Evidence for Geometry-Dependent Universal Fluctuations of the Kardar-Parisi-Zhang Interfaces in Liquid-Crystal Turbulence

Abstract

We provide a comprehensive report on scale-invariant fluctuations of growing interfaces in liquid-crystal turbulence, for which we recently found evidence that they belong to the Kardar-Parisi-Zhang (KPZ) universality class for 1+1 dimensions [Takeuchi and Sano in Phys. Rev. Lett. 104:230601, 2010; Takeuchi et al. in Sci. Rep. 1:34, 2011]. Here we investigate both circular and flat interfaces and report their statistics in detail. First we demonstrate that their fluctuations show not only the KPZ scaling exponents but beyond: they asymptotically share even the precise forms of the distribution function and the spatial correlation function in common with solvable models of the KPZ class, demonstrating also an intimate relation to random matrix theory. We then determine other statistical properties for which no exact theoretical predictions were made, in particular the temporal correlation function and the persistence probabilities. Experimental results on finite-time effects and extreme-value statistics are also presented. Throughout the paper, emphasis is put on how the universal statistical properties depend on the global geometry of the interfaces, i.e., whether the interfaces are circular or flat. We thereby corroborate the powerful yet geometry-dependent universality of the KPZ class, which governs growing interfaces driven out of equilibrium.

Appendix: Simulations of Dyson’s Brownian Motion

In Sect. 3.5 we have experimentally shown that the spatial two-point correlation of the circular and flat interfaces is indeed given, asymptotically, by that of the temporal correlation of the Airy₂ and Airy₁ processes, respectively. Theoretically, however, this correspondence is expected to be far beyond; the spatial profile of the interfaces itself is considered to be statistically equivalent to the locus of the Airy processes, or, for the curved interfaces, to that of the largest-eigenvalue dynamics in Dyson’s Brownian motion of GUE random matrices. Our analysis on spatial correlation of the interfaces may therefore shed light also on the temporal correlation of these stochastic processes. In this context, particularly interesting, and not investigated yet to our knowledge, is their temporal persistence property, which may be realized as the spatial persistence of the interfaces studied in Sect. 3.8. This correspondence might be mathematically not obvious, because the asymptotic equivalence in the moments of the Airy₂ process and those of the largest eigenvalue in Dyson’s Brownian motion for GUE has not been rigorously proved yet [10]. Here, performing direct simulations of Dyson’s Brownian motion, we shall indeed probe this bridge to the interface problem at the level of the persistence property, which should also help interpret our experimental results shown in Sect. 3.8.

Dyson’s Brownian motion is defined as the time evolution of the eigenvalues of a random matrix, taken from GUE or GOE here, whose independent elements exhibit uncorrelated Ornstein-Uhlenbeck processes [61]. Specifically, for an N×N Hermitian or symmetric matrix M, we consider the process

$$ \frac{\mathrm{d} M(t)}{\mathrm{d} t} = -\gamma M(t) + \varXi(t), $$

(31)

where γ is a constant scalar and Ξ(t) is a matrix with independent white-noise elements Ξ _ij (t) preserving the same symmetry as M; these matrix elements satisfy 〈Ξ _ij (t)〉=0, 〈Ξ _ii (t)Ξ _i′i′(t′)〉=δ _ii′ δ(t−t′) for the diagonal elements and $\langle[\operatorname{Re}\varXi_{ij}(t)][\operatorname{Re}\varXi _{i'j'}(t')] \rangle$ ($=\langle[\operatorname{Im}\varXi_{ij}(t)][\operatorname{Im}\varXi _{i'j'}(t')] \rangle$ for GUE) =(1/2)δ _ii′ δ _jj′ δ(t−t′) for the nondiagonal elements i>j. The one-point distribution for M(t) defined thereby remains to be that for GUE or GOE at any time t. We then focus on the largest eigenvalue λ ₁(t) of the matrix M(t), which is rescaled as

(32)

(33)

for GUE and GOE, respectively. The factors for this rescaling are determined, after Bornemann et al. [10], in such a way that λ _GUE(t′) and λ _GOE(t′) have the same values of the covariance and its derivative at zero as the Airy₂ and Airy₁ processes, respectively. Note that the factors for the GOE case are different from those used by Bornemann et al. [10] because of our somewhat unconventional definition for the Airy₁ process, which is however useful in the context of the growing interfaces (see footnote 8 on page 17).

