The stochastic porous medium equation in one dimension

TL;DR

This paper investigates the one-dimensional stochastic porous medium equation with additive noise, predicting growth exponents via functional RG, and confirming these with simulations that reveal anomalous scaling and multiscaling phenomena.

Contribution

It provides the first detailed analysis of the stochastic PME in one dimension, combining theoretical predictions with numerical validation and uncovering multiscaling effects.

Findings

Predicted growth exponents match numerical simulations.

Discovered anomalous scaling with a local exponent.

Identified multiscaling due to broad height difference distributions.

Abstract

We study the porous medium equation (PME) in one space dimension in presence of additive non-conservative white noise, and interpreted as a stochastic growth equation for the height field of an interface. We predict the values of the two growth exponents $\alpha$ and $\beta$ using the functional RG. Extensive numerical simulations show agreement with the predicted values for these exponents, however they also show anomalous scaling with an additional "local" exponent $\alpha_{\rm loc}$, as well as multiscaling originating from broad distributions of local height differences. The stationary measure of the stochastic PME is found to be well described by a random walk model, related to a Bessel process. This model allows for several predictions about the multiscaling properties.