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

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-ParisiZhang 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.

Scaling properties of one-dimensional driven-dissipative condensates

“…This is compatible with our findings: in fact, in order to unambiguously reveal KPZ scaling in generic cases, relevant to experiments of exciton-polariton condensates, one needs to simulate large enough systems. Here we have simulated systems with sizes up to L = 2 17 and confirm from both the phase correlations (cf. Sec.…”

“…As we are interested in the parameter regime with both K c and u c being positive, the Benjamin-Feir unstable region is not relevant for the current investigation. However, it can be relevant if one is interested in turbulence of the bosonic fluid in the presence of external noise [17]. Moreover, a similar stochastic dynamical equation, the so-called stochastic Gross-Pitaevskii equation [18,19], is used to describe, e.g.…”

Functional renormalization group approach to the Kraichnan model

“…Nevertheless there is not still a quantitative prediction for the anomalous exponent of the structure functions and it may turn useful to address this question in a simpler setting. Let us stress that the FRG has been proved to be a solid framework in similar problems such as the scaling exponents of the Kardar-Parisi-Zhang equation and its correlation functions in the strong coupling regime [21,22] (see [23] for the Burgers' equation).…”

Non-thermal fixed point in a holographic superfluid

“…In the context of non-relativistic Bose gases as well as of relativistic scalar and gauge theories it was shown that superfluid turbulence, related to characteristic distributions of vortex defects [29,30] or more general non-linear excitations [31][32][33], can be interpreted in terms of non-thermal fixed points. 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 [34,35].…”

“…Within that framework, the scaling features have been interpreted in terms of the more general concept of non-thermal fixed points [25][26][27]. This concept associates stationary points in the time evolution of non-thermally scaling correlation functions with fixed points in a renormalisation-group sense [26,34,35]. In the vicinity of those fixed points, correlation functions assume spatial and temporal scaling behaviour.…”