2024/12/05 by Tadahisa Funaki, Cláudio Landim, Funaki, Tadahisa +3 · 1 citation
Computer Science · Physics and Astronomy · #Chaos control and synchronization #FOS: Mathematics #FOS: Physical sciences #Nonlinear Dynamics and Pattern Formation #Probability (math.PR) #Quantum chaos and dynamical systems #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.2412.04015
openalex publication_date 2024/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we find a scaling limit of the space-time mass fluctuation field of Glauber + Kawasaki particle dynamics around its hydrodynamic mean curvature interface limit. Here, the Glauber rates are scaled by K=KN, the Kawasaki rates by N2 and space by 1/N. We start the process so that the interface Γt formed is stationary that is, Γt is `flat'. When the Glauber rates are balanced on Td, Γt=Γ=\x: x1=0\ is immobile and the hydrodynamic limit is given by ρ(t,v) = ρ+ for v1∈ (0,1/2) and ρ(t,v)= ρ- for v1∈ (-1/2,0) for all t≥ 0, where v=(v1,…,vd)∈ Td identified with [-1/2,1/2)d. Since in the formation the boundary region about the interface has width O(1/√(KN)), we will scale the v1 coordinate in the fluctuation field by √(KN) so that the scaling limit will capture information `near' the interface. We identify the fluctuation limit as a Gaussian field when KN\uparrow ∞ and KN= O(√(log(N))) in d≤ 2. In the one dimensional case, the field limit is given by \bf e(v1) Bt where Bt is a Brownian motion and \bf e is the normalized derivative of a decreasing `standing wave' solution ϕ of ∂2v1 ϕ- V'(ϕ)=0 on R, where V' is the homogenization of the Glauber rates. In two dimensions, the limit is \bf e(v1)Zt(v2) where Zt is the solution of a one dimensional stochastic heat equation. The appearance of the function \bf e(⋅) in the limit field indicates that the interface fluctuation retains the shape of the transition layer ϕ.