Dynamical crossover in invasion percolation

Abstract

The dynamical properties of the invasion percolation on the square lattice are investigated with an emphasis on the geometrical properties on the growing cluster of infected sites. The exterior frontier of this cluster forms a critical loop ensemble (CLE), whose length (l), the radius (r) and also roughness (w) fulfill the finite-size scaling hypothesis. The dynamical fractal dimension of the CLE defined as the exponent of the scaling relation between l and r is estimated to be Df = 1.76 ± 0.04. By studying the autocorrelation functions of these quantities we show importantly that there is a crossover between two time regimes, in which these functions change behavior from power-law at the small times, to exponential decay at long times. In the vicinity of this crossover time, these functions are estimated by log-normal functions. We also show that the increments of the considered statistical quantities, which are related to the random forces governing the dynamics of the observables undergo an anticorrelation/correlation transition at the time that the crossover takes place.

Invasion percolation in short-range and long-range disorder background

“…The process goes ahead until two opposite boundaries are touched by S(m max ). In the ordinary IP a phase transition occurs at this point to a phase where the invader fills the space, where IP shows power-law behavior [14]. This IP has the same properties as the ordinary site percolation at the critical point.…”

“…The holes of the system are obtained using the Hoshen Kopelman (HK) algorithm [18]. Importantly, using HK algorithm we extracted and analyzed the largest hole for which the fractal dimension is obtained to be D S f (ordinary IP) = 1.33 ± 0.01 as expected [14].…”

“…The other examples are three-dimensional IP [9], IP in correlated porous media [10,11], fractal growth dependence on the coordination number [12], and many applications in the * morteza.nattagh@gmail.com reservoir engineering [13]. Recently it was shown that, apart from the dynamical power-law behaviors, IP shows a dynamical crossover during which the autocorrelations change sign [14]. For a good review on the theoretical and experimental development of IP see [15].…”