Asymptotic and effective coarsening exponents in surface growth models

TL;DR

This paper analyzes surface growth models with perpetual coarsening, distinguishing between finite-time effective exponents and true asymptotic exponents, and explains why asymptotic behavior is unreachable by direct numerical integration.

Contribution

It provides a theoretical interpretation of coarsening exponents in surface growth models, clarifying the difference between effective and asymptotic exponents and their observability.

Findings

Asymptotic exponents appear at very large times beyond direct numerical reach.

Effective exponents are finite-time phenomena influenced by model dynamics.

The study clarifies the interpretation of numerical coarsening exponents.

Abstract

We consider a class of unstable surface growth models, z_t = -\partial_x J, developing a mound structure of size lambda and displaying a perpetual coarsening process, i.e. an endless increase in time of lambda. The coarsening exponents n, defined by the growth law of the mound size lambda with time, lambda=t^n, were previously found by numerical integration of the growth equations [A. Torcini and P. Politi, Eur. Phys. J. B 25, 519 (2002)]. Recent analytical work now allows to interpret such findings as finite time effective exponents. The asymptotic exponents are shown to appear at so large time that cannot be reached by direct integration of the growth equations. The reason for the appearance of effective exponents is clearly identified.