The kirkwood-salsburg equations for random continuum percolation

We develop two different hierarchies of Kirkwood-Salsburg equations for the connectedness functions of random continuum percolation. These equations are derived by writing the Kirkwood-Salsburg equations for the distribution functions of thes-state continuum Potts model (CPM), taking thes1 limit, and forming appropriate linear combinations. The first hierarchy is satisfied by a subset of the connectedness functions used in previous studies. It gives rigorous, low-order bounds for the mean number of clusters n _c and the two-point connectedness function. The second hierarchy is a closed set of equations satisfied by the generalized blocking functions, each of which is defined by the probability that a given set of connections between particles is absent. These auxiliary functions are shown to be a natural basis for calculating the properties of continuum percolation models. They are the objects naturally occurring in integral equations for percolation theory. Also, the standard connectedness functions can be written as linear combinations of them. Using our second Kirkwood-Salsburg hierarchy, we show the existence of an infinite sequence of rigorous, upper and lower bounds for all the quantities describing random percolation, including the mean cluster size and mean number of clusters. These equations also provide a rigorous lower bound for the radius of convergence of the virial series for the mean number of clusters. Most of the results obtained here can be readily extended to percolation models on lattices, and to models with positive (repulsive) pair potentials.     

Impurity diffusion in a harmonic potential and nonhomogeneous temperature

Classical blockmodel is known as the simplest among models of networks with community structure. The model can be also seen as an extremely simply example of interconnected networks. For this reason, it is surprising that the percolation transition in the classical blockmodel has not been examined so far, although the phenomenon has been studied in a variety of much more complicated models of interconnected and multiplex networks. In this paper we derive the self-consistent equation for the size the global percolation cluster in the classical blockmodel. We also find the condition for percolation threshold which characterizes the emergence of the giant component. We show that the discussed percolation phenomenon may cause unexpected problems in a simple optimization process of the multilevel network construction. Numerical simulations confirm the correctness of our theoretical derivations.     

Statistical description of glass-forming alloys with chemical interaction: Application to Al-R systems

The statistical model for describing network-forming systems, developed in our previous works, is applied to study of metallic alloys with chemical bonding. The model is based on the representation of the sum of statistical weights over all possible configurations for a thermoreversible network in the form of a functional integral over a scalar field. The mean-field solution of the model is derived, and for particular case of a binary alloy having single element of chemical short-range order A₂B-type, thermodynamic and structural properties have been analyzed. This analysis allows to plot the temperature-concentration phase diagram of the model representing two immiscibility gap meeting in the distectic point. It is shown that at some temperatures and concentrations, geometry percolation of the network of chemical bonds and thus a sol-gel transition may take place. The critical percolation line was plotted in common with phase diagram. Then, the structural transitions, glass-forming ability and magnetic properties of Al-R alloys are discussed in the frames of this conception. It is proposed that the range of easy glass formation is confined on the left by the minimal concentration for the sol-gel transition and on the right by the concentration corresponding to the fractal-to-Euclidian crossover in the structure of percolation cluster. Finally, the abnormal growth of Al-REM magnetic susceptibility occurring above melting point of Al₂R compound is also explained.     

Network Forming Fluids: Yukawa Square-Well <Emphasis Type="Italic">m</Emphasis>-Point Model

Thermal and connectivity properties of the Yukawa square-well m-point (YSWmP) model of the network forming fluid are studied using solution of the multidensity Ornstein-Zernike and connectedness Ornstein-Zernike equations supplemented by the associative mean spherical approximation (AMSA). The model is represented by the multicomponent mixture of Yukawa hard spheres with m_s^am_{s}^{a} square-well sites, located on the surface of each hard sphere. To validate the accuracy of the theory, computer simulation is used to calculate the structure, thermodynamic and connectivity properties of the one-component YSW4P version of the model which is compared against corresponding theoretical data. In addition, connectivity properties of the model were studied using Flory-Stockmayer (FS) theory. Predictions of the AMSA for the thermal properties of the model (radial distribution functions (RDF), internal energy, pressure, fractions of the particles in different bonding states) are in good agreement with computer simulation predictions. Similarly, good agreement was found for the connectedness RDF (CRDF), except for the statepoints located close to the percolation threshold, where the theory fails to reproduce the long-range behavior of the CRDF. Results of both theories (AMSA and FS) for the mean cluster size are reasonably accurate only at low degrees of association. Predictions of the FS theory for the percolation lines are in a good agreement with computer simulation predictions. AMSA predictions of percolation are much less accurate, where corresponding percolation lines are located at a temperatures approximately 25% lower then those calculated using computer simulation.     

