Universal correlators and distributions as experimental signatures of (2 + 1)-dimensional Kardar-Parisi-Zhang growth

We examine height-height correlations in the transient growth regime of the 2 + 1 Kardar-Parisi-Zhang (KPZ) universality class, with a particular focus on the spatial covariance of the underlying two-point statistics, higher-dimensional analog of the 1 + 1 KPZ Class Airy1 process. Making comparison to AFM kinetic roughening data in 2d organic thin films, we use our universal 2 + 1 KPZ spatial covariance to extract key scaling parameters for this experimental system. Additionally, we explore the i) height, ii) local roughness, and iii) extreme value distributions characteristic of these oligomer films, finding compelling agreement in all instances with our numerical integration of the KPZ equation itself. Finally, investigating nonequilibrium relaxation phenomena exhibited by 2 + 1 KPZ Class models, we have unearthed a universal KPZ ageing kinetics. In experiments with ample data in the time domain, our 2 + 1 KPZ Euler temporal covariance will allow a quick, independent estimate of the central KPZ scaling parameter. editor’s choice Copyright c © EPLA, 2014 Introduction. – Recent years have witnessed spectacular advances [1] on both experimental and theoretical fronts in the nonequilibrium statistical mechanics of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions. On the experimental side, Takeuchi and coworkers [2] have verified in extraordinary detail that the stochastic, interfacial fluctuations in turbulent liquid crystals are governed by the Tracy-Widom (TW) limit distributions [3], known long ago to Prähofer and Spohn [4], as well as Johannson [5], in their seminal studies of the polynuclear and single-step growth models, canonical members of the KPZ universality class. Beyond the universal TW height fluctuations, these experiments also provided strong evidence for the underlying 2-pt spatial correlations, known in the flat geometry to be set by the covariance of the Airy1 process [6]. Impressive theoretical efforts emerged first on curved [7], then flat KPZ problems, the latter due to Calabrese and Le Doussal [8]. Shortly thereafter, Imamura and Sasamoto [9], in a technical tour de force, extracted the stationary-state statistics characteristic of the 1 + 1 KPZ Class, dictated by the Baik-Rains limit distribution [10]. Subsequent work on the KPZ stationary state by Takeuchi has established a precursor experimental signature [11], providing access to dynamics in this elusive KPZ regime. Numerical studies [12–14], invoking KPZ scaling theory [15], built upon the Krug-Meakin (KM) toolbox [16], and inspired by early efforts [17,18] to distill essential KPZ distributions, have done much to bolster our understanding of universality within the 1+1 KPZ Class; they inform the experimental findings and, furthermore, complement the powerful, but often quite model-specific mathematical developments. Here, we step up to the 2 + 1 KPZ Class, addressing the business of higher-dimensional stochastic growth. We begin with the Kardar-Parisi-Zhang equation [19], which characterizes the fluctuations of the height h(x, t) of a kinetically roughened interface: ∂th = ν∇h+ 1 2 λ(∇h) +√Dη, in which ν, λ and D are phenomenological parameters, the last setting the strength of the stochastic noise η. For the flat KPZ Class geometry in a system size of lateral dimension L, the width w of the fluctuating interface grows with time t as w(t, L) = √ (〈h2〉 − 〈h〉2) ∼ tF (L/t), where β, χ and z=χ/β are, respectively, the early-time roughness, saturation width, and dynamic exponents, and F the universal Family-Vicsek scaling function [20]. HalpinHealy [21,22] has made a large-scale numerical integration of the 2 + 1 KPZ equation, deep in the nonlinear regime (λ = 20), with L = 10 and exhaustive averaging now, with this work, reaching ∼103 runs. This 2 + 1