Kardar-Parisi-Zhang Interfaces with Inward Growth

We study the ($1+1$)-dimensional Kardar-Parisi-Zhang (KPZ) interfaces growing inward from ring-shaped initial conditions, experimentally and numerically, using growth of a turbulent state in liquid-crystal electroconvection and an off-lattice Eden model, respectively. To realize the ring initial condition experimentally, we introduce a holography-based technique that allows us to design the initial condition arbitrarily. Then, we find that fluctuation properties of ingrowing circular interfaces are distinct from those for the curved or circular KPZ subclass and, instead, are characterized by the flat subclass. More precisely, we find an asymptotic approach to the Tracy-Widom distribution for the Gaussian orthogonal ensemble and the ${\text{Airy}}_1$ spatial correlation, as long as time is much shorter than the characteristic time determined by the initial curvature. Near this characteristic time, deviation from the flat KPZ subclass is found, which can be explained in terms of the correlation length and the circumference. Our results indicate that the sign of the initial curvature has a crucial role in determining the universal distribution and correlation functions of the KPZ class.

Figure 1

(a) Schematic of the optical setup. Nd:YAG: neodymium-doped yttrium aluminum garnet, SLM: spatial light modulator, CCD: charge-coupled device camera, LED: light-emitting diode. The whole setup is placed in an isothermal chamber as in Ref. [10]. (b), (c) Images of DSM2, growing from a “KPZ” initial condition (b) and a ring with $R_0=1342\mu \mathrm{m}$ (c). For (c), the elapsed time after laser shooting is indicated below each image. The scale bar is 1 mm. The dotted line in the right figure indicates the estimated initial condition. See, also, Movies S1 and S2 [28].

Figure 2

Skewness and kurtosis. The values for ${\chi }_1$ and ${\chi }_2$ are shown by the dashed and dotted lines, respectively. (a) Experimental results. Statistical errors are indicated by the error bars, shown at the first and last data points. (b) Numerical results. The arrows indicate decreasing initial curvature.

Figure 3

Rescaled fluctuation properties. (a)–(d) Mean rescaled velocity $⟨p(x,t)⟩$ (a), (b) and the variance $⟨q(x,t{)}^2{⟩}_c$ (c), (d) for the LC experiment (a), (c) and the Eden model (b), (d). The statistical errors are shown by the error bars at the first and last data points in (a), (c). The confidence interval associated with the estimation of the nonuniversal parameters is indicated by the shaded area in (c), while it is smaller than the marker size in (a). The arrows in (b), (d) indicate decreasing initial curvature. The horizontal lines indicate the mean or variance of ${\chi }_1$ (dashed lines) and ${\chi }_2$ (dotted lines). (e), (f) The spatial correlation function $C_s(\zeta ,t)$ for the LC experiment with $R_0=1241\mu \mathrm{m}$ (e), and for the Eden model with $N=16000$ (f). The solid and dashed lines indicate the ${\text{Airy}}_1$ and ${\text{Airy}}_2$ correlation function, respectively.

Figure 4

The mean rescaled velocity $⟨p(x,t)⟩$ (a), (b) and the variance $⟨q(x,t{)}^2{⟩}_c$ (c), (d) against rescaled time $\tau =v_{\infty }t/R_0$ for the Eden numerics (a), (c) and for the LC experiment (b), (d). For the experimental data, the errors are indicated in the same way as Figs. 3 and 3. The black solid lines are the spline-curve fit of the Eden data with $t\ge 10^3$. The dashed lines show the mean or variance of ${\chi }_1$.