The stochastic porous medium equation in one dimension

TL;DR

The paper develops a functional renormalization group treatment of the one-dimensional stochastic porous medium equation with additive noise and demonstrates that the global growth exponents are α = ε/(1+s) and z = 2 − ε(s−1)/(s+1) (with the exact z = 2α + d relation), while numerically confirming a robust Family–Vicsek-type scaling for the interface width. A key finding is the emergence of anomalous, multifractal scaling at small scales, captured by a local roughness exponent α_loc(q) that depends on q and on s, particularly for s>1. A central contribution is mapping the stationary interface to a continuum random-walk model with diffusion D(h) ∼ |h|^{s−1}, which reduces to a Bessel-process in the large-scale limit and yields detailed predictions for increment distributions, two-point height statistics, and multifractal exponents. The combination of FRG, random-walk modeling, and extensive numerics provides a quantitative theory for scaling, multiscaling, and stationary statistics of the SPME in 1D, with potential extensions to brain dynamics and other nonlinear diffusion problems.

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 $α$ and $β$ 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 $α_{\rm loc}$, 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.

Figures (13)

• Figure 1: Examples of stationary interfaces for $s=3$, $\nu=1$ (blue) and $s=0.5$, $\nu=0.25$ (red) $L=500$ . The latter has been rescaled by a factor $5$ for aesthetic purposes. For $|h_n|$ away from zero, the interface becomes rougher when $s<1$ and flatter when $s>1$. In the inset, the mean square increment of the interface height, $\langle (h_{n+1}-h_n)^2 \rangle$ at fixed value of $h_n$, is compared to the mean square jumps $1/D(h_n)$ of the random walk defined in Eq. \ref{['eq:RW']} (dashed lines) SM. We observe a perfect agreement at large $h_n$.
• Figure 2: Numerical solution of Eq.\ref{['RG2main']} illustrating the collapse of $D_u(h)$. Main panel, $s>1$: Starting from $D_0(h) =2 \sqrt{4 +h^2}$ which corresponds to $s=2$ and $a=1/(1+s)=1/3$. As $u$ increases, the solutions converge towards a self-similar fixed point, shown by the black dashed lines, which corresponds to $F(H)$ (with $b^2=1.612$), calculated numerically from \ref{['eqn:F']}. Inset, $s<1$: Starting from $D_0(h) = \frac{1}{2}(1/4^4+h^2)^{-1/4}$ corresponding to $s=0.5$ and $a=2/3$. The dotted line correspond to the predicted fixed point (with $b^2=0.62$).
• Figure 3: Main Panel: Family-Vicksek collapse \ref{['Family-Vicsek']} for $d=1$ interfaces with $s = 0.5$ (continuous red line) and $s = 3$ (blue line) for several system sizes $L$. The exponents $\alpha$ and $z$ are those obtained from the RG \ref{['RGexponentsavecd']}, leading to a successful collapse of the width onto a single curve. The black dotted line corresponds to the predicted short time growth $\sim u^\beta$, with $\beta = \alpha/z = 1/(3+s)$. The data for $s=3$ has been shifted to the left by a factor $10$ to improve the data presentation. Inset: Moments $\langle |\Delta|^q \rangle$ of the increments $\Delta = h_{n+\ell}-h_n$ for $1 \le n \le L-\ell$ and $s=0.5$. Here, $q=1,2,4$ from bottom to top. We obtain a succesful collapse using the scaling proposed in \ref{['eq:anomscalstatq']} with $\alpha_{\rm loc} = 1/2$.
• Figure 4: PDF of $h_n$ averaged over $n \in [L-p,L]$ with $L=500$ and $p=50$, with boundary condition $h_0=10$ and $D(h)=1+h^2$ ($s=3$) for (i) the stationary SPME (orange), (ii) the discrete RW model (blue), (iii) the analytical prediction (black) from the continuum diffusion model (using the two point transition probability in \ref{['eq:propagatorAbsh']}) (note that there is no fitting parameter). Inset: Distribution of the height $h_{n+1}$ conditioned to a given $h_n$ for stationary interfaces of the SPME (Histogram). It is compared to the prediction for the random walk defined in Eq.\ref{['eq:RW']}, which is a Gaussian of mean $h_n$ and variance $1/D(h_n)$ (Continuous lines)
• Figure 5: Integral curves in the phase-plane $(X,Y)$. Left panel for $a=1/3$ and right panel for $a=-1/3$. There are 5 singular points A,B (in the plane) and C,D,E (at infinity). Note that for $a<0$ and $a>0$ the points C and E exchange their positions. Each curve connecting two singular points corresponds to a self-similar solution $F(H)$. The behavior at starting singular point gives asymptotic of $F(H)$ for $H \to 0$ and at the ending point for $H \to \infty$. Only curves starting at point A give the solutions finite at $H=0$.