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

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 renormalization-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 $\frac{5}{3}$ for the energy cascade in superfluid turbulence is obtained analytically. We also get 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 ultracold atomic gases.

Figure 1

Non-Gaussian fixed points: a graphical representation of the values of the bare exponents ${\alpha }_1$ and ${\alpha }_2$, defined in Eq. (42) in blue and red for $d=1$ and black and red for $d\ne 1$, which characterize the non-Gaussian fixed points. The scaling exponents of the bare stochastic Burgers equation, as described by $S\left[\mathbf{v}\right]$ [Eq. (10)], are related by ${\alpha }_1={\alpha }_2+4$ and are shown as a black line. The blue dot at $({\alpha }_1,{\alpha }_2)=(1,-2)$ corresponds to the fixed point investigated in Refs. [45, 48], while the top half of the blue line (for ${\alpha }_1>0$) represents the set of points found in Ref. [45] for different types of forcing, corresponding to a bare action with exponents given by the dotted black line (see the end of Sec. 4d2). In all dimensions we find a continuum of fixed points, and a further, new fixed point that may arise if the fluid is forced on large scales (red dot). Along the blue line we find that the scaling exponent satisfies ${\eta }_1=2-{\alpha }_1/2$ while it is ${\eta }_1=2-{\alpha }_1/2+d$ along the black line. ${\eta }_2$ is related to ${\eta }_1$ through Eq. (33).

Figure 2

The range of values of ${\eta }_1$ [defined in Eqs. (25) and related to ${\eta }_2$ in Eq. (33) and to $\chi$ and $\eta$ in Eqs. (29)], which correspond to UV-convergent non-Gaussian fixed points for different spatial dimensions $d$ (white area). The stripe at $d=1$ only applies to this single dimension. In the regions shaded in light and dark gray, any potential fixed point is UV divergent such that the RG flow integrals must be regularized in the UV. The blue dots correspond to the average literature values for the exponent ${\eta }_1$ (cf. Refs. [101, 102, 103, 104, 105, 106, 107]). Their values, as given in Table I of Ref. [48], are ${\eta }_1=1.5 \left(d=1\right),{\eta }_1=2.379\left(15\right) \left(d=2\right)$, and ${\eta }_1=3.300\left(12\right) \left(d=3\right)$. The vertical blue line marks the range of possible exponents found in Ref. [45] for different forcings, with the exponent $\beta$, defined in Eq. (2), chosen between 0 and 2. The dark gray shaded area marks the range of ${\eta }_1$ corresponding to the sets $({\alpha }_1,{\alpha }_2)$ shown as the blue and black solid lines at ${\alpha }_2=-2$ and 0 in Fig. 1, respectively. It continues accordingly also for ${\eta }_1>\frac{7}{2}$. In-between the two dotted lines there is a direct cascade of kinetic energy. In the black area at the left top, the dynamical critical exponent $z$ is negative. See Sec. 4d3 for further details.

Figure 3

The scaling exponent of ${\epsilon }_{\text{kin}}\left(p\right)\sim p^{-\xi }$ for $d=1$, for different values of $\beta$. The gray area corresponds to values excluded for UV-finite fixed points. The linearly rising part corresponds to the values of ${\eta }_1$ associated with the blue line of Fig. 1, and the saturation value to the upper blue dot [cf. Eq. (59)].

Figure 4

Acoustic turbulence in a $d=2$ (upper panel) and $d=3$ (lower panel) dilute superfluid Bose gas: shown are occupation-number spectra of the late-stage evolution of a closed system after an initial quench, briefly after the last vortex-antivortex pair $\left(d=2\right)$ or vortex ring $\left(d=3\right)$ has disappeared (cf. Fig. 15 of Ref. [40]). The distributions show a snapshot of a time evolution in a closed system. The blue solid lines correspond to the predictions obtained in this work, given in Eq. (68), the black solid lines mark the canonical scaling $n\left(p\right)\sim p^{-d-1}$. The latter scaling is expected on geometrical grounds, e.g., results from randomly distributed plane-wave density depression waves or solitons [40]. It is emphasized that the comparatively large deviations ($\sim p^{-2.8}$ for $d=2$ and $\sim p^{-3.6}$ for $d=3$) are found independently and contrast the small deviation from the canonical scalings (66) of quantum turbulent spectra in the presence of vortices. The radial momentum is given in lattice units $k={\left[2{\sum }_{i=1}^d{sin}^2(2\pi n_{i}a/L)\right]}^{1/2}$, with $n_i\in \mathbb{Z},-L/\left(2a\right)\le n_i\le L/\left(2a\right),a$ being the grid constant and $L$ its side length. $k_{\xi }=2sin[\pi /\left(2\xi \right)]$ is the momentum corresponding to the inverse healing length.