2021/08/07 by Anya Katsevich, Katsevich, Anya
Materials Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Material Dynamics and Properties #Probability (math.PR) #Spectroscopy and Quantum Chemical Studies #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2108.03527
openalex publication_date 2021/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We derive the PDE governing the hydrodynamic limit of a Metropolis rate\ncrystal surface height process in the "rough scaling" regime introduced by\nMarzuola and Weare. The PDE takes the form of a continuity equation, and the\nexpression for the current involves a numerically computed multiplicative\ncorrection term similar to a mobility. The correction accounts for the fact\nthat, unusually, the local equilibrium distribution of the process is not a\nlocal Gibbs measure even though the global equilibrium distribution is Gibbs.\nWe give definitive numerical evidence of this fact, originally suggested in\nGao, et. al., Pure and Applied Analysis (2021). In that paper, an approximate\nPDE -- our PDE, but without the correction term -- was derived for the limit of\nthe Metropolis rate process under the assumption of a local Gibbs distribution.\nOur main contribution is to present a numerical method to compute the corrected\nmacroscopic current, which is given by a function of the third spatial\nderivative of the height profile. Our method exploits properties of the local\nequilibrium (LE) state of the third order finite difference process. We find\nthat the LE state of this process is not only useful for deriving the PDE; it\nalso enjoys nonstandard properties which are interesting in their own right.\nNamely, we demonstrate that the LE state is a "rough LE", a novel kind of LE\nstate discovered in our recent work on an Arrhenius rate crystal surface\nprocess.\n