Random Bond Percolation Research Articles

Redefinition of site percolation in light of entropy and the second law of thermodynamics.

In this article, we revisit random site and bond percolation in a square lattice, focusing primarily on the behavior of entropy and the order parameter. In the case of traditional site percolation, we find that both the quantities are zero at p=0, revealing that the system is in the perfectly ordered and in the disordered state at the same time. Moreover, we find that entropy with 1-p, which is the equivalent counterpart of temperature, first increases and then decreases again, but we know that entropy with temperature cannot decrease. However, bond percolation does not suffer from either of these two problems. To overcome this, we propose an alternative definition for site percolation where we occupy sites to connect bonds and we measure cluster size by the number of bonds connected by occupied sites. This resolves all the problems without affecting any of the existing known results.

Explosive percolation on a scale-free multifractal weighted planar stochastic lattice.

In this article, we investigate explosive bond percolation (EBP) with the product rule, formally known as the Achlioptas process, on a scale-free multifractal weighted planar stochastic lattice. One of the key features of the EBP transition is the delay, compared to the corresponding random bond percolation (RBP), in the onset of the spanning cluster. However, when it happens, it happens so dramatically that initially it was believed, although ultimately proved wrong, that explosive percolation (EP) exhibits a first-order transition. In the case of EP, much effort has been devoted to resolving the issue of its order of transition and almost no effort has been devoted to finding the critical point, critical exponents, etc., to classify it into universality classes. This is in sharp contrast to the situation for classical random percolation. We do not even know all the exponents of EP for a regular planar lattice or for an Erdös-Renyi network. We first find the critical point p_{c} numerically and then obtain all the critical exponents, β, γ, and ν, as well as the Fisher exponent τ and the fractal dimension d_{f} of the spanning cluster. We also compare our results for EBP with those for RBP and find that all the exponents of EBP obey the same scaling relations as do those for RBP. Our findings suggest that EBP is not special except for the fact that the exponent β is unusually small compared to that for RBP.

Network Anatomy Controlling Abrupt-like Percolation Transition

We virtually dissect complex networks in order to understand their internal structure, just as doctors do with the bodies of animals. Our novel method classifies network links into four categories: bone, fat, cartilage, and muscle, based on network connectivity. We derive an efficient percolation strategy from this new viewpoint of network anatomy, which enables abrupt-like percolation transition through removal of a small amount of cartilage links, which play a crucial role in network connectivity. Furthermore, we find nontrivial scaling laws in the relationships between four types of links in each cluster and evaluate power exponents, which characterize network structures as seen in the real large-scale network of trading business firms and in the Erdős-Rényi network. Finally, we observe changes in the transition point for random bond percolation process, demonstrating that the addition of muscle links enhances network robustness, while fat links are irrelevant. These findings aid in controlling the percolation transition for an arbitrary network.

Critical Behaviors and Universality Classes of Percolation Phase Transitions on Two-Dimensional Square Lattice* *Supported by the National Natural Science Foundation of China under Grant No. 11121403

We have investigated both site and bond percolation on two-dimensional lattice under the random rule and the product rule respectively. With the random rule, sites or bonds are added randomly into the lattice. From two candidates picked randomly, the site or bond with the smaller size product of two connected clusters is added when the product rule is taken. Not only the size of the largest cluster but also its size jump are studied to characterize the universality class of percolation. The finite-size scaling forms of giant cluster size and size jump are proposed and used to determine the critical exponents of percolation from Monte Carlo data. It is found that the critical exponents of both size and size jump in random site percolation are equal to that in random bond percolation. With the random rule, site and bond percolation belong to the same universality class. We obtain the critical exponents of the site percolation under the product rule, which are different from that of both random percolation and the bond percolation under the product rule. The universality class of site percolation differs different from that of bond percolation when the product rule is used.

Two-dimensional percolation at the free water surface and its relation with the surface tension anomaly of water.

The percolation temperature of the lateral hydrogen bonding network of the molecules at the free water surface is determined by means of molecular dynamics computer simulation and identification of the truly interfacial molecules analysis for six different water models, including three, four, and five site ones. The results reveal that the lateral percolation temperature coincides with the point where the temperature derivative of the surface tension has a minimum. Hence, the anomalous temperature dependence of the water surface tension is explained by this percolation transition. It is also found that the hydrogen bonding structure of the water surface is largely model-independent at the percolation threshold; the molecules have, on average, 1.90 ± 0.07 hydrogen bonded surface neighbors. The distribution of the molecules according to the number of their hydrogen bonded neighbors at the percolation threshold also agrees very well for all the water models considered. Hydrogen bonding at the water surface can be well described in terms of the random bond percolation model, namely, by the assumptions that (i) every surface water molecule can form up to 3 hydrogen bonds with its lateral neighbors and (ii) the formation of these hydrogen bonds occurs independently from each other.

