Research On Percolation Models Based On Monte Carlo Simulation

There is a unique phenomenon in nature,for example, dry coffer soaked, the flow of oil in the porous rock, forest fire, epidemic clisea.se, ore. The connon feature of such problems is that in these system (here exist two dislinet macro-state: permeable—impermeable, conductive—nonconductive. full forest fire—the fire is out after burning part of the forest, outbreaks of disease in all populations—stop after spreading in the part of population, and which macroscopic state will appear is controlled by a microscopic parameter (occupying probability or particle concentration). Because this kind of phenomena includes a process similar to that something gradually penetrating, it is named ’percolation’. And the established mathematical or physical models for studying percolation phenomena are collectively referred to as ’percolation model’In statistical physics point of view, the two macro-state of percolation mod-el can be considered as two phases of system, when the microscopic parameter p (occupation probability or concentration) gradually change and across a par-ticular value pc, system will change from one phase into another phase The above-mentioned pc can be understood as a phase transition point. Therefore, although initially the percolation model were proposed as a pure mathematical models, it was studied as a physical model later. These findings can help peo-ple more in-depth understanding of the physical essence of phase transitions and critical phenomena in nature. In general, although the rules of percolation model are usually simple, problems of percolation are not easy to study. So far. ex-act solutions obtained by analytical method are extremely rare, especially for three-dimensional percolation models. Under this background, numerical method becomes the most dependent research tool, in particular Monte Carlo simulation (Monte Carlo simulation). In this work we study the two of the most basic percolation models by means of Monte Carlo simulation (MC):isotropic percolation (IP) and directed perco-lation (DP). More specifically, we study the bond and the site percolation on three-dimensional simple cubic lattice (SC), and the directed bond and directed site percolation on (d+1)(d=2from7) dimensional simple cubic and body-centered cubic lattice (BCC). Due to that the numerical means only can research those systems with finite size, but the phrases’phase transition’or’phase transition point’are meaningful only for those infinite size system (thermody-namic limit), another tool-finite-size scaling (FSS) is applied.By extensive simulation and the FSS data analysis. this work obtain the improved estimates on percolation threshold pc, and part of our results for the high dimensional models are presented for the first time. Accuracy of the es-timates of critical exponents are also improved more or less. In the study of three-dimensional isotropic percolation, the critical behavior of a wide variety of sampled observables is verified. Among these quantities, the wrapping probabili-ties are the focus of our attention. In particular, our data shows that the leading correction exponent for some wrapping probabilities is approx-2, rather than the commonly believed universal value≈-1.1. In the study of directed percola-tion, a very efficient simulation scheme (proposed by P. Grassberger) is adopted. It is worth noting that the variance reduction method is extremely useful in high dimensions, which can significantly save CPU time and computing resources com-pared to the use of direct simulation, if we set the same target accuracy for them. In order to reduce, one fitting parameter, a dimensionless ratio Qt is defined and sampled, and is used to determine the percolation threshold pc. In addition,we study the probability distribution of two direct observations in DP:the occupied (wet) sites number N and a revised gyration radius R’ at time t. In each of the two probability distribution functions, there involves only one critical exponent.Finally, wo also study the geometric structure of percolation cluster. By de-composing the critical configuration of the bond percolation on periodic square lattice into three types:branches、junction, nonbridge, and deleting the branches. we obtain the so-called leaf-free configuration. The further deletion of junction-s from leaf-free configuration, we obtain the so-called bridge-free configuration. An analysis of the fractal dimension of these configuarations and the density of various types of bonds yieds some interesting results. For example, the deletion of all branches (the fraction≈43%of occupied boinds) don’t, alter the fractal dimension of cluster, and when system size L is large, both density of the bridges and nonbridges approach to1/4.