Linear fluctuation of interfaces in Glauber-Kawasaki dynamics
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915049425076224 |
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| author | Funaki, Tadahisa Landim, Claudio Sethuraman, Sunder |
| author_facet | Funaki, Tadahisa Landim, Claudio Sethuraman, Sunder |
| contents | 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 $K=K_N$, the Kawasaki rates by $N^2$ and space by $1/N$. We start the process so that the interface $Γ_t$ formed is stationary that is, $Γ_t$ is `flat'. When the Glauber rates are balanced on $T^d$, $Γ_t=Γ=\{x: x_1=0\}$ is immobile and the hydrodynamic limit is given by $ρ(t,v) = ρ_+$ for $v_1\in (0,1/2)$ and $ρ(t,v)= ρ_-$ for $v_1\in (-1/2,0)$ for all $t\ge 0$, where $v=(v_1,\ldots,v_d)\in T^d$ identified with $[-1/2,1/2)^d$. Since in the formation the boundary region about the interface has width $O(1/\sqrt{K_N})$, we will scale the $v_1$ coordinate in the fluctuation field by $\sqrt{K_N}$ so that the scaling limit will capture information `near' the interface.
We identify the fluctuation limit as a Gaussian field when $K_N\uparrow \infty$ and $K_N= O(\sqrt{\log(N)})$ in $d\leq 2$. In the one dimensional case, the field limit is given by ${\bf e}(v_1) B_t$ where $B_t$ is a Brownian motion and ${\bf e}$ is the normalized derivative of a decreasing `standing wave' solution $ϕ$ of $\partial^2_{v_1} ϕ- V'(ϕ)=0$ on $R$, where $V'$ is the homogenization of the Glauber rates. In two dimensions, the limit is ${\bf e}(v_1)Z_t(v_2)$ where $Z_t$ is the solution of a one dimensional stochastic heat equation. The appearance of the function ${\bf e}(\cdot)$ in the limit field indicates that the interface fluctuation retains the shape of the transition layer $ϕ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04015 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear fluctuation of interfaces in Glauber-Kawasaki dynamics Funaki, Tadahisa Landim, Claudio Sethuraman, Sunder Probability Statistical Mechanics 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 $K=K_N$, the Kawasaki rates by $N^2$ and space by $1/N$. We start the process so that the interface $Γ_t$ formed is stationary that is, $Γ_t$ is `flat'. When the Glauber rates are balanced on $T^d$, $Γ_t=Γ=\{x: x_1=0\}$ is immobile and the hydrodynamic limit is given by $ρ(t,v) = ρ_+$ for $v_1\in (0,1/2)$ and $ρ(t,v)= ρ_-$ for $v_1\in (-1/2,0)$ for all $t\ge 0$, where $v=(v_1,\ldots,v_d)\in T^d$ identified with $[-1/2,1/2)^d$. Since in the formation the boundary region about the interface has width $O(1/\sqrt{K_N})$, we will scale the $v_1$ coordinate in the fluctuation field by $\sqrt{K_N}$ so that the scaling limit will capture information `near' the interface. We identify the fluctuation limit as a Gaussian field when $K_N\uparrow \infty$ and $K_N= O(\sqrt{\log(N)})$ in $d\leq 2$. In the one dimensional case, the field limit is given by ${\bf e}(v_1) B_t$ where $B_t$ is a Brownian motion and ${\bf e}$ is the normalized derivative of a decreasing `standing wave' solution $ϕ$ of $\partial^2_{v_1} ϕ- V'(ϕ)=0$ on $R$, where $V'$ is the homogenization of the Glauber rates. In two dimensions, the limit is ${\bf e}(v_1)Z_t(v_2)$ where $Z_t$ is the solution of a one dimensional stochastic heat equation. The appearance of the function ${\bf e}(\cdot)$ in the limit field indicates that the interface fluctuation retains the shape of the transition layer $ϕ$. |
| title | Linear fluctuation of interfaces in Glauber-Kawasaki dynamics |
| topic | Probability Statistical Mechanics |
| url | https://arxiv.org/abs/2412.04015 |