Explosive percolation on the Bethe lattice

Based on self-consistent equations of the order parameter P∞ and the mean cluster size S, we develop a self-consistent simulation method for arbitrary percolation on the Bethe lattice (infinite homogeneous Cayley tree). By applying the self-consistent simulation to well-known percolation models, random bond percolation, and bootstrap percolation, we obtain prototype functions for continuous and discontinuous phase transitions. By comparing key functions obtained from self-consistent simulations for Achlioptas models with a product rule and a sum rule to the prototype functions, we show that the percolation transition of Achlioptas models on the Bethe lattice is continuous regardless of details of growth rules.

Temperature dependence of the lateral hydrogen bonded clusters of molecules at the free water surface

Molecular dynamics simulation of the liquid–vapour interface of SPC/E water have been performed on the canonical (N,V,T) ensemble in twelve temperatures ranging from 300 to 500K using systems of two different sizes in order to locate the percolation threshold of the lateral hydrogen bonding network of surface waters. This infinite network, present at ambient conditions, is found to break up at about 450K, i.e., 200K below the critical temperature of the model, and hence the breakup of this two dimensional hydrogen bonding network at the liquid surface well precedes that of the three dimensional hydrogen bonding network in the bulk phase. We found that surface percolation can be described as a random bond percolation, i.e., all water molecules can form up to 3 lateral hydrogen bonds with their neighbours in the surface layer, and the formation or breaking of any of these possible hydrogen bonds is independent of that of all the other ones.

Monte Carlo study of percolation on disordered triangular lattices

A simple model for amorphous solids, consisting of a mixed bond triangular lattice with a fraction of attenuated bonds randomly distributed (which simulate the presence of defects in the surface), is studied here by using computational simulation. The degree of disorder of the surface is tunable by selecting the values of (1) the fraction of regular [attenuated] bonds ρ [ 1 − ρ ] ( 0 ≤ ρ ≤ 1 ) and (2) the factor r , which is defined as the ratio between the value of the conductivity associated to an attenuated bond and that corresponding to a regular bond ( 0 ≤ r ≤ 1 ). The results obtained show how the percolation properties of the disordered system are modified with respect to the standard random bond percolation problem ( r = 0 ).

Simple models for granular force networks

A remarkable feature of static granular matter is the distribution of force along intricate networks. Even regular inter-particle contact networks produce wildly inhomogeneous force networks where certain “chains” of particles carry forces far larger than the mean. Although the full information of each inter-particle contact requires analysis of the stress tensor, potentially including friction and elasticity, there is a rich tradition of approximating these interactions with simple, scalar models that focus on the geometry of the system. In this paper, we briefly review past theoretical approaches to understanding the geometry of force networks. We then investigate the structure of experimentally derived granular force networks using a simple algorithm to obtain corresponding graphs. We compare our observations with the results of geometric models, including random bond percolation, which show similar spatial distributions without enforcing vector force balance. Our findings suggest that some aspects of the mean geometry of granular force networks may be captured by these simple descriptions.

Percolation of polyatomic species with the presence of impurities

In this paper, the percolation of (a) linear segments of size k and (b) k-mers of different structures and forms deposited on a square lattice contaminated with previously adsorbed impurities have been studied. The contaminated or diluted lattice is built by randomly selecting a fraction of the elements of the lattice (either bonds or sites) which are considered forbidden for deposition. Results are obtained by extensive use of finite size scaling theory. Thus, in order to test the universality of the phase transition occurring in the system, the numerical values of the critical exponents were determined. The characteristic parameters of the percolation problem are dependent not only on the form and structure of the k-mers but also on the properties of the lattice where they are deposited. A phase diagram separating a percolating from a nonpercolating region is determined as a function of the parameters of the problem. A comparison between random site and random bond percolation in the presence of impurities on the lattice is presented.

Stress relaxation of near-critical gels.

The time-dependent stress relaxation for a Rouse model of a cross-linked polymer melt is completely determined by the spectrum of eigenvalues of the connectivity matrix. The latter has been computed analytically for a mean-field distribution of cross-links. It shows a Lifshitz tail for small eigenvalues and all concentrations below the percolation threshold, giving rise to a stretched exponential decay of the stress relaxation function in the sol phase. At the critical point the density of states is finite for small eigenvalues, resulting in a logarithmic divergence of the viscosity and an algebraic decay of the stress relaxation function. Numerical diagonalization of the connectivity matrix supports the analytical findings and has furthermore been applied to cluster statistics corresponding to random bond percolation in two and three dimensions.

Nonisotropic effective-medium approximation for diffusion problems in random media

We present the nonisotropic effective-medium approximation to solve diffusion problems in a two-dimensional anisotropic random media. The problem has been worked out by introducing a generalization of the well-known effective-medium approximation. A set of coupled nonlinear self-consistent equations must be solved to find the effective rates in each direction. We have considered (analytically) some particular models in short and large frequency limits. The dc conductivity is also compared against the isotropic case. The ac conductivity and Cole-Cole diagrams for the nonisotropic random bond percolation model have been analyzed in terms of the physical parameters that characterize the anisotropy and the disorder in the media. A monoparametric nonisotropic bond disordered model $(\ensuremath{\alpha}$ model) has also been worked out to show the applicability of the present approach in the context of weak or strong disorder. Such a model of disorder leads the system to show a quasi-one-dimensional behavior, proper of nonisotropic materials.

