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

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 $d>1$.