Universality in coupled stochastic Burgers systems with degenerate flux Jacobian

TL;DR

This work extends nonlinear fluctuating hydrodynamics to two-component stochastic Burgers systems with a degenerate flux Jacobian, demonstrating that the dynamical exponent remains $z=\tfrac{3}{2}$ while the scaling functions depend on the inter-component couplings. By analyzing both continuum equations and a two-lane lattice gas, and establishing a cyclicity condition that preserves a Gaussian stationary measure, the authors show a robust universality between continuum and lattice models at special couplings, with a precise mapping $\lambda=2-1/\sqrt{\gamma}$ linking lattice and continuum parameters. Numerical simulations confirm the $t^{2/3}$ scaling for spacetime correlators and reveal coupling-dependent deviations from the one-component KPZ scaling function, while time-integrated current fluctuations follow Baik-Rains-like statistics only at certain limits. Mode-coupling theory provides a qualitative framework for the observed behavior, predicting a nontrivial dependence of scaling functions on $\lambda$ and pointing to rich two-component universality beyond the standard KPZ paradigm. Overall, the study highlights both the universality of the dynamical exponent and the model-specificity of scaling functions in degenerate two-component stochastic systems, suggesting several avenues for further theoretical and numerical exploration.

Abstract

In our contribution we study stochastic models in one space dimension with two conservation laws. One model is the coupled continuum stochastic Burgers equation, for which each current is a sum of quadratic non-linearities, linear diffusion, and spacetime white noise. The second model is a two-lane stochastic lattice gas. As distinct from previous studies, the two conserved densities are tuned such that the flux Jacobian, a $2 \times 2$ matrix, has coinciding eigenvalues. In the steady state, investigated are spacetime correlations of the conserved fields and the time-integrated currents at the origin. For a particular choice of couplings the dynamical exponent 3/2 is confirmed. Furthermore, at these couplings, continuum stochastic Burgers equation and lattice gas are demonstrated to be in the same universality class.

Figures (9)

• Figure 1: Correlators $S_{11}$ and $S_{22}$ obtained from numerical simulations of the coupled stochastic Burgers equations in Eq. \ref{['3.1']} with $b=3$ and $\lambda=0$ using the discretizations of Eqs. \ref{['2.16']} and \ref{['2.18']}. The system size is $L=4096$ and the number of independent simulations is $10^5$. The covariance of the initial correlator is $\langle \phi_{\alpha,j}\phi_{\beta,0} \rangle = \delta_{\alpha\beta}\delta_{j0}$. The sum rule, $\sum_j S_{11}(j,t) = \sum_j S_{22 }(j,t) = 1$, is satisfied within a numerical accuracy of $1\%$.
• Figure 2: Probability density functions (PDFs) of (a) $\zeta_1$ and (b) $\zeta_2$ for the coupled KPZ equations \ref{['3.3']} at $\lambda=0$ for different values of the parameter $b$ and a sequence of times $t$. The system size is $L=8192$ with periodic boundary conditions. The total number of simulation runs is $5\times 10^4$. For both $\zeta_1$ and $\zeta_2$ we set $\Gamma_\perp=b$. The corresponding PDFs collapse on a single curve. The variances for $\zeta_1$ and $\zeta_2$ are $1.545$ and $0.943$, respectively. Comparison is with the corresponding quantities at $\lambda=1$.
• Figure 3: Illustrating the two-lane lattice gas model 2012-popkov-schutz. The lattice sites are indexed by $(\alpha, n)$ where $\alpha=1, 2$, $n \in \mathbb{Z}$.
• Figure 4: Plots of the correlator $\textsf{S}_{\alpha\beta}$ for the two lanes in the lattice-gas model with $\gamma = 0.25$. The system size is $L=2^{16}$ and the number of independent simulations is $4 \times 10^5$. The initial measure is provided in Eq. \ref{['4.2']} with $\mu = \tilde{\mu} = (\log\gamma)/2$ corresponding to the density $1/2$ in each of the lanes. In accordance with the sum rule, within a numerical accuracy of $1\%$, $\sum_n \textsf{S}_{11} (n,t) = 1 / 4$ and $\sum_n \textsf{S}_{12} (n,t) = 1 /12$, where $1/4,1/12$ is the value at $t =0$.
• Figure 5: Plots of PDFs for $\Theta_1$ and $\Theta_2$ in the two-lane lattice gas model for $\gamma=0.25$. The initial configurations are chosen from the stationary measure \ref{['4.2']} at exact half-filling in each lane. (a) is a mirror image of (b).