The KPZ Equation of Kinetic Interface Roughening: A Variational Perspective

Abstract

Interfaces of rather different natures—as, e.g., bacterial colony or forest fire boundaries, orsemiconductor layers grown by different methods (MBE, sputtering, etc.)—are self-affinefractals, and feature scaling with universal exponents (depending on the substrate’s dimensionalityd and global topology, as well as on the driving randomness’ spatial andtemporal correlations but not on the underlying mechanisms). Adding lateral growth asan essential (non-equilibrium) ingredient to the known equilibrium ones (randomnessand interface relaxation), the Kardar–Parisi–Zhang (KPZ) equation succeeded in finding(via the dynamic renormalization group) the correct exponents for flat d = 1 substratesand (spatially and temporally) uncorrelated randomness. It is this interplay which givesrise to the unique, non-Gaussian scaling properties characteristic of the specific, universaltype of non-equilibrium roughening. Later on, the asymptotic statistics of process h(x)fluctuations in the scaling regime was also analytically found for d = 1 substrates. Ford > 1 substrates, however, one has to rely on numerical simulations. Here we reviewa variational approach that allows for analytical progress regardless of substrate dimensionality.After reviewing our previous numerical results in d = 1, 2, and 3 on the timeevolution of one of the functionals—which we call the non-equilibrium potential (NEP)—aswell as its scaling behavior with the nonlinearity parameter λ, we discuss the stochasticthermodynamics of the roughening process and the memory of process h(x) in KPZ andin the related Golubovi´c–Bruinsma (GB) model, providing numerical evidence for thesignificant dependence on initial conditions of the NEP’s asymptotic behavior in bothmodels. Finally, we highlight some open questions.