Anomalous scaling at non-thermal fixed points of Burgers' and Gross-Pitaevskii turbulence

Jan M. Pawlowski; Steven Mathey; Thomas Gasenzer

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Anomalous scaling at non-thermal fixed points of Burgers' and Gross-Pitaevskii turbulence

Steven Mathey, Thomas Gasenzer, Jan M. Pawlowski

Quantum Gases · Submitted 29 May 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this way, the canonical Kolmogorov exponent 5/3 for the energy cascade in superfluid turbulence is obtained analytically.

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arXiv ID1405.7652 (version 2)

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CitationJan M. Pawlowski, Steven Mathey, Thomas Gasenzer. "Anomalous scaling at non-thermal fixed points of Burgers' and Gross-Pitaevskii turbulence." arXiv:1405.7652 (2014). https://pith.science/paper/2BAPO7IQ

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Scaling in the dynamical properties of complex many-body systems has been of strong interest since turbulence phenomena became the subject of systematic mathematical studies. In this article, dynamical critical phenomena far from equilibrium are investigated with functional renormalisation group equations. The focus is set on scaling solutions of the stochastic driven-dissipative Burgers equation and their relation to solutions known in the literature for Burgers and Kardar-Parisi-Zhang dynamics. We furthermore relate superfluid as well as acoustic turbulence described by the Gross-Pitaevskii model to known analytic and numerical results for scaling solutions. In this way, the canonical Kolmogorov exponent 5/3 for the energy cascade in superfluid turbulence is obtained analytically. We also get first results for anomalous exponents of acoustic and quantum turbulence. These are consistent with existing experimental data. Our results should be relevant for future experiments with, e.g., exciton-polariton condensates in solid-state systems as well as with ultra-cold atomic gases.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Scaling in the dynamical properties of complex many-body systems has been of strong interest since turbulence phenomena became the subject of systematic mathematical studies.	background 1.00	0.03	0.02	asserted 1.00
s1	In this article, dynamical critical phenomena far from equilibrium are investigated with functional renormalisation group equations.	method 0.71	0.82	0.03	asserted 0.93
s2	The focus is set on scaling solutions of the stochastic driven-dissipative Burgers equation and their relation to solutions known in the literature for Burgers and Kardar-Parisi-Zhang dynamics.	contribution 0.91	0.44	0.06	asserted 0.79
s3	We furthermore relate superfluid as well as acoustic turbulence described by the Gross-Pitaevskii model to known analytic and numerical results for scaling solutions.	contribution 0.97	0.76	0.07	asserted 0.97
s4	In this way, the canonical Kolmogorov exponent 5/3 for the energy cascade in superfluid turbulence is obtained analytically.	result 1.00	0.90	0.33	asserted 1.00
s5	We also get first results for anomalous exponents of acoustic and quantum turbulence.	result 0.98	0.95	0.15	asserted 0.96
s6	These are consistent with existing experimental data.	interpretation 0.62	0.54	0.08	hedged possibility 0.58
s7	Our results should be relevant for future experiments with, e.g., exciton-polariton condensates in solid-state systems as well as with ultra-cold atomic gases.	future work 0.61	0.16	0.08	hedged possibility 0.85

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Original abstract

Scaling in the dynamical properties of complex many-body systems has been of strong interest since turbulence phenomena became the subject of systematic mathematical studies. In this article, dynamical critical phenomena far from equilibrium are investigated with functional renormalisation group equations. The focus is set on scaling solutions of the stochastic driven-dissipative Burgers equation and their relation to solutions known in the literature for Burgers and Kardar-Parisi-Zhang dynamics. We furthermore relate superfluid as well as acoustic turbulence described by the Gross-Pitaevskii model to known analytic and numerical results for scaling solutions. In this way, the canonical Kolmogorov exponent 5/3 for the energy cascade in superfluid turbulence is obtained analytically. We also get first results for anomalous exponents of acoustic and quantum turbulence. These are consistent with existing experimental data. Our results should be relevant for future experiments with, e.g., exciton-polariton condensates in solid-state systems as well as with ultra-cold atomic gases.

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States the same result 2 papers

It is shown that for k<2pi/l, the energy spectrum is very similar to the Kolmogorov's -5/3 law which is the most important statistical property of the conventional turbulence, where k is the wave number of the Fourier component of the velocity field and 1 the average intervortex spacing.

Energy Spectrum of Superfluid Turbulence without Normal Fluid · 0.82
The obtained energy spectrum is consistent with the Kolmogorov law.

Kolmogorov spectrum of superfluid turbulence: numerical analysis of the Gross-Pitaevskii equation with the small scale dissipation · 0.80

Papers citing it · 8 sentences without a confident label

Compares with it 1 paper

It is worth noting that scaling regimes similar to those found in the holographic superfluid have been observed already in the non-dissipative GP model, both in two and three dimensions [ref] .

Quantum Turbulence in a Three-Dimensional Holographic Superfluid · Scaling behavior for different initial conditions · 0.87

Cites it as background 7 papers

General universality classes of such non-thermal fixed points are expected to emerge from a renormalisation-group analysis which includes scaling in space and time [ref] .

