Dynamic Scaling of Non-Euclidean Interfaces

Abstract

The dynamic scaling of curved interfaces presents features that are strikingly different from those of the planar ones. Spherical surfaces above one dimension are flat because the noise is irrelevant in such cases. Kinetic roughening is thus a one-dimensional phenomenon characterized by a marginal logarithmic amplitude of the fluctuations. Models characterized by a planar dynamical exponent z > 1, which include the most common stochastic growth equations, suffer a loss of correlation along the interface, and their dynamics reduce to that of the radial random deposition model in the long time limit. The consequences in several applications are discussed, and we conclude that it is necessary to reexamine some experimental results in which standard scaling analysis was applied.

Dynamics of curved interfaces

“…The rest of the paper is organized as follows: in section II we study growing radial interfaces and section III is concerned with spherical interfaces; in both cases we consider a broader range of relaxation mechanisms than in the previous work [23]. Section IV is devoted to different geometries of physical interest, which are introduced in this work for the sake of completeness, and finally, in section V we draw our main conclusions and discuss possible lines for future research.…”

“…3. The effect of stochasticity in this 1 + 1d system was estimated logarithmic on average [23]. This result was obtained by means of a small noise expansion of the radial random deposition equation at first order in the fluctuation intensity.…”

“…Since we are decomposing the solution in the small noise expansion (6) the statistical independence among the solution and the noise holds at every order. We know the value of the correlator [23] …”

“…Dilution refers to the fact that the already deposited material becomes distributed in a larger area, and it thus gets diluted inside the material being deposited at the time. Herein we will limit ourselves to large spatial scale properties of solutions to equations whose planar counterparts are linear, and we will generalize our previous results [23]. The numerical study of the Eden model in [18] suggests that dilution is not taking place in this particular model; using this result as a benchmark we will phenomenologically neglect dilution in the equations under consideration.…”

“…We will perform now a more detailed analysis of this phenomenon using Edwards-Wilkinson (EW) dynamics. The radial EW equation is the following Itô equation [16,17,23]…”