The Local Equilibrium State of a Crystal Surface Jump Process in the Rough Scaling Regime

TL;DR

This paper studies a novel local equilibrium state of a crystal surface jump process under a nonstandard 'rough' scaling regime, revealing unique non-smooth properties and their impact on convergence to hydrodynamic limits.

Contribution

It introduces and analyzes a new 'rough' local equilibrium state arising in a nonstandard scaling regime for crystal surface models, contrasting it with traditional smooth states.

Findings

The rough LE state exhibits non-smooth variation across lattice sites.

Numerical and analytical analysis shows different convergence behaviors compared to smooth LE states.

The rough LE state influences the limits and convergence mechanisms in the hydrodynamic limit.

Abstract

We investigate the local equilibrium (LE) distribution of a crystal surface jump process as it approaches its hydrodynamic (continuum) limit in a nonstandard scaling regime introduced by Marzuola and Weare. The atypical scaling leads to a local equilibrium state whose structure is novel, to the best of our knowledge. The distinguishing characteristic of the new, "rough" LE state is that the ensemble average of single lattice site observables do not vary smoothly across lattice sites. We investigate numerically and analytically how the rough LE state affects the convergence mechanism via three key limits, and show that by comparison, more standard, "smooth" LE states satisfy stronger versions of these limits.