Non-Linear Stochastic Equations with Calculable Steady States

Abstract

We consider generalizations of the Kardar--Parisi--Zhang equation that accomodate spatial anisotropies and the coupled evolution of several fields, and focus on their symmetries and non-perturbative properties. In particular, we derive generalized fluctuation--dissipation conditions on the form of the (non-linear) equations for the realization of a Gaussian probability density of the fields in the steady state. For the amorphous growth of a single height field in one dimension we give a general class of equations with exactly calculable (Gaussian and more complicated) steady states. In two dimensions, we show that any anisotropic system evolves on long time and length scales either to the usual isotropic strong coupling regime or to a linear-like fixed point associated with a hidden symmetry. Similar results are derived for textural growth equations that couple the height field with additional order parameters which fluctuate on the growing surface. In this context, we propose phenomenological equations for the growth of a crystalline material, where the height field interacts with lattice distortions, and identify two special cases that obtain Gaussian steady states. In the first case compression modes influence growth and are advected by height fluctuations, while in the second case it is the density of dislocations that couples with the height.

arXiv comment9 pages, revtex4

References in corpus (9)

• Analytical Solution of the Off-Equilibrium Dynamics of a Long Range Spin-Glass Model
• Nonequilibrium Dynamics of Interfaces and Lines
• Similarity-Detection and Localization
• Phase ordering and roughening on growing films
• An Exactly Solved Model of Three Dimensional Surface Growth in the Anisotropic KPZ Regime
• Dynamic Scaling of Coupled Nonequilibrium Interfaces
• A Model for Growth of Binary Alloys with Fast Surface Equilibration
• Universality classes in anisotropic non-equilibrium growth models
• Ordering Phenomena on Growing Films

Cited by in corpus (5)

• Strong anisotropy in two-dimensional surfaces with generic scale invariance: Gaussian and related models
• Dynamical Inequality in Growth Models
• Exponent Inequalities in Dynamical Systems
• The stationary AKPZ equation: logarithmic superdiffusivity
• Three-dimensional Gaussian fluctuations of non-commutative random surfaces along time-like paths