{
  "id": "2412.04015",
  "version": "v1",
  "published": "2024-12-05T09:51:03.000Z",
  "updated": "2024-12-05T09:51:03.000Z",
  "title": "Linear fluctuation of interfaces in Glauber-Kawasaki dynamics",
  "authors": [
    "Tadahisa Funaki",
    "Claudio Landim",
    "Sunder Sethuraman"
  ],
  "categories": [
    "math.PR",
    "cond-mat.stat-mech"
  ],
  "abstract": "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 $\\Gamma_t$ formed is stationary that is, $\\Gamma_t$ is `flat'. When the Glauber rates are balanced on $T^d$, $\\Gamma_t=\\Gamma=\\{x: x_1=0\\}$ is immobile and the hydrodynamic limit is given by $\\rho(t,v) = \\rho_+$ for $v_1\\in (0,1/2)$ and $\\rho(t,v)= \\rho_-$ 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 $\\phi$ of $\\partial^2_{v_1} \\phi - V'(\\phi)=0$ 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 $\\phi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-05T09:51:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear fluctuation",
      "glauber-kawasaki dynamics",
      "glauber rates",
      "hydrodynamic mean curvature interface limit",
      "space-time mass fluctuation field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}