Non-Thermal Fixed Point in a Holographic Superfluid · Introduction · 0.84
Thus, stochastic growth of roughening interfaces, for which the equation was initially introduced [ref] , shares common features with systems as dissimilar as, for instance, the Burgers equation in hydrodynamics [ref] , in random media [ref] , random matrices [ref] , the dynamics of Bose gases [ref] or active fluids [ref] .

KPZ equation with short-range correlated noise: emergent symmetries and non-universal observables · Introduction · 0.89
For related developments in the context of cosmology see also [ref] , for non-equilibrium fRG applications with a standard momentum cutoff see e.g. [ref] .

Flowing with the Temporal Renormalisation Group · Introduction · 1.00

What it cites · 86 sentences without a confident label

Uses the method or tool of 6 papers

In the following we apply the fixed point approach developed in [ref] for momentum and frequency dependent correlation functions.

Infrared behaviour and fixed points in Landau gauge QCD · Fixed Points of the Renormalisation-group flow · 0.98
We set up functional renormalisation group (RG) flow equations to look for non-perturbative fixed points, applying ideas from Refs. [ref] in the context of classical hydrodynamic turbulence [ref] .

Far-from-equilibrium quantum many-body dynamics · Introduction · 0.95
We set up functional renormalisation group (RG) flow equations to look for non-perturbative fixed points, applying ideas from Refs. [ref] in the context of classical hydrodynamic turbulence [ref] .

Renormalization Group Analysis of Turbulent Hydrodynamics · Introduction · 0.95

Uses data or values from 2 papers

Within the numerical precision it was found that $ζ_Q=4$ in $d=2$ and $ζ_Q=5$ in $d=3$ [ref] .

Nonthermal fixed points, vortex statistics, and superfluid turbulence in an ultracold Bose gas · Strong wave and quantum turbulence · 0.94
In order to make contact with scaling in acoustic turbulence in a superfluid, we insert the average literature values for $η_1$, as given in Table I. of Ref. [ref] , cf. also eta_range, which correspond to a forcing potential field delta-correlated in space ($β= 2$), into kin_energy and obtain $ϵ_kin(p) ∼p^-ξ$, with

Nonperturbative renormalization group for the stationary Kardar-Parisi-Zhang equation: scaling functions and amplitude ratios in 1+1, 2+1 and 3+1 dimensions · Acoustic turbulence in a superfluid · 1.00

Compares with 10 papers

We find that, for the direct energy cascade, the strong-wave-turbulence scaling [ref] of $E_P(p)$ is equivalent to the classical Kolmogorov law [ref] , with an intermittency correction $2η_P/3$.

Matter Wave Turbulence: Beyond Kinetic Scaling · Strong wave and quantum turbulence · 0.93
We compare the predictions epsilon_scaling with the scalings found in [ref] .

Nonthermal fixed points, vortex statistics, and superfluid turbulence in an ultracold Bose gas · Acoustic turbulence in a superfluid · 1.00
Furthermore, the perturbative results of [ref] were recovered, and the form of the scaling function $g(x)$ derived compared well with the exact results of Refs. [ref] .

The Kardar-Parisi-Zhang equation and universality class · Burgers and KPZ turbulence · 0.99

Cites as support 2 papers

This can be attributed to Galilean invariance which prohibits an anomalous scaling of the velocity field [ref] : Since the non-linearity of $E$ is part of the advective derivative of the fluid velocity, it must scale in the same way as the partial time derivative.

Nonperturbative renormalization group for the stationary Kardar-Parisi-Zhang equation: scaling functions and amplitude ratios in 1+1, 2+1 and 3+1 dimensions · Fixed-point equations · 0.98
This can be attributed to Galilean invariance which prohibits an anomalous scaling of the velocity field [ref] : Since the non-linearity of $E$ is part of the advective derivative of the fluid velocity, it must scale in the same way as the partial time derivative.

Asymptotic properties of turbulent magnetohydrodynamics · Fixed-point equations · 0.98

Cites as background 32 papers

We begin with a brief summary of the results of Ref. [ref] .

Matter Wave Turbulence: Beyond Kinetic Scaling · Strong wave and quantum turbulence · 0.83
A key result is that nearly degenerate Bose gases in $d=2,3$ dimensions, quenched parametrically close to the Bose-Einstein condensation (in $d=2$ BKT) transition, can evolve quickly to a quasi-stationary state exhibiting critical scaling [ref] and slowing-down behaviour [ref] .

Critical Dynamics of a Two-dimensional Superfluid near a Non-Thermal Fixed Point · Strong wave and quantum turbulence · 0.95
A key result is that nearly degenerate Bose gases in $d=2,3$ dimensions, quenched parametrically close to the Bose-Einstein condensation (in $d=2$ BKT) transition, can evolve quickly to a quasi-stationary state exhibiting critical scaling [ref] and slowing-down behaviour [ref] .

Nonthermal fixed points, vortex statistics, and superfluid turbulence in an ultracold Bose gas · Strong wave and quantum turbulence · 0.95

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