Symmetry-based nonlinear fluctuating hydrodynamics in one dimension

TL;DR

The paper presents a symmetry-based construction of nonlinear fluctuating hydrodynamics (NFH) for one-dimensional systems with generic homogeneous nearest-neighbor interactions, deriving NFH from conservation laws and symmetry rather than microscopic details. A one-loop renormalization-group analysis reveals a nontrivial fixed point with dynamical exponent $z=3/2$, shared by both sound and heat modes, and extensive simulations show space–time correlations collapse toward the universal Prähofer–Spohn KPZ scaling function. Finite-size effects are captured by a $1/L$ offset, and robustness checks confirm the KPZ universality across system size, sound velocity, nonlinear couplings, and diffusion constants. The work reconciles universal transport in 1D nonequilibrium systems with a principled, symmetry-based framework and clarifies limitations of previous NFH formulations.

Abstract

We present a symmetry-based formulation of nonlinear fluctuating hydrodynamics (NFH) for one-dimensional many-particle systems with generic homogeneous nearest-neighbor interactions. We derive the hydrodynamic equations solely from symmetry and conservation principles, ensuring full consistency with thermalization. Using the dynamic renormalization group, we show that both the sound and heat modes share the dynamical exponent $z=3/2$. Extensive numerical simulations of the derived NFH equations confirm this exponent and further reveal that both modes are close to the universal KPZ scaling function, namely the Prähofer-Spohn function. These findings establish a unified, symmetry-based framework for understanding universal transport and fluctuation phenomena in one-dimensional nonequilibrium systems, independent of microscopic details.

Figures (12)

• Figure 1: Numerical results for the dynamical scaling of the space–time correlation $\mathbb{S}^\alpha(x,t)$ obtained from simulations of Eq. (\ref{['EOM phi']}) with parameters $D_0 = D_s = \lambda_{1,2,3,4} = 1$, $c_s = 0.1$ and system size $L=8192$. A constant offset $\mathcal{C}_\alpha$ is added to account for finite-size effects. (a) Power-law decay of $\mathbb{S}^\alpha(x+\alpha c_s t=0,t)+\mathcal{C}_\alpha$ on a log–log scale in the comoving frame; the dashed line of slope $-2/3$ corresponds to the KPZ exponent. (b) Dynamical exponent $z$ as a function of time $t$ extracted from the local slope in (a); the dashed line indicates the KPZ value $z=3/2$. (c), (e) Spatial profiles of $\mathbb{S}^\alpha(x,t)+\mathcal{C}_\alpha$ at various times, plotted against the comoving coordinate $x+\alpha c_s t$. (d), (f) Data collapse of (c) and (e) based on Eq. (\ref{['eq:modified_scaling']}), revealing the universal scaling functions $f_\alpha$. The Prähofer–Spohn (black dashed) and $3/2$ Lévy (blue dashed) functions are shown for comparison. Additional numerical results are presented in Figs. \ref{['emfig1']}–\ref{['emfig3']} in the End Matter.
• Figure 2: Self-energy diagrams. The left panel shows the $\Sigma$ loop corresponding to Eq. (\ref{['Sigma_EM']}), and the right panel shows the $\Xi$ loop corresponding to Eq. (\ref{['Xi_EM']}). The solid and wavy lines denote $\phi^\alpha$ and $\pi^\alpha$, respectively.
• Figure 3: Diagrams of vertex corrections.
• Figure 4: Numerical results for the time correlation function $\langle \phi^{\alpha}(x,t)\phi^{\beta}(x,0)\rangle$. The parameters are identical to those in Fig. \ref{['fig1']}. In contrast to Fig. \ref{['fig1']}(a), which shows the autocorrelations ($\alpha=\beta$), this figure highlights the cross-correlations ($\alpha \neq \beta$), while also displaying the autocorrelations for reference. No constant offset $\mathcal{C}_{\alpha}$ is added, and the comoving frame ($x+\alpha c_s t=0$) is not used. The dashed line indicates a slope of $-2/3$, corresponding to the KPZ exponent.
• Figure 5: Dependence of the dynamical critical exponent $z$ on the sound velocity $c_s$, where $z$ is obtained from the data collapse analysis of $\mathbb{S}^{\alpha}(x,t)$. The error bars are comparable to the symbol size.