# The KPZ Equation of Kinetic Interface Roughening: A Variational Perspective

## TL;DR

This paper explores the KPZ equation, a model for interface roughening, using a variational approach to better understand its behavior across different dimensions.

## Contribution

The paper introduces a variational approach for analyzing the KPZ equation, enabling analytical progress in any substrate dimensionality.

## Key findings

- The non-equilibrium potential (NEP) shows distinct scaling behavior with the nonlinearity parameter λ in different dimensions.
- Numerical evidence shows that the NEP's asymptotic behavior strongly depends on initial conditions in both KPZ and GB models.
- The variational approach provides insights into the stochastic thermodynamics of interface roughening processes.

## Abstract

Interfaces of rather different natures—as, e.g., bacterial colony or forest fire boundaries, or semiconductor layers grown by different methods (MBE, sputtering, etc.)—are self-affine fractals, and feature scaling with universal exponents (depending on the substrate’s dimensionality d and global topology, as well as on the driving randomness’ spatial and temporal correlations but not on the underlying mechanisms). Adding lateral growth as an essential (non-equilibrium) ingredient to the known equilibrium ones (randomness and interface relaxation), the Kardar–Parisi–Zhang (KPZ) equation succeeded in finding (via the dynamic renormalization group) the correct exponents for flat d=1 substrates and (spatially and temporally) uncorrelated randomness. It is this interplay which gives rise to the unique, non-Gaussian scaling properties characteristic of the specific, universal type 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. For d>1 substrates, however, one has to rely on numerical simulations. Here we review a 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 time evolution of one of the functionals—which we call the non-equilibrium potential (NEP)—as well as its scaling behavior with the nonlinearity parameter λ, we discuss the stochastic thermodynamics of the roughening process and the memory of process h(x) in KPZ and in the related Golubović–Bruinsma (GB) model, providing numerical evidence for the significant dependence on initial conditions of the NEP’s asymptotic behavior in both models. Finally, we highlight some open questions.