d-Dimensional KPZ Equation as a Stochastic Gradient Flow in an Evolving Landscape: Interpretation and Time Evolution of Its Generating Functional

Abstract

The deterministic KPZ equation has been recently formulated as a gradient flow. Its non-equilibrium analog of a free energy-the "non-equilibrium potential" [h], providing at each time the landscape where the stochastic dynamics of h(x,t) takes place-is however unbounded, and its exact evaluation involves all the detailed histories leading to h(x,t) from some initial configuration h 0 (x,0). After pinpointing some implications of these facts, we study the time behavior of [h] t (the average of [h] over noise realizations at time t) and show the interesting consequences of its structure when an external driving force F is included. The asymptotic form of the time derivative [h] is shown to be valid for any substrate dimensionality d, thus providing a valuable tool for studies in d > 1.