Coarse grained approach for universality classification of discrete models

TL;DR

This paper introduces a novel method to analytically determine coarse-grained coefficients of continuous equations from discrete models, validated on KPZ class models, bridging discrete and continuous universality classifications.

Contribution

The paper presents a new analytical approach to derive coarse-grained coefficients from discrete models using translations and interface growth velocity, applicable to universality class analysis.

Findings

Analytical coefficients agree with Monte Carlo simulations.

Method successfully applied to KPZ and crossover models.

Provides a bridge between discrete models and continuous equations.

Abstract

Discrete and continuous models belonging to a universality class share the same linearities and (or) nonlinearities. In this work, we propose a new approach to calculate coarse grained coefficients of the continuous differential equation from discrete models. We apply small constant translations in a test space and show how to obtain these coefficients from the transformed average interface growth velocity. Using the examples of the ballistic deposition (BD) model and the restricted solid-on-solid (RSOS) model, both belonging to the Kardar-Parisi-Zhang (KPZ) universality class, we demonstrate how to apply our approach to calculate analytically the corresponding coefficients of the KPZ equation. Our analytical nonlinear coefficients are in agreement with numerical results obtained by Monte Carlo tilted simulations. In addition to the BD and the RSOS we study a competitive RSOS model that shows crossover between the KPZ and Edwards-Wilkinson universality classes.