Percolation transition in two-dimensional ± J Ising spin glasses

The percolation properties of geometrical clusters are investigated for the Ising spin glasses in the square and triangular lattices with the asymmetric weights of ferromagnetic and antiferomagnetic bonds. By Monte Carlo simulation, we obtain the phase diagram of the percolation transition temperature as a function of the weight of ferromagnetic bonds. At each transition temperature, we estimate the critical exponents ν and γ which agree well with those exponents belonging to the universality class of the random bond percolation except for the pure ferromagnetic case.

In situ fluorescence experiments to test the reliability of random bond and site bond percolation models during sol-gel transition in free-radical crosslinking copolymerization

Sol-gel phase transition during the free-radical crosslinking copolymerization of methyl methacrylate and ethylene glycol dimethacrylate was studied using the steady-state fluorescence method. Copolymerization was performed both in the absence and presence of toluene at 75°C. Sol-gel phase transitions were monitored by observing the direct intensity of an excited pyrene methyl pivalate during in situ fluorescence experiments. Random bond and site bond percolation models were employed to quantify the results of bulk and solution polymerizations, respectively, during the sol-gel phase transition. Gel points were determined and the β exponents were found to be around 0.37 and 0.38 during gelation in bulk and solution polymerizations, respectively.

Fixed-cluster acceleration algorithm for spin systems.

A recently introduced cluster acceleration algorithm for spin models [J. Machta et al., Phys. Rev. Lett. 75, 2792 (1995)] based on invasion percolation is generalized to temperatures away from the critical point using concepts of random bond percolation. The generalized algorithm is equivalent to an optimized version of the Swendsen-Wang cluster algorithm [R. H. Swendsen and J. S. Wang, Phys. Rev. Lett. 58, 86 (1987)], and we demonstrate its success and speed at all temperatures for the $d=2$ Ising model. We argue that the previously discussed connection to self-organized criticality may be viewed as one limit of the fixed-cluster algorithm.

Crossover from percolation to self-organized criticality.

We include immunity against fire into the self-organized critical forest-fire model. When the immunity assumes a critical value, clusters of burnt trees are identical to percolation clusters of random bond percolation. As long as the immunity is below its critical value, the asymptotic critical exponents are those of the original self-organized critical model, i.e., the system performs a crossover from percolation to self-organized criticality. We present a scaling theory and computer simulation results.

Gelation rates of poly(vinyl alcohol) solution

Apparent gelation rates t −1 gel, defined as the reciprocal of the gelation time, are measured for poly(vinyl alcohol) solutions as functions of polymer concentration, C p, and quenching temperature, T Q. Here the measurements are limited to the case of homogeneous gelation where T Q is below 0°C. The critical gelation concentration, C∗ t = ∞ , extrapolated to t −1 gel = 0 is independent of T Q, and is identical to the polymer chain overlap concentration, C∗ Rg . In order to represent the degree of polymer chain overlap a reduced concentration, ø = (C p - C∗ Rg ) C∗ Rg , is introduced. An experimental relation, t −1 gel = A( T Q) ø 2, is obtained where A(T Q) is the temperature factor. The ø dependence of the equation can be explained using the chain overlap concept for gelation and random bond percolation theory, and the temperature factor A(T Q) is analysed by the crystal nucleation theory.

Nucleation near the spinodal in long-range Ising models

Properties of metastable long-range Ising models (LRIMs) are studied for deep quenches near the mean-field spinodal with Monte Carlo simulations using Glauber dynamics. The theory of spinodal-assisted nucleation is found to agree well with the data. Nucleating droplets are shown to have the same structure as large clusters in random long-range bond percolation.

Current distribution on a three-dimensional, bond-diluted, random-resistor network at the percolation threshold

Current and logarithm-current distributions on a three-dimensional random-bond percolation cubic network were studied at the percolation threshold by computer simulations. Predictions of a hierarchical model that combine fractal structure and randomness agree with our numerical simulations. In the thermodynamic limit the logarithm-current distribution exhibits ann(ln(i))∼i1/3 dependence below some characteristic currentic. This distribution may scale with lni/lnL, but the data are insufficient to make this a definite conclusion. Due to the small range of lnL considered, a study of the moments does not reveal this behavior and a study of the distribution itself is required.

The potts model on bethe lattices

We analyze the properties of theq-state ferromagnetic Potts model for realq. The nature of the phase transition at the critical point is first-order forq≠2, and second-order forq=2. The random-bond percolation limitq→1, and its second-order-like transition, are not related to the previous behaviour since they arise from non-stable phases of the system. It is suggested that this property characterizes the model on high-dimensional lattices, too.