Surface phase transitions of multiple-site associating fluids

Surface phase transitions of Lennard–Jones (LJ) based two- and four-site associating fluids have been studied for various associating strengths using grand-canonical transition matrix Monte Carlo simulations. Our results suggest that, in the case of a smooth surface, represented by a LJ 9-3-type potential, multiple-site associating fluids display a prewetting transition within a certain temperature range. However, the range of the prewetting transition decreases with increasing associating strength and increasing number of sites on the fluid molecules. With the addition of associating sites on the surface, a quasi-2D vapor–liquid transition may appear, which is observed at a higher surface site density for weaker associating fluids. The prewetting transition at lower associating strength is found to shift towards the quasi-2D vapor–liquid transition with increasing surface site density. However, for highly associating fluids, the prewetting transition is still intact, but shifts slightly towards the lower temperature range. Adsorption isotherms, chemical potentials and density profiles are used to characterize surface phase transitions.     

On the maximal noise for stochastic and QCD travelling waves

Using the relation of a set of nonlinear Langevin equations to reaction–diffusion processes, we note the existence of a maximal strength of the noise for the stochastic travelling wave solutions of these equations. Its determination is obtained using the field-theoretical analysis of branching-annihilation random walks near the directed percolation transition. We study its consequence for the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation. For the related Langevin equation modeling the quantum chromodynamic nonlinear evolution of gluon density with rapidity, the physical maximal-noise limit may appear before the directed percolation transition, due to a shift in the travelling-wave speed. In this regime, an exact solution is known from a coalescence process. Universality and other open problems and applications are discussed in the outlook.     

Viscoelasticity of single wall carbon nanotube suspensions

We investigate the viscoelastic properties of an associating rigid rod network: aqueous suspensions of surfactant stabilized single wall carbon nanotubes (SWNTs). The SWNT suspensions exhibit a rigidity percolation transition with an onset of solidlike elasticity at a volume fraction of 0.0026; the percolation exponent is 2.3+/-0.1. At large strain, the solidlike samples show volume fraction dependent yielding. We develop a simple model to understand these rheological responses and show that the shear dependent stresses can be scaled onto a single master curve to obtain an internanotube interaction energy per bond approximately 40k(B)T. Our experimental observations suggest SWNTs in suspension form interconnected networks with bonds that freely rotate and resist stretching. Suspension elasticity originates from bonds between SWNTs rather than from the stiffness or stretching of individual SWNTs.     

Connectedness percolation of elongated hard particles in an external field

A theory is presented of how orienting fields and steric interactions conspire against the formation of a percolating network of, in some sense, connected elongated colloidal particles in fluid dispersions. We find that the network that forms above a critical loading breaks up again at higher loadings due to interaction-induced enhancement of the particle alignment. Upon approach of the percolation threshold, the cluster dimensions diverge with the same critical exponent parallel and perpendicular to the field direction, implying that connectedness percolation is not in the universality class of directed percolation.     

Evolution of a Magnetoresonance Line Near to a Limit of Percolation in Dilute Magnetics

In paper the results of numerical modeling of a magnetic resonance in dilute magnetics near to a threshold of a percolation are discussed. The classical equation of motion of magnetic moments is used in view of an exchange interaction such as RKKI and imitation of spin-phonon interaction by Monte-Carlo method. It is shown, that cluster structure of a magnetic and threshold of percolation are determined by critical distance, on which there is a change of a sign of an exchange interaction. In an examination of percolation phase transition the jump change of breadth of a line of a magnetic resonance is set, that can form the basis for experimental definition of a threshold percolation and parameters of an exchange interaction by methods of a radiospectroscopy.     

