d-Dimensional KPZ Equation as a Stochastic Gradient Flow in an Evolving Landscape: Interpretation and Time Evolution of Its Generating Functional - Wio, Horacio S.; Rodríguez, Miguel A.; Gallego, Rafael; Revelli, Jorge A.; Alés, Alejandro; Deza, Roberto R. (Frontiers in Physics, 2017)
The deterministic KPZ equation has been recently formulated as a gradient flow. Its nonequilibrium analog of a free energy—the "nonequilibrium potential'' Φ[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 from some initial configuration h(x ⃗,0) to a final one h(x ⃗,t). After pinpointing some implications of these facts, we study the time behavior of 〈Φ[h] 〉 (the average of Φ[h] over noise realizations at time t) and show the interesting consequences of its structure when an external driving force F is included (i.e. the KPZ behavior as an activation-like process). The asymptotic form of the time derivative Φ ̇[h] is shown to be valid for any substrate dimensionality d, thus providing a valuable tool for studies in d>1.