We numerically integrate Eq. (31) using the standard numerical scheme for the Ornstein-Uhlenbeck process [33], which is exact for any finite time step Δt. We then compute the temporal persistence probability P _±(ζ) for the fluctuations of λ _GUE(t′) and λ _GOE(t′) as functions of a period ζ over which the sign of positive or negative fluctuations remains unchanged. Since the measurement of the persistence probability is influenced by the choice of the discrete time step Δt, we use in the following Δt=10⁻² N ^−1/3 and 10⁻³ N ^−1/3 and check that the results are not affected in a significant way. Concerning the other parameters, we fix γ=1 without loss of generality, and compare N=64 and 256 to confirm that our results reflect the property in the asymptotic limit N→∞.

Figure 21 shows the results obtained from more than 1000 and 50 independent simulations for N=64 and 256, respectively, both of length 10⁶ time steps. It clearly shows that, for both GUE and GOE and for both positive and negative fluctuations, the persistence probability decays exponentially within our statistical accuracy, P _±(ζ)∼exp(−κ _± ζ), as we have found for the spatial persistence of the interface fluctuations (Fig. 15). Concerning the coefficient κ _±, we find here

(34)

in the unit defined by Eqs. (32) and (33). This shows no significant difference between the positive and negative fluctuations for both cases, which is also confirmed from the ratio P ₋(ζ)/P ₊(ζ) [inset of Fig. 21(b) for GUE]. In contrast, the values of κ _± seem to be slightly different between GUE and GOE, but this turns out to result from the different normalizations of the time in Eqs. (32) and (33). Measuring the duration τ in the original time unit of Dyson’s Brownian motion, we find that the persistence probabilities P _±(τ) for GUE and GOE overlap reasonably well for both positive and negative fluctuations [insets of Fig. 21(c, d)].

Fig. 21

Temporal persistence probability P _±(ζ) for Dyson’s Brownian motion of GUE (a, b) and GOE (c, d) random matrices. The four series of colored symbols in the main panels correspond to different N and Δt as shown in the legend of the panel (a). The dashed lines are guides for the eyes indicating the estimated values of κ _± for each case. The inset of the panel (b) shows the ratio P ₋(ζ)/P ₊(ζ) for the four data series shown in the panels (a, b). The insets of the panels (c, d) display the data for both GUE (blue and green) and GOE (red and turquoise) with N=64 in the original time unit of Dyson’s Brownian motion, which overlap reasonably well within statistical accuracy. (Color figure online)

The appropriate rescaled time units used to define κ _± allow us to make a direct comparison to the experimental values $\kappa^{(\mathrm{s})}_{\pm}$ for the spatial persistence of the growing interfaces. For the circular case, we have found $\kappa^{(\mathrm{s})}_{+} = 1.07(8)$ and $\kappa^{(\mathrm{s})}_{-} = 0.87(6)$, to be compared with the GUE values κ ₊=0.90(8) and κ ₋=0.90(6). While the values for $\kappa^{(\mathrm{s})}_{-}$ and κ ₋ are in good agreement, we notice that those for $\kappa^{(\mathrm{s})}_{+}$ and κ ₊ seem to be slightly different. In particular, the apparent asymmetry between the positive and negative spatial persistence in the experiment is not reproduced in the temporal persistence of GUE Dyson’s Brownian motion. Two possibilities can be considered; our estimates for $\kappa^{(\mathrm{s})}_{+}$ and κ ₊ are not sufficiently precise and/or affected by finite-time effects and they actually take the same value, or the spatial profile of the circular interfaces and the locus of the largest eigenvalue in GUE Dyson’s Brownian motion are not equivalent at the level of the persistence property. One of the authors’ simulations of an off-lattice Eden model give $\kappa^{(\mathrm{s})}_{+} = 0.90(2)$ and $\kappa^{(\mathrm{s})}_{-} = 0.89(4)$ [92] and thus support the former possibility, but we do not single out either of them at present. This should be clarified by further study with better precision, preferably with numerical estimation of the persistence probability for the Airy processes, for which Bornemann’s method for evaluating Fredholm determinants [9, 10] could be utilized.