KPZ dynamics from a variational perspective: potential landscape, time behavior, and other issues

TL;DR

This paper explores the deterministic KPZ equation as a gradient flow within a nonequilibrium potential landscape, analyzing its properties, time behavior, and effects of external forces across different substrate dimensions.

Contribution

It introduces a variational perspective on KPZ dynamics through a nonequilibrium potential, examining its moments and asymptotic behavior for any substrate dimension.

Findings

The NEP is unbounded and depends on the entire history of the system.

Time derivatives of the NEP's moments reveal specific signatures of the dynamics.

The asymptotic form of the NEP's time derivative is valid for any substrate dimension.

Abstract

The deterministic KPZ equation has been recently formulated as a gradient flow, in a nonequilibrium potential (NEP) \[\Phi[h(\mathbf{x},t)]=\int\mathrm{d}\mathbf{x}\left[\frac{\nu}{2}(\nabla h)^2-\frac{\lambda}{2}\int_{h_0(\mathbf{x},0)}^{h(\mathbf{x},t)}\mathrm{d}\psi(\nabla\psi)^2\right].\] This NEP---which provides at time $t$ the landscape where the stochastic dynamics of $h(\mathbf{x},t)$ takes place---is however unbounded, and its exact evaluation involves all the detailed histories leading to $h(\mathbf{x},t)$ from some initial configuration $h_0(\mathbf{x},0)$. After pinpointing some consequences of these facts, we study the time behavior of the NEP's first few moments and analyze its signatures when an external driving force $F$ is included. We finally show that the asymptotic form of the NEP's time derivative $\dot\Phi[h]$ turns out to be valid for any substrate dimensionality $d$, thus providing a valuable tool for studies in $d>1$.