Universality Classes with Strong Coupling in Conserved Surface Roughening: Explicit vs Emergent Symmetries

TL;DR

The paper addresses strong coupling in conserved 1D surface roughening by introducing a one-parameter family of stochastic equations indexed by $n$, generalizing the stochastic Burgers equation. It combines a one-loop dynamic renormalization group analysis with targeted numerical simulations for $n=1,2,3$ to map out how hyperscaling and potential Galilean-like relations depend on $n$ and parity. The results show robust hyperscaling $2\alpha+d=z$ across cases, with even $n$ largely consistent with a Galilean-like constraint ($\alpha+z=n+1$) when vertex renormalization is negligible, and odd $n$ indicating vertex corrections and emergent symmetry at the strong-coupling fixed point; the height fluctuations remain symmetric with non-Gaussian kurtosis. These findings connect to known conserved-phase models such as cKPZ and the porous-medium equation, offering insights into real systems like thin films and active matter, and suggesting experimental contexts where such universality classes may be observed.

Abstract

The occurrence of strong coupling or nonlinear scaling behavior for kinetically rough interfaces whose dynamics are conserved, but not necessarily variational, remains to be fully understood. Here we formulate and study a family of conserved stochastic evolution equations for one-dimensional interfaces, whose nonlinearity depends on a parameter n, thus generalizing that of the stochastic Burgers equation, whose behavior is retrieved for n=0. This family of equations includes as particular instances a stochastic porous medium equation and other continuum models relevant to various hard and soft condensed matter systems. We perform a one-loop dynamical renormalization group analysis of the equations, which contemplates strong coupling scaling exponents that depend on the value of $n$ and may or may not imply vertex renormalization. These analytical expectations are contrasted with explicit numerical simulations of the equations with n=1,2, and 3. For odd n, numerical stability issues have required us to generalize the scheme originally proposed for n=0 by T. Sasamoto and H. Spohn. Precisely for n=1 and 3, and at variance with the n=0 and 2 cases (whose numerical exponents are consistent with non-renormalization of the vertex), numerical strong coupling exponent values are obtained which suggest vertex renormalization, akin to that reported for the celebrated conserved KPZ equation. We also study numerically the statistics of height fluctuations, whose probability distribution function turns out (at variance with cKPZ) to have zero skewness for long times and at saturation, irrespective of the value of n. However, the kurtosis is non-Gaussian, further supporting the conclusion on strong coupling asymptotic behavior. The zero skewness seems related with space symmetries of the n=0 and 2 equations, and with an emergent symmetry at the strong coupling fixed point for odd values of n.

Figures (5)

• Figure 1: Deterministic ($D_n=0$) solutions of Eq. \ref{['eq:n']} (left panels) and their PSDs (right panels), for $n=0,1,2,$ and 3, top to bottom. In each panel, the solid blue line corresponds to the initial condition and the dash-dotted red line corresponds to the numerical solution at time $t_f$. Top to bottom, $n=0$, $\lambda_0=2^6$, $B_0=2^{-6}$, $t_f=2^{-1}$ (first row); $n=1$, $\lambda_1=2^6$, $B_1=2^{0}$, $t_f=2^0$ (second row); $n=2$, $\lambda_2=2^8$, $B_2=2^{-2}$, $t_f=2^{-2}$ (third row); $n=3$, $\lambda=2^8$, $B_3=2^{1.4}$, $t_f=2^{-4}$ (fourth row). In all cases, the initial condition is $\phi(x,t=0)=A\sin(k_0 x)$ with $A=0.01$ and $k_0=1$.
• Figure 2: Same as Fig. \ref{['fig:detPsin-n1234']} but for $n$-independent simulation parameter values $\lambda_n=2^8$, $B_n=2^{-\frac{1}{2}}$, and $t_f=2^{-2}$. Results are for $n=0$, $n=1$, $n=2$, and $n=3$, top to bottom. The initial condition is again $\phi(x,t=0)=A\sin(k_0 x)$ with $A=0.01$ and $k_0=1$.
• Figure 3: Results for numerical simulation of Eq. \ref{['eq:n']} for $n=1$, namely, Eq. \ref{['eq:n=1']}. Parameters are: $L=64$, $T_t=2^8$, $B_1=2^0$, $\lambda_1=2^2$, $D_1=2^{-14}$, and $\phi_0=2^{-2}$. Averages are over 300 realizations of the noise. Top left, roughness $W(t)$ over time. Top right, PSD at different times. Top right inset, collapse of the PSD data. Bottom left, skewness (lower symbols) and kurtosis (upper symbols) for different times. Bottom right, histogram of height fluctuations at saturation to steady state. Middle left, morphology during linear growth. Middle right, morphology at saturation. Measurements are taken at times $t=\Delta t \cdot 2^N$ with $N=0,1,2,3,\ldots$ Blue (red) dots and lines correspond to early (late) times. Dashed purple and solid green reference straight lines on the top panels have the slopes indicated by the indicated scaling exponent values. Solid and dashed lines in the bottom right panel correspond to two different Gaussian distributions.
• Figure 4: Results for numerical simulation of Eq. \ref{['eq:n']} for $n=2$, namely, Eq. \ref{['eq:n=2']}. Parameters: 256 elements of size $\Delta x=2^{0}$, $2^{20}$ time steps of duration $\Delta t=2^{-9}$, $B_2=\lambda_2=D_2=1$, and $\phi_0=0$. Averages are over 200 realizations of the noise. The descriptions for all the panels are analogous to those of Fig. \ref{['fig:N1-Sto']}.
• Figure 5: Results for numerical simulation of Eq. \ref{['eq:n']} for $n=3$, namely, Eq. \ref{['eq:n=3']}. Parameters: 256 elements of size $\Delta x=2^{-1}$, $2^{27}$ time steps of duration $\Delta t=2^{-10}$, $B_3=2^{-2}$, $\lambda_3=2^2$, $D_3=2^{-14}$, and and $\phi_0=2^{-1}$. Averages are over 700 realizations of the noise. The descriptions for all the panels are analogous to those of Figs. \ref{['fig:N1-Sto']}.