Universality in driven systems with a multiply-degenerate umbilic point

TL;DR

The paper investigates universality in weakly hyperbolic driven systems by studying a multilane TASEP with an umbilic manifold where multiple mode velocities coincide. It develops an effective mode-coupling theory for the degenerate umbilic mode and a nondegenerate companion mode, validating predictions via lattice Monte Carlo and continuous nonlinear fluctuating hydrodynamics simulations. The results establish a robust dynamical exponent $z_u=3/2$ for the umbilic mode and reveal a universal umbilic scaling function along the umbilic line, with the nondegenerate mode following a $z_s=5/3$ scaling under a specific condition; increasing degeneracy $K$ modifies the umbilic function but suggests convergence toward KPZ in the large-$K$ limit. These findings point to novel universality classes for long-lived hydrodynamic modes with equal characteristic velocities and highlight the utility of two-mode MCT in capturing the coupled dynamics of degenerate and nondegenerate modes.

Abstract

We investigate a driven particle system, a multilane asymmetric exclusion process, where the particle number in every lane is conserved, and stationary state is fully uncorrelated. The phase space has, starting from three lanes and more, an umbilic manifold where characteristic velocities of all the modes but one coincide, thus allowing us to study a weakly hyperbolic system with arbitrarily large degeneracy. We then study space-time fluctuations in the steady state, at the umbilic manifold, which are expected to exhibit universal scaling features. We formulate an effective mode-coupling theory (MCT) for the multilane model within the umbilic subspace and test its predictions. Unlike in the bidirectional two-lane model with an umbilic point studied earlier, here we find a robust $z=3/2$ dynamical exponent for the umbilic mode. The umbilic scaling function, obtained from Monte-Carlo simulations, for the simplest 3-lane scenario, appears to have an universal shape for a range of interaction parameters. Remarkably, the shape and dynamic exponent of the non-degenerate mode can be analytically predicted on the base of effective MCT, up to non-universal scaling factor. Our findings suggest the existence of novel universality classes with dynamical exponent $3/2$, appearing in long-lived hydrodynamic modes with equal characteristic velocities.

Figures (7)

• Figure 1: Allowable hops for the three-lane model ($K+1=3$) with their rates (\ref{['eq:rates']}). Particles with the same color hop with the same rates, e.g. hopping of black-coloured particles is forbidden by hardcore exclusion.
• Figure 2: Characteristic velocities $c_1,c_2$, stemming from the degenerate umbilic mode, along trajectories $\vec{\rho}(d)$ in phase space of densities, passing through (or near) the umbilic line, on a manifold orthogonal to it. The parametrization is chosen so that $d=0$ corresponds to the minimal distance to the umbilic line. Dashed curves correspond to a trajectories passing through the UP while for solid curves a small mismatch $\epsilon=0.005$ is employed. Explicitly, for dashed curves, $(\rho_1,\rho_2,\rho_3) =(0.3,0.3,0,3) + d(1,1,-2)/\sqrt{6}$ while for solid curves, $(\rho_1,\rho_2,\rho_3) =(0.3+\epsilon,0.3-\epsilon,0.3) + d(1,1,-2)/\sqrt{6}$. Other parameters: $K=2$, $a=0.4$. The third characteristic velocity $c_3$ of the nondegenerate mode is not shown. The curves show the typical topology of an isolated umbilic point, see also the main text.
• Figure 3: Characteristic velocities in the vicinity of an UP with fourfold degeneracy. Two solid curves given by $\vec{\rho}(\epsilon,d) - (\rho, \rho, \rho, \rho, \rho) = d(1,1,-2,0,0) + \epsilon (0,0,0,1,-1)$ with $\epsilon =10^{-3}$ cross the dashed curves $\vec{\rho}(0,s)$ at the origin, indicating a non-isolated UP. Other parameters: $K=4$, $a=0.4$, $\rho=0.3$.
• Figure 4: Data collapse for the umbilic mode from lattice Monte Carlo simulations with $z_u = 3/2$, $c_u = 5/8$ and $E_u = 2.12$. Shaded regions show two standard deviation neighborhoods. Black curves show the best-fit KPZ (Prähofer-Spohn) scaling function (dashed curve) and the (scaled) $S_{11}$ correlator of Ref. 2025Spohn (full curve) at $X=Y=-1$. Simulation parameters: $K=2$, $a=2$, $\rho = 1/4$, system size $N = 10^6$, $n_{\textrm{samples}} = 10^3$.
• Figure 5: Data collapse for the heat mode from lattice Monte Carlo simulations with $z_s = 5/3$, $c_s = 7/4$ and $E_s = 1.30$. Shaded regions show two standard deviation neighborhoods. Best collapse of the data is obtained for $z_s \approx 1.645$. Black curve shows the maximally asymmetric Levy stable $\frac{5}{3}$ distribution \ref{['5/3LeviStable']}. Simulation parameters as in Fig. \ref{['FigUmbilicLattice']}.