Gaussian statistics as an emergent symmetry of the stochastic Burgers equation

TL;DR

This paper demonstrates that the stochastic Burgers equation exhibits emergent Gaussian fluctuations in its asymptotic behavior, revealing an unexpected symmetry that contrasts with the non-Gaussian Tracy-Widom fluctuations of related models.

Contribution

The study uncovers the emergence of Gaussian statistics and symmetry in the stochastic Burgers equation through renormalization group analysis and numerical simulations, challenging prior expectations.

Findings

Gaussian fluctuations occur in the asymptotic regime

Odd-order cumulants cancel exactly

Excess kurtosis vanishes for large systems

Abstract

Symmetries play a conspicuous role in the large-scale behavior of critical systems. While in equilibrium they allow to classify asymptotics into different universality classes, out of equilibrium they can emerge, some times unexpectedly, 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 for the physical field. This robustness of Gaussian behavior contradicts naive expectations, due to the detailed relation ---including the same set of symmetries--- 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. All odd-order cumulants cancel exactly, while the excess kurtosis vanishes for large systems, indeed consistent with Gaussian behavior.