Functional renormalisation approach to driven dissipative dynamics

TL;DR

This paper develops a non-perturbative functional renormalisation group approach to analyze stationary scaling states in driven dissipative systems like Burgers' and Gross-Pitaevskii equations, revealing new fixed points and scaling relations relevant to turbulence and Bose gases.

Contribution

It introduces a Galilei-invariant approximation within the functional RG framework to study non-perturbative fixed points, capturing vorticity and turbulence features beyond existing literature.

Findings

Identifies a continuum of RG fixed points in all spatial dimensions.

Derives a new scaling relation applicable to ultra-cold Bose gases and KPZ dynamics.

Confirms anomalous scaling corrections through numerical simulations.

Abstract

I investigate stationary scaling states of Burgers' and Gross-Pitaevskii equations (GPE). The path integral representation of the steady state of the stochastic Burgers equation is used in order to investigate the scaling solutions of the system at renormalisation group (RG) fixed points. I employ the functional RG in order to access the non-perturbative regime. I devise an approximation that respects Galilei invariance and is designed to resolve the frequency and momentum dependence of low order correlation functions. I establish a set of RG fixed point equations for inverse propagators with an arbitrary frequency and momentum dependence. In all spatial dimensions they yield a continuum of fixed points as well as an isolated one. These results are fully compatible with the existing literature for $d=1$. For $d\neq1$ however results of the literature focus almost exclusively on irrotational solutions while the solutions that my approximation can capture contain necessarily vorticity and are closer to Navier-Stokes turbulence. Non-equilibrium steady states of ultra-cold Bose gases coupled to external baths of energy and particles such as exciton-polariton condensates are related to Kardar-Parisi-Zhang (KPZ) dynamics. I postulate that the scaling that we obtain in this context applies as well to far-from-equilibrium quasi-stationary states (non-thermal fixed points) of the closed system described by the GPE. I translate results found in the KPZ literature to their corresponding dual in the ultra-cold Bose gas set-up. I find that this provides a new scaling relation which can be used to analytically identify the classical Kolmogorov -5/3 exponent and its anomalous correction. Moreover I estimate the anomalous correction to the scaling exponent of the compressible part of the kinetic energy spectrum of the Bose gas which is confirmed by numerical simulations of the GPE.