随机多孔介质逾渗模型渗透率的临界标度性质

研究了一类非零键渗透率满足均匀分布的随机多孔介质逾渗模型-数值计算了该模型系统渗透率在临界点处的标度指数-结果表明该指数并不能看作是普适常数,而与均匀分布的参数有关-这意味着即使非零键渗透率值的概率密度函数满足负一阶矩存在条件,系统渗透率在逾渗临界点处的标度指数仍然依赖于分布函数的具体参数,并不是常数-这一数值结果与Sahimi对此问题的结论不同- 关键词： 逾渗 随机多孔介质 标度指数 渗透率     

Bootstrap percolation in a polluted environment

Let a low densityp of sites on the lattice Z² be occupied, remove a proportionq of them, and call the remaining sites empty. Then update this configuration in discrete time by iteration of the following synchronous rule: an empty site becomes occupied by contact with at least two occupied nearest neighbors, while occupied and removed sites nerver change their states. Ifq/p ² is large most sites remain unoccupied forever, while ifq/p ² is small, this dynamics eventually makes most sites occupied. This demonstrates how sensitive the usual bootstrap percolation rule (theq=0 case) is to the pollution of space.     

基于有向渗流理论的关联微博转发网络信息传播研究

微博网络的快速性、爆发性和时效性, 以及用户复杂的行为模式, 使得研究其信息传播模型及影响因素成为网络舆情的热点方向. 利用压缩映射定理, 分析不动点迭代过程的收敛条件, 得到有向网络信息传播过程的渗流阈值和巨出向分支的数值解法; 通过可变同配系数生成模型, 分析关联特征对信息传播的影响; 最后利用微博转发网络数据进行仿真对比实验. 结果表明: 虽然四类关联特征同时体现出同配、异配特征, 但信息传播结果更多受入度-入度相关性、入度-出度相关性影响; 通过删除少量节点的方法, 提取边同配比例, 验证大部分节点的四类关联特征呈现一致性.     

Infinite-scale percolation in a new type of branching diffusion process

We give an account and (basically) a solution of a new class of problems synthesizing percolation theory and branching diffusion processes. They lead to a novel type of stochastic process, namely branching processes with diffusion on the space of parameters distinguishing the branching particles from each other.On leave from L. D. Landau Institute for Theoretical Physics, Moscow, Russia.     

On the Fractal dimension and correlations in percolation theory

We discuss the fractal dimension of the infinite cluster at the percolation threshold. Using sealing theory and renormalization group we present an explicit expression for the two-point correlation function within percolation clusters. The fractal dimension is given by direct integration of this function.See especially Ref. 1 for a discussion of the general aspects of percolation.     

Discrete network models for the low-field Hall effect near a percolation threshold: Theory and simulations

The critical behavior of the weak-field Hall effect near a percolation threshold is studied with the help of two discrete random network models. Many finite realizations of such networks at the percolation threshold are produced and solved to yield the potentials at all sites. A new algorithm for doing that was developed that is based on the transfer matrix method. The site potentials are used to calculate the bulk effective Hall conductivity and Hall coefficient, as well as some other properties, such as the Ohmic conductivity, the size of the backbone, and the number of binodes. Scaling behavior for these quantities as power laws of the network size is determined and values of the critical exponents are found.School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel     

On the universality of geometrical and transport exponents of rigidity percolation

We develop a three-parameter position-space renormalization group method and investigate the universality of geometrical and transport exponents of rigidity (vector) percolation in two dimensions. To do this, we study site-bond percolation in which sites and bonds are randomly and independently occupied with probabilitiess andb, respectively. The global flow diagram of the renormalization transformation is obtained which shows that thegeometrical exponents of the rigid clusters in both site and bond percolation belong to the same universality class, and possibly that of random (scalar) percolation. However, if we use the same renormalization transformation to calculate the critical exponents of the elastic moduli of the system in bond and site percolation, we find them to be very different (although the corresponding values of the correlation length exponent are the same). This indicates that the critical exponent of the elastic moduli of rigidity percolation may not be universal, which is consistent with some of the recent numerical simulations.     

