Direct Evidence for Universal Statistics of Stationary Kardar-Parisi-Zhang Interfaces

The nonequilibrium steady state of the one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) universality class has been studied in-depth by exact solutions, yet no direct experimental evidence of its characteristic statistical properties has been reported so far. This is arguably because, for an infinitely large system, infinitely long time is needed to reach such a stationary state and also to converge to the predicted universal behavior. Here we circumvent this problem in the experimental system of growing liquid-crystal turbulence, by generating an initial condition that possesses a long-range property expected for the KPZ stationary state. The resulting interface fluctuations clearly show characteristic properties of the 1D stationary KPZ interfaces, including the convergence to the Baik-Rains distribution. We also identify finite-time corrections to the KPZ scaling laws, which turn out to play a major role in the direct test of the stationary KPZ interfaces. This paves the way to explore unsolved properties of the stationary KPZ interfaces experimentally, making possible connections to nonlinear fluctuating hydrodynamics and quantum spin chains as recent studies unveiled relation to the stationary KPZ.

Figure 1

Typical snapshots of a flat (a) and a Brownian (b) interface, separating the metastable DSM1 (gray) and growing DSM2 regions (black). $h_{\mathrm{lab}}(x,t_{\mathrm{lab}})$ denotes the position of the upper interface in the laboratory frame, at time $t_{\mathrm{lab}}$ from the laser emission. $t$ and $h(x,t)$ are defined as follows: $t\equiv t_{\mathrm{lab}}$ and $h(x,t)\equiv h_{\mathrm{lab}}(x,t_{\mathrm{lab}})-⟨h(x,t_{\mathrm{lab}}^{\mathrm{init}}){⟩}_x$ for the flat case (a), $t\equiv t_{\mathrm{lab}}-t_{\mathrm{lab}}^{\mathrm{init}}$ and $h(x,t)\equiv h(x,t_{\mathrm{lab}})-h(x,t_{\mathrm{lab}}^{\mathrm{init}})$ for the Brownian case (b). See also Movies S1 and S2 [30].

Figure 2

Parameter estimation for the flat interfaces. (a) $d⟨h⟩/dt$ against $t^{-2/3}$ (main panel) and $⟨h^2{⟩}_c{t}^{-2/3}$ against $t^{-2/3}$ (inset). The dashed lines show the results of linear regression. (b) Histograms of the rescaled height $q(x,t)$ at different $t$ (legend). Agreement with the GOE Tracy-Widom (TW) distribution is confirmed. BR stands for the Baik-Rains distribution.

Figure 3

Evaluation of the Brownian interfaces. (a) $C_{h_{\mathrm{lab}}}(\ell ,t_{\mathrm{lab}})/\ell$ against $\ell$ for different $t_{\mathrm{lab}}$ (indicated in the legends) for the Brownian (main panel) and flat (inset) interfaces. The black stars indicate the results of direct evaluation of the computer-generated images used for the holograms. (b) $d⟨h⟩/dt$ against $t^{-4/3}$ (main panel) and $t^{-2/3}$ (inset) for the Brownian interfaces. The dashed line in the main panel shows the result of linear regression.

Figure 4

Main results of the Brownian interface experiments. (a) Histograms of the rescaled height $q(x,t)$ at different $t$ (legend). The data are found to converge to the Baik-Rains (BR) distribution, as shown quantitatively in Section 2 of the Supplemental Material and Fig. S2 [30]. GOE TW stands for the GOE Tracy-Widom distribution. (b) Two-point correlation function $g^{\prime \prime }(y)$. The experimental data are evaluated by $[\xi (t{)}^2/(\mathrm{\Gamma }t{)}^{2/3}]2⟨(\partial h_{\mathrm{lab}}/\partial x)(\ell +x,t+t_{\mathrm{lab}}^{\mathrm{init}})(\partial h_{\mathrm{lab}}/\partial x)(x,t_{\mathrm{lab}}^{\mathrm{init}})⟩$ with different $t$ (legend). The black curve indicates Prähofer and Spohn’s exact solution [40, 41]. (c) Rescaled two-time function $C_t(t_1,t_2)/C_t(t_2,t_2)$ for different $t_2$ (legend). The data are found to converge to Ferrari and Spohn’s exact solution [38] (black curve), as shown quantitatively in Section 3 of the Supplemental Material and Fig. S3 [30]. (d) Persistence probability $P_{\pm }(t_1,t_2)$ for different $t_1$ (legend). For visibility, $P_{-}(t_1,t_2)$ is shifted by factor 5. The dashed line is a guide for eyes indicating the power law $t^{-2/3}$ for ${\mathrm{FBM}}_{1/3}$. The inset shows $P_{\pm }(t_1,t_2)\mathrm{\Delta }{t}^{2/3}$.