Stochastic Growth Equations and Reparametrization Invariance

TL;DR

This paper introduces a reparametrization invariant framework for deriving and analyzing stochastic surface growth equations, capturing physical mechanisms and symmetries often lost in traditional approaches, and applicable beyond standard approximations.

Contribution

It presents a novel reparametrization invariant method to derive and analyze stochastic growth equations, revealing symmetries and conservation laws beyond traditional small gradient expansions.

Findings

Derivation of continuum growth equations from lattice models

Identification of physical processes underlying growth terms

Explicit display of conservation laws and symmetries

Abstract

It is shown that, by imposing reparametrization invariance, one may derive a variety of stochastic equations describing the dynamics of surface growth and identify the physical processes responsible for the various terms. This approach provides a particularly transparent way to obtain continuum growth equations for interfaces. It is straightforward to derive equations which describe the coarse grained evolution of discrete lattice models and analyze their small gradient expansion. In this way, the authors identify the basic mechanisms which lead to the most commonly used growth equations. The advantages of this formulation of growth processes is that it allows one to go beyond the frequently used no-overhang approximation. The reparametrization invariant form also displays explicitly the conservation laws for the specific process and all the symmetries with respect to space-time transformations which are usually lost in the small gradient expansion. Finally, it is observed, that the knowledge of the full equation of motion, beyond the lowest order gradient expansion, might be relevant in problems where the usual perturbative renormalization methods fail.