Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals

We investigate growing interfaces of topological-defect turbulence in the electroconvection of nematic liquid crystals. The interfaces exhibit self-affine roughening characterized by both spatial and temporal scaling laws of the Kardar-Parisi-Zhang theory in $1+1$ dimensions. Moreover, we reveal that the distribution and the two-point correlation of the interface fluctuations are universal ones governed by the largest eigenvalue of random matrices. This provides quantitative experimental evidence of the universality prescribing detailed information of scale-invariant fluctuations.

Figure 1

Growing DSM2 cluster. (a) Images. Indicated below is the elapsed time after the emission of laser pulses. (b) Snapshots of the interfaces taken every 5 s in the range $2\mathrm{s}\le t\le 27\mathrm{s}$. The gray dashed circle shows the mean radius of all the droplets at $t=27\mathrm{s}$. The coordinate $x$ at this time is defined along this circle.

Figure 2

Scaling of the width $w(l,t)$ and the height-difference correlation function $C(l,t)$. (a, b) Raw data of $w(l,t)$ (a) and $C(l,t{)}^{1/2}$ (b) at different times $t$. The length scale $l$ is varied up to $2\pi ⟨R⟩$ and $\pi ⟨R⟩$, respectively. (c) Time evolution of the overall width $W(t)$ and the plateau level $C_{\mathrm{pl}}(t{)}^{1/2}$ of the correlation function. (d) Collapse of the data in (a) showing the Family-Vicsek scaling [Eq. (1)]. The dashed lines are guides for the eyes showing the KPZ scaling.

Figure 3

Parameter estimation. (a) Growth rate $d⟨R⟩/dt$ averaged over 1.0 s against $t^{-2/3}$. The $y$ intercept of the linear regression (dashed line) provides an estimate of $\lambda$. (b) $C(l,t)/l$ against $l$ for different times $t$. Inset: nominal estimates of $A$ obtained from $w(l,t)$ (blue bottom symbols) and $C(l,t)$ (green top symbols) as functions of $t$ (see text).

Figure 4

Local radius distributions. (a) Cumulants $⟨R^n{⟩}_c$ vs $t$. The dashed lines are guides for the eyes showing the indicated powers. (b) Skewness $⟨R^3{⟩}_c/⟨R^2{⟩}_c^{3/2}$ and kurtosis $⟨R^4{⟩}_c/⟨R^2{⟩}_c^2$. The dashed and dotted lines indicate the values of the skewness and the kurtosis of the GUE and GOE TW distributions. (c) Local radius distributions as functions of $q\equiv (R-\lambda t)/(A^2\lambda t/2{)}^{1/3}$. The dashed and dotted lines show the GUE and GOE TW distributions, respectively. (d) Differences in the cumulants of $q$ and ${\chi }_{\mathrm{GUE}}$. The dashed line indicates $⟨q^n{⟩}_c=⟨{\chi }_{\mathrm{GUE}}^n{⟩}_c$. Inset: the same data for $n=1$ in logarithmic scales. The dashed line is a guide for the eyes.

Figure 5

Two-point correlation function $C_2(l,t)$ plotted in the rescaled axes $u\equiv (Al/2)(A^2\lambda t/2{)}^{-2/3}$ and $C_2^{\prime }(u)\equiv C_2(l)/(A^2\lambda t/2{)}^{2/3}$. The dashed line indicates the theoretical prediction $g_2(u)$. Inset: integral of the rescaled correlation function $C_2^{\mathrm{int}}\equiv {\int }_0^{\infty }{C}_2^{\prime }(u)du$ as a function of time $t$. The difference from the theoretical counterpart $g_2^{\mathrm{int}}\equiv {\int }_0^{\infty }{g}_2(u)du$ is shown. The dashed line is a guide for the eyes.