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

2016/08/23 by Shen, Hao, Weber, Hendrik
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1608.06556

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 as in a result by Mourrat-Weber but an additional coupling argument is needed to show convergence of the linearized dynamics.

Related

• Convergence of Glauber dynamic on Ising-like models with Kac interaction to Φ²ⁿ₂
• Convergence of the two-dimensional dynamic Ising-Kac model to Φ⁴₂
• The dynamical Ising-Kac model in 3D converges to Φ⁴₃
• Local persistense and blocking in the two dimensional Blume-Capel Model
• Surfaces of Constant Temperature for Glauber Dynamics