二分网上的靴襻渗流

靴襻渗流最早应用于统计物理学中研究磁铁因非磁性杂质导致磁有序的降低并最终消失的现象. 随着复杂网络研究的深入, 许多学者展开网络上的靴襻渗流研究. 在自然界中, 许多系统自然呈现出二分结构, 二分网络是复杂网络中的一种重要的网络模式. 本文通过建立动力学方程和计算机仿真模拟的方法研究二分网上的靴襻渗流, 关注的参数是二分网中两类节点初始的活跃比例和活跃阈值, 分别用f₁, f₂和Ω₁, Ω₂表示, 得到二分网两类节点终态活跃比例随初始活跃比例的变化会发生相变等结论. 同时 验证了动力学方程与仿真模拟的一致性.     

Erdös Rényi随机网络上爆炸渗流模型相变性质的数值模拟研究

基于改进的Newman和Ziff算法以及有限尺寸标度理论, 通过对表征渗流相变特征物理量的序参量、平均集团尺寸、二阶矩、标准偏差及尺寸不均匀性的数值模拟, 分析研究了Erdös Rényi随机网络上Achlioptas爆炸渗流模型的相变性质.研究表明: 尽管序参量表现出了不连续相变的特征, 但序参量以及其他特征物理量仍具有连续相变的幂律标度行为.因此严格地说, Erdös Rényi随机网络中的爆炸渗流相变是一种奇异相变, 它既不是标准的不连续相变, 又与常规随机渗流表现出的连续相变处于不同的普适类. 关键词： Erdös Rényi随机网络 爆炸渗流模型 相变 幂律标度行为     

Locally ordered regions in the phase transition in the systems with a finite-range correlated quenched disorder

The locally ordered regions (LOR) in the phase transition in disordered systems are studied. There are two parts in this paper. One part is to report our numerical results on the one-dimensional saddle point equation of the Ginzburg-Landau Hamiltonian with random temperature in the presence of an ordering field. The disordered system is modelled as a lattice, on which each cell has a local reduced temperature. The random part of the local reduced temperature is distributed in the Gaussian form. The one-dimensional saddle point equation is solved numerically. The average, the fluctuation and the correlation length of the solution are calculated. The scaling relations for these quantities with the temperature, the ordering field and the disorder strength are derived. The numerical data are fitted with the scaling relations well. Another part is to discuss qualitatively the phase diagram of the finite-range correlated disordered systems. There are two proposed classes for the phase transition in connection with the LOR. One class is described by the percolative scenario, in which the phase transition is inhomogeneous. In the percolative scenario the percolation of the LOR dominates the phase transition. In another class, the phase transition is homogeneous, and can be described by the renormalization group (RG) with replica symmetry breaking (RSB). In the RG with RSB, there is nothing to do with the percolation of LOR. We shall show that these two theories, which seem contradictory, may describe two parts of the whole phase diagram. Whether the phase transition is homogeneous or inhomogeneous depends on the interaction between the LOR. If the interaction between the LOR is strong enough, the phase transition is percolative and inhomogeneous. If the interaction between the LOR is weak, the phase transition is homogeneous. The interaction between the LOR is discussed with the numerical solution on the saddle point equation.     

Resonant localized states and quantum percolation on random chains with power-law-diluted long-range couplings

We investigate the nature of one-electron eigenstates in power-law-diluted chains for which the probability of occurrence of a bond between sites separated by a distance r decays as p(r) = p/r(1+σ). Using an exact diagonalization scheme and a phenomenological finite-size scaling analysis, we determine the quantum percolation transition phase diagram in the full parameter space (p,σ). We show that the density of states displays singularities at some resonance energies associated with degenerate eigenstates localized in a pair of sites with special symmetries. This model is shown to present an intermediate phase for which there is classical percolation but no quantum percolation. Quantum percolation only takes place for σ < 0.78, a value larger than the corresponding one for the Anderson transition in long-ranged coupled chains with random diagonal disorder. The fractality of critical wavefunctions is also characterized.     

