Nonequilibrium Free Energy-Like Functional for the KPZ Equation

TL;DR

This paper derives a thermodynamic-like potential for the KPZ equation, enabling the analysis of its properties and stationary distribution, challenging previous beliefs about its non-existence.

Contribution

It provides the first derivation of a nonequilibrium free energy-like functional for the KPZ equation, facilitating new insights into its invariance and stationary distribution.

Findings

Exact stationary probability distribution obtained for arbitrary dimensions.

Supports the conjecture that no critical dimension exists in the strong coupling regime.

Proves global shift invariance properties of the KPZ functional.

Abstract

Opposing to a (common) belief against the existence of a thermodynamic-like potential for the KPZ equation, here we present a derivation for such a functional. With its knowledge we prove some global shift invariance properties previously conjectured by other authors. The procedure could be extended in order to derive a more general form of such a functional leading to other known related nonlinear kinetic equations. Exploiting the KPZ's functional, and for arbitrary dimension, we have obtained the exact form of the stationary probability distribution function and have shown a couple of examples of how it is possible to exploit it in order to obtain relevant results like finding support to the conjecture that in the strong coupling regime a critical dimension doesn't exists.