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

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2016.00052

DOI=10.3389/fphy.2016.00052

ISSN=2296-424X

ABSTRACT=<p>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[<italic>h</italic>], providing at each time the landscape where the stochastic dynamics of <italic>h</italic>(<bold>x</bold>,<italic>t</italic>) takes place—is however <italic>unbounded</italic>, and its exact evaluation involves all the detailed histories leading to <italic>h</italic>(<bold>x</bold>,<italic>t</italic>) from some initial configuration <italic>h</italic><sub>0</sub>(<bold>x</bold>,0). After pinpointing some implications of these facts, we study the time behavior of 〈Φ[<italic>h</italic>]〉<sub><italic>t</italic></sub> (the average of Φ[<italic>h</italic>] over noise realizations at time <italic>t</italic>) and show the interesting consequences of its structure when an external driving force <italic>F</italic> is included. The asymptotic form of the time derivative <inline-formula><mml:math id="M64" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mrow><mml:mo>Φ</mml:mo></mml:mrow><mml:mo>˙</mml:mo></mml:mover><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula> is shown to be valid for <italic>any</italic> substrate dimensionality <italic>d</italic>, thus providing a valuable tool for studies in <italic>d</italic> &gt; 1.</p>