Dynamical inequality in growth models

¹ Department of Mathematics, Kings College London - Strand, London WC2R 2LS, UK, EU
² Department of Physics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University Tel Aviv 69978, Israel

^a eytan.katzav@kcl.ac.uk
^b bricki@netvision.net.il

Received: 21 June 2011
Accepted: 2 August 2011

Abstract

A recent exponent inequality is applied to a number of dynamical growth models. Many of the known exponents for models such as the Kardar-Parisi-Zhang (KPZ) equation are shown to be consistent with the inequality. In some cases, such as the molecular beam equation, the situation is more interesting, where the exponents saturate the inequality. As the acid test for the relative strength of four popular approximation schemes we apply the inequality to the exponents obtained for two non-local KPZ systems. We find that all methods but one, the self-consistent expansion, violate the inequality in some regions of parameter space. To further demonstrate the usefulness of the inequality, we apply it to a specific model, which belongs to a family of models in which the inequality becomes an equality. We thus show that the inequality can easily yield results, which otherwise have to rely either on approximations or general beliefs.