We investigate solutions, by using the Green function approach, for a system governed by a non-Markovian Fokker-Planck equation and subjected to a Comb structure. This structure consists of the axis of structure as the backbone and fingers which are attached perpendicular to the axis. For this system, we consider an arbitrary initial condition, in the presence of time dependent diffusion coefficients and spatial fractional derivative, and analyze the connection to the anomalous diffusion.

The long-range bond-percolation problem, on a linear chain (d = 1), in the presence of diluted sites (with an occupancy probability p s for an active site) is studied by means of a Monte Carlo simulation. The occupancy probability for a bond between two active sites i and j, separated by a distance r ij is given by p ij = <img width=32 height=32 id="_x0000_i1026" src="../../img/revistas/bjp/v33n3/a32img01.gif" align=absbottom>, where p represents the usual occupancy probability between nearest-neighbor sites. This model allows one to analyse the competition between long-range bonds (which enhance percolation) and diluted sites (which weaken percolation). By varying the parameter a (a > 0), one may find a crossover between a nonextensive regime and an extensive regime; in particular, the cases a = 0 and a ® ¥ represent, respectively, two well-known limits, namely, the mean-field (infinite-range bonds) and first-neighbor-bond limits. The percolation order parameter, P¥, was investigated numerically for different values of a and p s. Two characteristic values of a were found, which depend on the site-occupancy probability p s, namely, a1(p s) and a2(p s) (a2(p s) > a1(p s) > 0). The parameter P¥ equals unit, "p > 0, for 0 < a < a1(p s) and vanishes, "p < 1, for a > a2(p s). In the interval a1(p s) < a < a2(p s), the parameter P¥ displays a familiar behavior, i.e., 0 for p < p c(a) and finite otherwise. It is shown that both a1(p s) and a2(p s) decrease with the inclusion of diluted sites. For a fixed p s, it is shown that a convenient variable, p* º p*(p, a, N), may be defined in such a way that plots of P¥ versus p* collapse for different sizes and values of a in the nonextensive regime.

A method for obtaining critical properties of physical systems is presented. Based on a recursive relation involving a physical parameter of the system, it drives the system spontaneously to the critical point, providing an efficient way to estimate critical properties. The method is illustrated for several ferromagnetic Ising systems on well-known Bravais lattices. A finite-size scaling approach is performed, by applying the method on lattices of different sizes. The efficiency of the method is confirmed by evaluating critical temperatures, as well as critical exponents, that turn up to be in good agreement with those available in the literature, with a relatively small computational effort.