Unveiling the different scaling regimes of the one-dimensional Kardar-Parisi-Zhang--Burgers equation using the functional renormalisation group

TL;DR

This work analyzes the scaling structure of the one-dimensional KPZ--Burgers equation by leveraging the functional renormalisation group (FRG) to identify an inviscid Burgers fixed point with dynamical exponent $z=1$ that governs the UV regime, alongside the KPZ fixed point with $z=3/2$ in the IR. The authors develop an advanced two-grid numerical scheme that couples small-$p$ (IR) and large-$p$ (UV) FRG flow equations, enabling a unified calculation of the correlation function $C(varpi,p)$ across all momenta and illuminating the crossover between fixed points. They quantify the extent of the IB regime as a function of the microscopic coupling $g_\Lambda = \lambda^2 D/\nu^3$, observing an UV transition from EW to IB scaling for large $g_\Lambda$ and confirming the IR KPZ scaling in all cases. The results provide a robust framework for studying crossovers in non-equilibrium scaling, with potential applications to turbulence and Navier–Stokes-like systems where UV–IR fixed-point competition shapes observable spectra.

Abstract

The Kardar-Parisi-Zhang (KPZ) equation is a celebrated non-linear stochastic equation featuring non-equilibrium scaling. Although in one dimension, its statistical properties are very well understood, a new scaling regime has been reported in recent numerical simulations. This new regime is characterised by a dynamical exponent $z=1$, markedly different from the expected one $z=3/2$ for the KPZ universality class, and it emerges when approaching the inviscid limit. The origin of this scaling has been traced down to the existence of a new fixed point, termed the inviscid Burgers (IB) fixed point, which was uncovered using the functional renormalisation group (FRG). The FRG equations can be solved analytically in the asymptotic regime of vanishing viscosity and large momenta, showing that indeed $z=1$ exactly at the IB fixed point. In this work, we set up an advanced method to numerically solve the full FRG flow equations in a certain approximation, which allows us to determine in a unified way the correlation function over the whole range of momenta, not restricted to some particular regime. We analyse the crossover between the different fixed points, and quantitatively determine the extent of the IB regime.

Figures (4)

• Figure 1: Sketch of the two grids. The blue area depicts the dimensionless grid at $\kappa=\Lambda$; the black dots symbolise the nodes of the dimensionful grid, the dots highlighted with blue are about to exit the small-$p$ grid and start the large-$p$ evolution. The green area depicts the dimensionless grid at $\kappa<\Lambda$, and the dots highlighted with green will exit at this value of $\kappa$.
• Figure 2: Half-frequency $\varpi_{1/2}(p)$ as a function of momentum $p$ calculated before (gray dots) and after (black dots) the large-$p$ evolution given by (\ref{['eq:largep']}), at (a) $\hat{g}_{\Lambda} = 0.1$ and (b) $\hat{g}_{\Lambda} = 500$. The different shades show the IB (green), KPZ (red) and EW (blue) universal scaling regimes. The plain lines are guidelines showing the expected power-laws. The red dotted line with $3/2$ slope in the intermediate region of panel (b) is provided for comparison.
• Figure 3: (a) dimensionful and (b) dimensionless correlation functions at small $\kappa$ with $\hat{g}_{\Lambda} = 0.1$; (c) dimensionful and (d) dimensionless correlation functions for $\hat{g}_{\Lambda} = 300$. The lines show the IB (green), KPZ (red) and EW (blue) scaling regimes, the dashed gray line shows the non-universal EW-like scaling.
• Figure 4: Width of the IB region (a) and its upper and lower bounds (b) as functions of the microscopic coupling $\hat{g}_{\Lambda}$ calculated before (gray dots) and after the large-$p$ evolution given by Eq. (\ref{['eq:largep']}) (black dots).