磁性颗粒复合体磁渗流区矫顽力异常的研究

制备了MnZn铁氧体/SiO₂颗粒复合体.研究了磁性颗粒复合体的有效磁导率μ、 比磁化强度σ以及矫顽力H_c随磁性颗粒含量的变化.研究发现，在MnZn铁氧体体积百分含 量为90%—98%的区域，复合体的有效磁导率μ的变化速率发生突变，出现磁渗流现象，从实验得到的体系磁渗流阈值V_c=97.9%.在磁渗流区，矫顽力表现出异常行为.结果表明 ，这种异常行为与复合体微观结构有着密切关系.在磁渗流前，矫顽力H_c的变化主要来 源于磁 关键词： 颗粒复合体 磁渗流 矫顽力     

Hypergeometric series in a series expansion of the directed-bond percolation probability on the square lattice

The asymmetric directed-bond percolation (ADBP) problem with an asymmetry parameterk is introduced and some rigorous results are given concerning a series expansion of the percolation probability on the square lattice. It is shown that the first correction term,d _n,1 (k) is expressed by Gauss' hypergeometric series with a variablek. Since the ADBP includes the ordinary directed bond percolation as a special case withk=1, our results give another proof for the Baxter-Guttmann's conjecture thatd _n,1(1) is given by the Catalan number, which was recently proved by Bousquet-Mélou. Direct calculations on finite lattices are performed and combining them with the present results determines the first 14 terms of the series expansion for percolation probability of the ADBP on the square lattice. The analysis byDlog Padé approximations suggests that the critical value depends onk, while asymmetry does not change the critical exponent of percolation probability.     

Fractal structure of equipotential curves on a continuum percolation model

We numerically investigate the electric potential distribution over a two-dimensional continuum percolation model between the electrodes. The model consists of overlapped conductive particles on the background with an infinitesimal conductivity. Using the finite difference method, we solve the generalized Laplace equation and show that in the potential distribution, there appear quasi-equipotential clusters   which approximately and locally have the same values as steps and stairs. Since the quasi-equipotential clusters have the fractal structure, we compute the fractal dimension of equipotential curves and its dependence on the volume fraction over [0,1]$[0,1]$. The fractal dimension in [1.00, 1.246] has a peak at the percolation threshold _pc$p_c$.     

Critical behavior of a dynamical percolation model

The critical behavior of the dynamical percolation model, which realizes the molecular-aggregation conception and describes the crossover between the hadronic phase and the partonic phase, is studied in detail. The critical percolation distance for this model is obtained by using the probability P∞ of the appearance of an infinite cluster. Utilizing the finite-size scaling method the critical exponents γ/v and T are extracted from the distribution of the average cluster size and cluster number density. The influences of two model related factors, I.e. The maximum bond number and the definition of the infinite cluster, on the critical behavior are found to be small.     

The existence of phase V in the Mandelbrot percolation process

Dekking and Meester defined six phases for a subclass of random Cantor sets consisting of those generated by Bernoulli random substitutions. They proved that the random Sierpinski carpet passed through all these phases asp tended from 0 to 1, but the were not able to prove the existencne of phase V in the Mandelbrot percolation process. In this paper, we accomplish the proof by improving their methods.Research supported by the Chinese Natural Science Foundation.     

Critical behavior of a dynamical percolation model

The critical behavior of the dynamical percolation model,which realizes the molecular-aggregation conception and describes the crossover between the hadronic phase and the partonic phase,is studied in detail. The critical percolation distance for this model is obtained by using the probability P∞ of the appearance of an infinite cluster. Utilizing the finite-size scaling method the critical exponents γ/ν and τ are extracted from the distribution of the average cluster size and cluster number density. The influences of two model related factors,i.e. the maximum bond number and the definition of the infinite cluster,on the critical behavior are found to be small.     

Trapping,percolation, and anomalous diffusion of particles in a two-dimensional random field

We analyze from first principles the advection of particles in a velocity field with HamiltonianH(x, y)=¯ V ₁ y–¯ V ₂ x+W ₁ (y)-W ₂ (x), whereW _i , i=1, 2, are random functions with stationary, independent increments. In the absence of molecular diffusion, the particle dynamics are very sensitive to the streamline topology, which depends on the mean-to-fluctuations ratio=max(|¯V₁¦/; ¦¯V₂|/), with =|W ₁ |^21/2=rms fluctuations. Remarkably, the model is exactly solvable for 1 and well suited for Monte Carlo simulations for all , providing a nice setting for studying seminumerically the influence of streamline topology on large-scale transport. First, we consider the statistics of streamlines for=0, deriving power laws for p_nc(L) and (L), which are, respectively, the escape probability and the length of escaping trajectories for a box of sizeL, L » 1. We also obtain a characterization of the statistical topography of the HamiltonianH. Second, we study the large-scale transport of advected particles with > 0. For 0 < < 1, a fraction of particles is trapped in closed field lines and another fraction undergoes unbounded motions; while for 1 all particles evolve in open streamlines. The fluctuations of the free particle positions about their mean is studied in terms of the normalized variablest ^– v/2[x(t)–x(t)] andt ^–v/2 [y(t)-(t)]. The large-scale motions are shown to be either Fickian (=1), or superdiffusive (=3/2) with a non-Gaussian coarse-grained probability, according to the direction of the mean velocity relative to the underlying lattice. These results are obtained analytically for 1 and extended to the regime 0<<1 by Monte Carlo simulations. Moreover, we show that the effective diffusivity blows up for resonant values of ) for which stagnation regions in the flow exist. We compare the results with existing predictions on the topology of streamlines based on percolation theory, as well as with mean-field calculations of effective diffusivities. The simulations are carried out with a CM 200 massively parallel computer with 8192 SIMD processors.     

