AUTHOR=Wio Horacio S., Rodríguez Miguel A., Gallego Rafael, Revelli Jorge A., Alés Alejandro, Deza Roberto R.
TITLE=d-Dimensional KPZ Equation as a Stochastic Gradient Flow in an Evolving Landscape: Interpretation and Time Evolution of Its Generating Functional
JOURNAL=Frontiers in Physics
VOLUME=4
YEAR=2017
PAGES=52
URL=https://www.frontiersin.org/article/10.3389/fphy.2016.00052
DOI=10.3389/fphy.2016.00052
ISSN=2296-424X
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” 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 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 〈Φ[h]〉_{t} (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. 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.