Percolation Transition in Fluids: Scaling Behavior of the Spanning Probability Functions

We examine different spanning probability functions (wrapping and crossing) near the percolation threshold of a supercritical square-well fluid and determine the threshold values of these probabilities, which may be universal for all fluids. It is shown that for a continuous system, over a wide range of system size, the wrapping probabilities can be described by universal scaling functions, whereas the crossing probabilities do not show such universal behavior over the same range of system size. The obtained universal functions for the wrapping probabilities can be used for an estimation of the percolation threshold in fluids in general. The results for the crossing probabilities allow us then to characterize large clusters in real fluids.

Figure 1

Wrapping probabilities $R_w^{(e)}$, $R_w^{(2)}$, and $R_w^{(3)}$ for the cluster which wraps the simulated system at least in any one, two, and three dimensions, respectively, as functions of the fluid density (left panel) and of the scaling variable $(\rho -{\rho }_c)L^{1/\nu }$ (right panel). Different curves and symbols for each spanning rule correspond to different system sizes. The percolation threshold (${\rho }_c\approx 0.2395$ at $T^{*}=1.2$) is indicated by the vertical line. Wrapping probabilities at the percolation threshold in an infinite system are shown by horizontal lines.

Figure 2

Wrapping probabilities $R_w^{(e)}$ (upper panel) and crossing probability $R_{\mathrm{cr}}^{(e)}$ (middle panel) at several densities (shown in legends) as a function of the inverse system size $L^{-1}$. The densities closest to the percolation threshold are marked by solid symbols. Crossing probabilities $R_{\mathrm{cr}}^{(e)}$, $R_{\mathrm{cr}}^{(2)}$, and $R_{\mathrm{cr}}^{(3)}$ at the percolation threshold are shown as a function of $L^{-1}$ in the lower panel. Symbols: simulated data; solid lines: fitting curves.

Figure 3

Crossing probabilities $R_{\mathrm{cr}}^{(e)}$, $R_{\mathrm{cr}}^{(2)}$, and $R_{\mathrm{cr}}^{(3)}$ for the cluster that crosses the simulated system at least in any one, two, and three dimensions (left panel), respectively, and their derivatives (right panel) as a function of the fluid density. The percolation threshold is shown by vertical lines. The crossing probabilities at the percolation threshold in an infinite system are shown by horizontal lines.