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

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.

arXiv commentrevised version, 12 pages, 4 figures

References in corpus (19)

• Exact evolution equation for the effective potential
• Non-Perturbative Renormalization Flow in Quantum Field Theory and Statistical Physics
• The Exact Renormalisation Group and Approximate Solutions
• The nonperturbative functional renormalization group and its applications
• Burgers Turbulence
• An Introduction to the Nonperturbative Renormalization Group
• An appetizer to modern developments on the Kardar-Parisi-Zhang universality class
• Flow Equations and BRS Invariance for Yang-Mills Theories
• Non-perturbative renormalization group for the Kardar-Parisi-Zhang equation
• A new method to solve the Non Perturbative Renormalization Group equations
• Non-perturbative renormalisation group for the Kardar-Parisi-Zhang equation: general framework and first applications
• Solutions of renormalization group flow equations with full momentum dependence
• Nonperturbative renormalization group for the stationary Kardar-Parisi-Zhang equation: scaling functions and amplitude ratios in 1+1, 2+1 and 3+1 dimensions
• Functional renormalisation group for turbulence
• Breaking of scale invariance in the time dependence of correlation functions in isotropic and homogeneous turbulence
• KPZ equation with short-range correlated noise: emergent symmetries and non-universal observables
• Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation
• The unpredicted scaling of the one-dimensional Kardar-Parisi-Zhang equation
• The inviscid fixed point of the multi-dimensional Burgers-KPZ equation