Coarse grained approach for volume conserving models

TL;DR

This paper presents a coarse-grained method to derive continuous stochastic differential equations for volume conserving surface models, revealing how symmetry affects the coefficients within the same universality class.

Contribution

It introduces a novel formalism to calculate all SDE coefficients from the stationary probability density, linking them directly to physical properties.

Findings

Calculated all coefficients for the symmetric model's KPZ equation.

Determined additional terms for the asymmetric model, including the diffusion coefficient.

Provided a theoretical and numerical framework connecting SDE coefficients with the SPDF.

Abstract

Volume conserving surface (VCS) models without deposition and evaporation, as well as ideal molecular-beam epitaxy models, are prototypes to study the symmetries of conserved dynamics. In this work we study two similar VCS models with conserved noise, which differ from each other by the axial symmetry of their dynamic hopping rules. We use a coarse-grained approach to analyze the models and show how to determine the coefficients of their corresponding continuous stochastic differential equation (SDE) within the same universality class. The employed method makes use of small translations in a test space which contains the stationary probability density function (SPDF). In case of the symmetric model we calculate all the coarse-grained coefficients of the related conserved Kardar-Parisi-Zhang (KPZ) equation. With respect to the symmetric model, the asymmetric model adds new terms which have to be analyzed, first of all the diffusion term, whose coarse-grained coefficient can be determined by the same method. In contrast to other methods, the used formalism allows to calculate all coefficients of the SDE theoretically and within limits numerically. Above all, the used approach connects the coefficients of the SDE with the SPDF and hence gives them a precise physical meaning.