Exact Analysis of Level-Crossing Statistics for (d+1)-Dimensional Fluctuating Surfaces

Abstract

We carry out an exact analysis of the average frequency ν + αx i in the direction x i of positiveslope crossing of a given level α such that, h(x, t) −h = α, of growing surfaces in spatial dimension d.Here, h(x, t) is the surface height at time t, andh is its mean value. We analyze the problem when the surface growth dynamics is governed by the Kardar-Parisi-Zhang (KPZ) equation without surface tension, in the time regime prior to appearance of cusp singularities (sharp valleys), as well as in the random deposition (RD) model. The total number N + of such level-crossings with positive slope in all the directions is then shown to scale with time as t d/2 for both the KPZ equation and the RD model. PACS number(s): 52.75.Rx, 68.35.Ct.

Surface roughness analysis of hydrophilic SiO2/TiO2/glass nano bilayers by the level crossing approach

“…Where As introduced in Ref [25][26][27][28][29][30][31], we can also define the generalized total number of crossings with positive slope, ) (q N tot  as described by…”

“…In addition, this method was also employed in studying the fluctuations of other systems with scale dependent complexity, such as: fluctuations of velocity fields in Burgers turbulence [29], Kardar-Parisi-Zhang equation in (d + 1)-dimensions [30], stock market [31]. As a new application, we have used the level crossing analysis to describe the effect of etching time on the hydrophilic degrees of SiO 2 /TiO 2 nano bi-layer system.…”

Level Crossing Analysis of the Magnetohydrodynamic Equation in (1+1)-Dimension

“…Recently, it has been shown that starting from the flat interface the MHD equation will develop shock singularity after time scale t * , where t * depends on the forcing properties as t * D −1/3 0 σ. 24,25,29 This means that for time scales before t * the relaxation contribution tends to zero when υ → 0. In this regime one can observe that the generating function equation is closed.…”

“…In this regime one can observe that the generating function equation is closed. [25][26][27]29 Let us define the generating function as below:…”

Mapping stochastic processes onto complex networks

“…This implies that the reconstruction also preserves the cascade nature of the turbulence series in scale. Finally, we note that knowledge of the weights w ij enables us to determine the level-crossing frequency of the time series [22,23,24]. The level crossing is characterized by the quantity ν + α , which is the average frequency of positive-slope crossing of a level α in a time series.…”