Stochastic porous-medium equation in one dimension
Maximilien Bernard, Andrei A. Fedorenko, Pierre Le Doussal, Alberto Rosso · Physical review. E · 2025
We study the porous medium equation (PME) in one space dimension in the presence of additive nonconservative 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 α and β using the functional renormalization group. Extensive numerical simulations show agreement with the predicted values for these exponents; however, they also show anomalous scaling with an additional local exponent α_{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.