Study On The Properties Of Holes In Percolation Model

Percolation model,as a simple and practical physical model,is widely used in physics,chemistry,biology,materials and engineering.In granular materials,for example,the reconstruction of the complex geometry discrete particle,internal microstructure characterization of multiphase granular materials,internal diffusion and anomalous diffusion behavior of granular materials,can be studied by constructing a suitable percolation model.Monte Carlo method is a commonly used and effective method to study percolation models,for different types of percolation model,many efficient algorithms have been proposed by researchers.Based on the percolation model,we employed Monte Carlo method to study the size distributions of the largest hole and the stochastic processes in it,and the content is divided into five parts:Firstly,some basic concepts of phase transition and critical phenomena,classical statistical physical models,and related applications of percolation theory in materials science are briefly introduced in this paper.Secondly,we introduce the basic principle of Monte Carlo simulation and some commonly used simulation algorithms.Thirdly,we simulate the bond percolation on the planar square lattice with periodic boundary conditions,and study the size distributions of the largest hole in the largest percolation cluster and backbone,together with the largest cluster and backbone.In the vicinity of the critical point,we can locate the pseudocritical point by the value of skewness and kurtosis of these size distributions;According to these size distribution characteristics,we find that different types of wrapping clusters or holes contribute to the total distribution differently;For the largest hole in the largest cluster,due to the duality,its total distribution and subdistributions of different wrapping types exhibit good symmetry nearby the critical point;We also find that these size distribution functions are universal at critical point,and their distribution forms can approximately be considered as a combination of one Gaussian function and two Gumbel functions.Fourthly,we study the stochastic processes in the largest hole,including percolation process,random walk and self-avoiding walks.Especially for random walk,we study the random walk in the largest cluster,the largest cluster hole,the largest backbone and the largest backbone hole,which are similar to the diffusion process of particles in different fractal structure conditions,and according to the power law relationship between walking distance and time<R2>?t2vRW,we get the variation tendency of the random walk exponents vRW along with the time.For the percolation process or self-avoiding walks in the hole,we approximately determine the scaling relation of the length of self-avoiding walks versus the system size.Finally,we make a brief conclusion of this dissertation and present some prospects for future research.