Mean Curvature Interface Limit from Glauber+Zero-Range Interacting Particles

• DOI: 10.1007/s00220-022-04424-8
• Publication Date: 2022-06-28T11:05:55Z
• Authors: Perla El Kettani; Tadahisa Funaki; Danielle Hilhorst; Hyunjoon Park; Sunder Sethuraman
• Keywords: [MATH.MATH-PR] Mathematics [math]/Probability [math.PR]; Mathematics - Analysis of PDEs; Probability (math.PR); FOS: Mathematics; 60K35, 82C22, 35K57, 35B40; [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]; 0101 mathematics; 01 natural sciences; Mathematics - Probability; Analysis of PDEs (math.AP)

We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of a class of Glauber+Zero-range particle systems. The Zero-range part moves particles while preserving particle numbers, and the Glauber part governs the creation and annihilation of particles and is set to favor two levels of particle density. When the two parts are simultaneously seen in certain different time-scales, the Zero-range part being diffusively scaled while the Glauber part is speeded up at a lesser rate, a mean-curvature interface flow emerges, with a homogenized `surface tension-mobility' parameter reflecting microscopic rates, between the two levels of particle density. We use relative entropy methods, along with a suitable `Boltzmann-Gibbs' principle, to show that the random microscopic system may be approximated by a `discretized' Allen-Cahn PDE with nonlinear diffusion. In turn, we show the behavior, especially generation and propagation of interface properties, of this `discretized' PDE.<br/>48 pages. Divided previous version into three parts which extend/revise results: arXiv:2112.13081 on continuous PDE estimates; arXiv:2112.13973 on discrete PDE Schauder estimates; and this one on the interacting particle system limit<br/>