### Abstract

A methodology is proposed for studying rare events in stochastic partial differential equations in systems that are so large that standard large deviation theory does not apply. The idea is to deduce the behavior of the original model by breaking the system into appropriately scaled subsystems that are sufficiently small for large deviation theory to apply but sufficiently large to be asymptotically independent from one another. The methodology is illustrated in the context of a simple one-dimensional stochastic partial differential equation. The application reveals a connection between the dynamics of the partial differential equation and the classical Johnson-Mehl-Avrami- Kolmogorov nucleation and growth model. It also illustrates that rare events are much more likely and predictable in large systems than in small ones due to the extra entropy provided by space.

Original language | English (US) |
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Pages (from-to) | 1023-1038 |

Number of pages | 16 |

Journal | Journal of Statistical Physics |

Volume | 131 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2008 |

### Fingerprint

### Keywords

- Large deviation theory
- Metastability
- Nucleation
- Phase transformation
- Rare events
- Small noise
- Spatially extended system
- Stochastic partial differential equation

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*131*(6), 1023-1038. https://doi.org/10.1007/s10955-008-9537-8

**Rare events in stochastic partial differential equations on large spatial domains.** / Vanden Eijnden, Eric; Westdickenberg, Maria G.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 131, no. 6, pp. 1023-1038. https://doi.org/10.1007/s10955-008-9537-8

}

TY - JOUR

T1 - Rare events in stochastic partial differential equations on large spatial domains

AU - Vanden Eijnden, Eric

AU - Westdickenberg, Maria G.

PY - 2008/6

Y1 - 2008/6

N2 - A methodology is proposed for studying rare events in stochastic partial differential equations in systems that are so large that standard large deviation theory does not apply. The idea is to deduce the behavior of the original model by breaking the system into appropriately scaled subsystems that are sufficiently small for large deviation theory to apply but sufficiently large to be asymptotically independent from one another. The methodology is illustrated in the context of a simple one-dimensional stochastic partial differential equation. The application reveals a connection between the dynamics of the partial differential equation and the classical Johnson-Mehl-Avrami- Kolmogorov nucleation and growth model. It also illustrates that rare events are much more likely and predictable in large systems than in small ones due to the extra entropy provided by space.

AB - A methodology is proposed for studying rare events in stochastic partial differential equations in systems that are so large that standard large deviation theory does not apply. The idea is to deduce the behavior of the original model by breaking the system into appropriately scaled subsystems that are sufficiently small for large deviation theory to apply but sufficiently large to be asymptotically independent from one another. The methodology is illustrated in the context of a simple one-dimensional stochastic partial differential equation. The application reveals a connection between the dynamics of the partial differential equation and the classical Johnson-Mehl-Avrami- Kolmogorov nucleation and growth model. It also illustrates that rare events are much more likely and predictable in large systems than in small ones due to the extra entropy provided by space.

KW - Large deviation theory

KW - Metastability

KW - Nucleation

KW - Phase transformation

KW - Rare events

KW - Small noise

KW - Spatially extended system

KW - Stochastic partial differential equation

UR - http://www.scopus.com/inward/record.url?scp=44349189777&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=44349189777&partnerID=8YFLogxK

U2 - 10.1007/s10955-008-9537-8

DO - 10.1007/s10955-008-9537-8

M3 - Article

AN - SCOPUS:44349189777

VL - 131

SP - 1023

EP - 1038

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 6

ER -