals, Inc.</description><identifier>ISSN: 0010-3640</identifier><identifier>EISSN: 1097-0312</identifier><identifier>DOI: 10.1002/cpa.20144</identifier><identifier>CODEN: CPAMAT</identifier><language>eng</language><publisher>Hoboken: Wiley Subscription Services, Inc., A Wiley Company</publisher><subject>Calculus of variations and optimal control ; Exact sciences and technology ; Limit theorems ; Mathematical analysis ; Mathematics ; Partial differential equations ; Probability and statistics ; Probability theory and stochastic processes ; Sciences and techniques of general use</subject><ispartof>Communications on pure and applied mathematics, 2007-03, Vol.60 (3), p.393-438</ispartof><rights>Copyright © 2006 Wiley Periodicals, Inc.</rights><rights>2007 INIST-CNRS</rights><rights>Copyright John Wiley and Sons, Limited Mar 2007</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c4304-48e01fc098db8f824a79a27a5647d1a6a337b582f06521e2af234d726fd901e83</citedby><cites>FETCH-LOGICAL-c4304-48e01fc098db8f824a79a27a5647d1a6a337b582f06521e2af234d726fd901e83</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://onlinelibrary.wiley.com/doi/pdf/10.1002%2Fcpa.20144$$EPDF$$P50$$Gwiley$$H</linktopdf><linktohtml>$$Uhttps://onlinelibrary.wiley.com/doi/full/10.1002%2Fcpa.20144$$EHTML$$P50$$Gwiley$$H</linktohtml><link.rule.ids>314,776,780,1411,27901,27902,45550,45551</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=18446274$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Kohn, Robert V.</creatorcontrib><creatorcontrib>Otto, Felix</creatorcontrib><creatorcontrib>Reznikoff, Maria G.</creatorcontrib><creatorcontrib>Vanden-Eijnden, Eric</creatorcontrib><title>Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation</title><title>Communications on pure and applied mathematics</title><addtitle>Comm. Pure Appl. Math</addtitle><description>We study the action minimization problem that is formally associated to phase transformation in the stochastically perturbed Allen‐Cahn equation. The sharp‐interface 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. 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| subjects | Calculus of variations and optimal control Exact sciences and technology Limit theorems Mathematical analysis Mathematics Partial differential equations Probability and statistics Probability theory and stochastic processes Sciences and techniques of general use |
| title | Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation |
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