d-dimensional KPZ equation as a stochastic gradient flow in an evolving landscape: interpretation, parameter dependence, and asymptotic form

H.S. WIO; M.RODRÍGUEZ; R. GALLEGO; J.A. REVELLI; A. ALÉS; R.R. DEZA

Frontiers in Physics: Mathematical Physics

Frontiers

Lugar: Laussane; Año: 2016

2296-424X

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 landscapewhere 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 finalone 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 Φ̇ [h]  is shown to be valid for any substrate dimensionality d, thus providing a valuable tool for studies in d>1.