Strong anisotropy in two-dimensional surfaces with generic scale invariance: Nonlinear effects

Abstract

We expand a previous study [Phys. Rev. E 86, 051611 (2012)] on the conditions for occurrence of strong anisotropy (SA) in the scaling properties of two-dimensional surfaces displaying generic scale invariance. There, a natural Ansatz was proposed for SA, which arises naturally when analyzing data from e.g. thin-film production experiments. The Ansatz was tested in Gaussian (linear) models of surface dynamics and in non-linear models, like the Hwa-Kardar (HK) equation [Phys. Rev. Lett. 62, 1813], which are susceptible of accurate approximations through the former. In contrast, here we analyze non-linear equations for which such type of approximations fail. Working within generically-scale-invariant situations, and as representative case studies, we formulate and study a generalization of the HK equation for conserved dynamics, and reconsider well-known systems, such as the conserved and the non-conserved anisotropic Kardar-Parisi-Zhang equations. Through the combined use of Dynamic Renormalization Group analysis and direct numerical simulations, we conclude that the occurrence of SA in two-dimensional surfaces requires dynamics to be conserved. We find that, moreover, SA is not generic in parameter space but requires, rather, specific shapes from the terms appearing in the equation of motion, whose justification needs detailed information on the dynamical process that is being modeled in each particular case.

Gaussian statistics as an emergent symmetry of the stochastic scalar Burgers equation

“…(1), which are frequently employed to explore critical dynamics far from equilibrium [28], we show in this article that the statistics of the evolving fluctuating field can differ from expectations based on straightforward analysis of the symmetries of its "microscopic" description. Indeed, taking the celebrated 1D Burgers equation [30] and its scalar 2D generalizations [31,32] as representative cases, we find that Gaussian statistics are more robust than might have been expected when the large-scale behavior is controlled by a nonlinearity which breaks the updown symmetry. Specifically, the stochastic scalar Burgers equation reads [33,34]…”

“…The scaling exponents of Eqs. ( 3) and ( 4) have been investigated analytically [31][32][33]35] and numerically [32,37,38], and are collected in Table I. Note, HK scaling is anisotropic, hence the different exponent values along the x and y directions, while α x /z x = α y /z y = β [37].…”

Strong-coupling phases of the anisotropic Kardar-Parisi-Zhang equation

“…The NLO approximation [47] noticeably reduces the complexity of the flow equations but the loop-integrals in Eqs. (33) are still four-dimensional integrals and numerically cumbersome. However, for a qualitative picture of the phase diagram it is sufficient to consider only the flow of the scale dependent couplings and to set all flowing functions to one:…”

“…It was shown by Chen and coworkers that the order parameter for the dynamics of a driven Bose-Einstein condensate maps onto a compact form of the AKPZ equation [9,10]. In contrast to the standard KPZ equation however, where by means of an exact solution in one dimension [11][12][13][14][15][16][17] and extensive simulations in higher dimensions [18][19][20][21][22][23][24][25][26][27][28] the phase diagram is relatively well understood, much less effort was spent on the AKPZ equation and most of the studies are limited to the weak-coupling regime [8,[29][30][31][32][33]. An understanding of the AKPZ equation which includes the strong-coupling behavior was missing until now.…”

Characterization of the anisotropy of rough surfaces: Crossing statistics

“…18,19 Recently, Guillemot et al have introduced a regularity parameter to quantify the degree of anisotropy of periodic structures on rough surfaces. 20 Several methods are based on field renormalization 19,21,22 and a well-known approach is the heightheight correlation function measurement followed by the analysis of directional dependency of the roughness exponents. 18,22 Even though previous research provides appropriate tools to find the direction of anisotropy, in very few of them the possibility of shedding lights on the nature of the anisotropy is provided.…”