About the fastest growth of the Order Parameter in models of percolation

The growth of the average size 〈_smax〉$〈s_{max}〉$ of the largest component at the percolation threshold _pc(N)$p_c\left(N\right)$ on a graph of size N$N$ has been defined as 〈_smax(_pc(N),N)〉∼N^χ$〈s_{max}(p_c\left(N\right),N)〉\sim N^{\chi }$. Here we argue that the precise value of the ‘growth exponent’ χ$\chi$ indicates the nature of percolation transition; χ<1$\chi <1$ or χ=1$\chi =1$ determines if the transition is continuous or discontinuous. We show that a related exponent η=1−χ$\eta =1-\chi$ which describes how the average maximal jump sizes in the Order Parameter decays on increasing the system size, is the single exponent that describes the finite-size scaling of a number of distributions related to the fastest growth of the Order Parameter in these problems. Excellent quality scaling analysis are presented for the two single peak distributions corresponding to the Order Parameters at the two ends of the maximal jump, the bimodal distribution constructed by the weighted average of these distributions and for the distribution of the maximal jump in the Order Parameter.     

An alternative proof of some percolation threshold (p c ) using the idea of self-organized criticality

Summary By means of a well-developed method in self-organized criticality, we can obtain the lower bound for the percolation threshold (p _c) of the corresponding site percolation problem. In some special cases, we have proved that such lower bounds are indeed the percolation thresholds. We can reproduce some well-known percolation thresholds of various lattices including the Cayley trees and Kock curves in this framework.     

Quantum to classical crossover under dephasing effects in a two-dimensional percolation model

Scaling theory predicts complete localization in d = 2 in quantum systems belonging to the orthogonal class(i.e., with timereversal symmetry and spin-rotation symmetry). The conductance g behaves as g^exp(-L/l) with system size L and localization length l in the strong disorder limit. However, classical systems can always have metallic states in which Ohm’s law shows a constant g in d=2. We study a two-dimensional quantum percolation model by controlling dephasing effects. The numerical investigation of g aims at simulating a quantum-to-classical percolation evolution. An unexpected metallic phase, where g increases with L, generates immense interest before the system becomes completely classical. Furthermore, the analysis of the scaling plot of g indicates a metal-insulator crossover.     

On the universality of crossing probabilities in two-dimensional percolation

Six percolation models in two dimensions are studied: percolation by sites and by bonds on square, hexagonal, and triangular lattices. Rectangles of widtha and heightb are superimposed on the lattices and four functions, representing the probabilities of certain crossings from one interval to another on the sides, are measured numerically as functions of the ratioa/b. In the limits set by the sample size and by the conventions and on the range of the ratioa/b measured, the four functions coincide for the six models. We conclude that the values of the four functions can be used as coordinates of the renormalization-group fixed point.     

On the phase transition to sheet percolation in random Cantor sets

Thed-dimensional random Cantor set is a generalization of the classical middle-thirds Cantor set. Starting with the unit cube [0, 1] ^d , at every stage of the construction we divide each cube remaining intoM ^d equal subcubes, and select each of these at random with probabilityp. The resulting limit set is a random fractal, which may be crossed by paths or (d–1)-dimensional sheets. We examine the critical probabilityp _s(M, d) marking the existence of these sheet crossings, and show that p_s(M,d)1–p_c(M ^d) asM, where p_c(M ^d) is the critical probability of site percolation on the lattice (M ^d) obtained by adding the diagonal edges to the hypercubic lattice ^d. This result is then used to show that, at least for sufficiently large values ofM, the phases corresponding to the existence of path and sheet crossings are distinct.