Scaling relations for contour lines of rough surfaces

Equilibrium and nonequilibrium growth phenomena, e.g., surface growth, generically yields self-affine distributions. Analysis of statistical properties of these distributions appears essential in understanding statistical mechanics of underlying phenomena. Here, we analyze scaling properties of the cumulative distribution of iso-height loops (i.e., contour lines) of rough self-affine surfaces in terms of loop area and system size. Inspired by the Coulomb gas methods, we find the generating function of the area of the loops. Interestingly, we find that, after sorting loops with respect to their perimeters, Zipf-like scaling relations hold for ranked loops. Numerical simulations are also provided in order to demonstrate the proposed scaling relations.

Figure 1

Small part of contour lines of a rough surface with size ${3000}^2$ and $H=0.5$; by zooming in on the picture one can see many small loops.

Figure 2

(Color online) Scaling relation for cumulative distribution of areas. Curves show $\frac{N_A}{A^{-d/2}}$ for rough surfaces with size ${4000}^2$ and different roughness exponents.

Figure 3

(Color online) Scaling relation for the coefficient of cumulative distribution of areas for rough surfaces with respect to system size for different values of $H$ as shown in the graph. Slops of the curves from top to bottom are given by $1.66\pm 0.04$, $1.60\pm 0.03$, $1.53\pm 0.03$, $1.46\pm 0.04$, and $1.37\pm 0.04$.

Figure 4

(Color online) Top: Moments $⟨A^p⟩$ of loop areas of rough surfaces versus system size; for $p=1$, 3, and 5. Here we have the exponents ${\theta }_1=2.02\pm 0.06$, ${\theta }_3=6.05\pm 0.55$, and ${\theta }_5=11.03\pm 1.06$. Bottom: ${\theta }_p$ versus $p$ for surfaces with size ${4000}^2$ with different roughness exponents $H$ as indicated on the graph.

Figure 5

(Color online) Top left $\left(H=0.3\right)$, right $\left(H=0.5\right)$, and bottom left $\left(H=0.3\right)$: for system size of $L=4000$, the curves of ranked loop perimeters divided by $n^{-D/d}$ vs rank numbers are shown for eight different realizations. Bottom right: the squares stand for the averaged $\xi$ (the numerical estimated exponent) over 200 realizations. The solid line shows the theoretical relation between $\xi$ and $H$, i.e., $\xi =\frac{D}{d}$.