Glauber dynamics of 2D Kac–Blume–Capel model and their stochastic PDE limits

Hao Shen; Hendrik Weber

doi:10.1016/j.jfa.2017.12.014

Glauber dynamics of 2D Kac–Blume–Capel model and their stochastic PDE limits

Hao Shen, Hendrik Weber

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

10 Citations (SciVal)

Abstract

We study the Glauber dynamics of a two dimensional Blume–Capel model (or dilute Ising model) with Kac potential parametrized by (β,θ) – the “inverse temperature” and the “chemical potential”. We prove that the locally averaged spin field rescales to the solution of the dynamical Φ⁴ equation near a curve in the (β,θ) plane and to the solution of the dynamical Φ⁶ equation near one point on this curve. Our proof relies on a discrete implementation of Da Prato–Debussche method [13] as in [33] but an additional coupling argument is needed to show convergence of the linearized dynamics.

Original language	English
Pages (from-to)	1321-1367
Journal	Journal of Functional Analysis
Volume	275
Issue number	6
Early online date	20 Jun 2018
DOIs	https://doi.org/10.1016/j.jfa.2017.12.014
Publication status	Published - 15 Sept 2018

Funding

We would like to thank Weijun Xu for many helpful discussions on phase coexistence models and the dynamical Φ 4 equations. H.S. was partially supported by the NSF through DMS-1712684 . Appendix A

Keywords

Blume–Capel
Glauber dynamics
Stochastic PDE

ASJC Scopus subject areas

Analysis

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10.1016/j.jfa.2017.12.014

http://wrap.warwick.ac.uk/96038/

Cite this

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title = "Glauber dynamics of 2D Kac–Blume–Capel model and their stochastic PDE limits",

abstract = "We study the Glauber dynamics of a two dimensional Blume–Capel model (or dilute Ising model) with Kac potential parametrized by (β,θ) – the “inverse temperature” and the “chemical potential”. We prove that the locally averaged spin field rescales to the solution of the dynamical Φ4 equation near a curve in the (β,θ) plane and to the solution of the dynamical Φ6 equation near one point on this curve. Our proof relies on a discrete implementation of Da Prato–Debussche method [13] as in [33] but an additional coupling argument is needed to show convergence of the linearized dynamics.",

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AU - Shen, Hao

AU - Weber, Hendrik

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N2 - We study the Glauber dynamics of a two dimensional Blume–Capel model (or dilute Ising model) with Kac potential parametrized by (β,θ) – the “inverse temperature” and the “chemical potential”. We prove that the locally averaged spin field rescales to the solution of the dynamical Φ4 equation near a curve in the (β,θ) plane and to the solution of the dynamical Φ6 equation near one point on this curve. Our proof relies on a discrete implementation of Da Prato–Debussche method [13] as in [33] but an additional coupling argument is needed to show convergence of the linearized dynamics.

AB - We study the Glauber dynamics of a two dimensional Blume–Capel model (or dilute Ising model) with Kac potential parametrized by (β,θ) – the “inverse temperature” and the “chemical potential”. We prove that the locally averaged spin field rescales to the solution of the dynamical Φ4 equation near a curve in the (β,θ) plane and to the solution of the dynamical Φ6 equation near one point on this curve. Our proof relies on a discrete implementation of Da Prato–Debussche method [13] as in [33] but an additional coupling argument is needed to show convergence of the linearized dynamics.

KW - Blume–Capel

KW - Glauber dynamics

KW - Stochastic PDE

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SP - 1321

EP - 1367

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 6

ER -

Glauber dynamics of 2D Kac–Blume–Capel model and their stochastic PDE limits

Abstract

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Keywords

ASJC Scopus subject areas

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