Density correlations

We have investigated the homogeneity of the density, by means of the same tools already used (correlation functions for the measure of correlation dimension $d_2$): the Fig. fig:c_sc shows the previously defined function $C_{B(y,\Delta y)}(R)$ for stripes at different density.

It appears again a clustering effect, with a correlation dimension ranging from $1$ (homogeneous stripes) to $0.2$ (highly clusterized stripes). In the figure it is also shown the very small distance region, $R<\sigma_B$, where homogeneity should be recovered. Since in our simulation, $\Delta y \approx \sigma_B$, we expect $d(y)=2$ in this region.

Figure: Cumulated correlation function C(R), as defined in the text, measured along stripes at different heights for the 2D Inclined Channel Model. In the inset is displayed the normalized number density profile with the position of the choose stripes. Here $N=500$, $N_w \approx 56$, $r=0.95$, $r_w=0.95$, $g_x=1$, $g_y=-2$ (i.e.: the inclination angle $\phi =\pi /6$). The dashed lines represent the power-law fits, the vertical dot-dashed line represent the width of the stripes $\Delta y$