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

The deterministic KPZ equation has been recently formulated as a gradient flow. Its non-equilibrium analog of a free energy-the non-equilibrium potential Phi[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 (t) (the average of Phi[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 Phi[h] is shown to be valid for any substrate dimensionality d, thus providing a valuable tool for studies in d > 1.