Width and extremal height distributions of fluctuating interfaces with window boundary conditions

Abstract

We present a detailed study of squared local roughness (SLRDs) and local extremal height distributions (LEHDs), calculated in windows of lateral size $l$, for interfaces in several universality classes, in substrate dimensions $d_s=1$ and $d_s=2$. We show that their cumulants follow a Family-Vicsek type scaling, and, at early times, when $\xi \ll l$ ($\xi$ is the correlation length), the rescaled SLRDs are given by log-normal distributions, with their $n$th cumulant scaling as $(\xi /l{)}^{(n-1)d_s}$. This give rise to an interesting temporal scaling for such cumulants ${⟨w_n⟩}_c\sim t^{{\gamma }_n}$, with ${\gamma }_n=2n\beta +(n-1)d_s/z=\left[2n+(n-1)d_s/\alpha \right]\beta$. This scaling is analytically proved for the Edwards-Wilkinson (EW) and Random Deposition interfaces, and numerically confirmed for other classes. In general, it is featured by small corrections and, thus, it yields exponents ${\gamma }_n$'s (and, consequently, $\alpha$, $\beta$ and $z$) in nice agreement with their respective universality class. Thus, it is an useful framework for numerical and experimental investigations, where it is, usually, hard to estimate the dynamic $z$ and mainly the (global) roughness $\alpha$ exponents. The stationary (for $\xi \gg l$) SLRDs and LEHDs of Kardar-Parisi-Zhang (KPZ) class are also investigated and, for some models, strong finite-size corrections are found. However, we demonstrate that good evidences of their universality can be obtained through successive extrapolations of their cumulant ratios for long times and large $l$'s. We also show that SLRDs and LEHDs are the same for flat and curved KPZ interfaces.

11 pages, 10 figures, 4 tables

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Cited by in corpus (5)

• Initial pseudo-steady state & asymptotic KPZ universality in semiconductor on polymer deposition
• Crossover from compact to branched films in electrodeposition with surface diffusion
• Local average height distribution of fluctuating interfaces
• Hallmarks of the Kardar-Parisi-Zhang universality class elicited by scanning probe microscopy
• Height distributions in interface growth: The role of the averaging process