Gaussian statistics as an emergent symmetry of the stochastic scalar Burgers equation

Abstract

Symmetries play a conspicuous role in the large-scale behavior of critical systems. In equilibrium they allow to classify asymptotics into different universality classes and, out of equilibrium, they sometimes emerge as collective properties which are not explicit in the "bare" interactions. Here we elucidate the emergence of an up-down symmetry in the asymptotic behavior of the stochastic scalar Burgers equation in one and two dimensions, manifested by the occurrence of Gaussian fluctuations even within the time regime controlled by nonlinearities. This robustness of Gaussian behavior contradicts naive expectations, due to the detailed relation -including the lack of up-down symmetry-between Burgers equation and the Kardar-Parisi-Zhang equation, which paradigmatically displays non-Gaussian fluctuations described by Tracy-Widom distributions. We reach our conclusions via a dynamic renormalization group study of the field statistics, confirmed by direct evaluation of the field probability distribution function from numerical simulations of the dynamical equation.

The derivative of the Kardar-Parisi-Zhang equation is not in the KPZ universality class

“…The Gaussian behavior obtained for Eq. ( 3) coincides with analytical DRG results which follow successful analyses of field statistics for the KPZ [45][46][47] and nonlinear-Molecular Beam Epitaxy (MBE) [48] equations, and for the scalar Burgers equation with non-conserved noise [49]. The method performs a partial RG transformation in which the equation is coarse-grained [50], while omitting the rescaling step [27,36].…”

“…We further obtain (see SM in Appendix A) u 4 c ∼ [ln(1/s)] 0.79 , hence the kurtosis, K → 0 as s → 0. Finally, as in [49], an exact symmetry induces u 2n+1 c = 0 ∀n, hence S = 0. More generally, the u-PDF is symmetric (unlike the TW or BR distributions) [49].…”

“…Finally, as in [49], an exact symmetry induces u 2n+1 c = 0 ∀n, hence S = 0. More generally, the u-PDF is symmetric (unlike the TW or BR distributions) [49]. Combined with the vanishing kurtosis, these results fully agree with the Gaussian statistics numerically found for u.…”

“…The method performs a partial RG transformation in which the equation is coarse-grained [50], while omitting the rescaling step [27,36]. Within a one-loop approximation [45][46][47][48][49] [see details in the SM (Appendix A)], w 2 = u 2 c ∝ R dk 1 . The variance of u scales as 1/L for L ≫ s, with s the lattice spacing, which agrees with the expected α = −1/2.…”

“…Hence, the symmetry of the (Gaussian) PDF is an emergent property of the large-scale behavior in Eq. ( 3), much as it is for Burgers equation with non-conserved noise (NCN) [49]. Akin to the latter, the symmetric field PDF in the nonlinear regime can be related with the behavior of the deterministic (viscous) Burgers equation, which is analytically known [53,54] to yield sawtooth profiles, symmetric, as Eq.…”