Non-Linear Stochastic Equations with Calculable Steady States

TL;DR

This paper explores generalized stochastic growth equations with anisotropies and multiple fields, deriving conditions for Gaussian steady states and analyzing their long-term behaviors in various dimensions and coupling scenarios.

Contribution

It introduces a class of generalized growth equations with calculable steady states and analyzes their asymptotic regimes and symmetries, extending understanding of anisotropic and coupled surface growth models.

Findings

Exact steady states are derived for certain growth equations.

Anisotropic systems evolve to isotropic or linear-like fixed points.

Coupled equations with dislocation density or lattice distortions can have Gaussian 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.