Langevin dynamics of the glass forming polymer melt: Fluctuations around the random phase approximation

In this paper the Martin-Siggia-Rose (MSR) functional integral representation is used for the study of the Langevin dynamics of a polymer melt in terms of collective variables: mass density and response field density. The resulting generating functional (GF) takes into account fluctuations around the random phase approximation (RPA) up to an arbitrary order. The set of equations for the correlation and response functions is derived. It is generally shown that for cases whenever the fluctuation-dissipation theorem (FDT) holds we arrive at equations similar to those derived by Mori-Zwanzig. The case when FDT in the glassy phase is violated is also qualitatively considered and it is shown that this results in a smearing out of the ideal glass transition. The memory kernel is specified for the ideal glass transition as a sum of all “water-melon” diagrams. For the Gaussian chain model the explicit expression for the memory kernel was obtained and discussed in a qualitative link to the mode-coupling equation. Received: 9 January 1998 / Revised: 24 April 1998 / Accepted: 2 July 1998     

Statistical description of the sol-gel transition in systems with thermoreversible chemical bonds

A general statistical model is proposed for describing network-forming systems. The model is based on the representation of the partition function for all possible configurations of a thermoreversible network in the form of a functional integral over a scalar field. According to this model, two types of first-order phase transitions can occur in the systems under consideration: macroscopic phase separation with the structural phase transition due to the change in the configuration of the spatial network and the sol-gel transition due to the formation of a thermoreversible percolation cluster consisting of bound structural units. A detailed analysis is performed of the thermodynamic and structural properties of a solution of monomers that have f functional groups and can form thermoreversible chemical bonds. The influence of specific features of the chemical and volume interactions on the phase diagram of the system is investigated. The mutual position of the sol-gel transition line and the phase diagram is determined for different model parameters. It is revealed that two substantially different regimes of the behavior of the sol-gel transition line in the “temperature-volume fraction of structural units” plane are observed with a change in the rigidity of chemical bonds.     

Static and dynamic heterogeneities in a model for irreversible gelation

We study the structure and the dynamics in the formation of irreversible gels by means of molecular dynamics simulation of a model system where the gelation transition is due to the random percolation of permanent bonds between neighboring particles. We analyze the heterogeneities of the dynamics in terms of the fluctuations of the self-intermediate scattering functions: in the sol phase close to the percolation threshold, we find that this dynamic susceptibility increases with the time until it reaches a plateau. At the gelation threshold this plateau scales as a function of the wave vector k as k(eta-2), with eta being related to the decay of the percolation pair connectedness function. At the lowest wave vector, approaching the gelation threshold it diverges with the same exponent gamma as the mean cluster size. These findings suggest an alternative way of measuring critical exponents in a system undergoing chemical gelation.     

Long-range connections, quantum magnets and dilute contact processes

In this article, we briefly review the critical behaviour of a long-range percolation model in which any two sites are connected with a probability that falls off algebraically with the distance. The results of this percolation transition are used to describe the quantum phase transitions in a dilute transverse Ising model at the percolation threshold p_c of the long-range connected lattice. In the similar spirit, we propose a new model of a contact process defined on the same long-range diluted lattice and explore the transitions at p_c. The long-range nature of the percolation transition allows us to evaluate some critical exponents exactly in both the above models. Moreover, mean field theory is valid for a wide region of parameter space. In either case, the strength of Griffiths McCoy singularities are tunable as the range parameter is varied.     

Coupled channel T-operator equations with connected kernels: (I). Three-body problem with pairwise interactions

Previous work on T-operator coupled equations for two-channel systems is generalized and applied to the problem of three bodies interacting via pair potentials. Sets of coupled, integral equations for the two-body arrangement channel T-operators are derived using a channel coupling array W, and the connectedness properties of the kernels of these equations are discussed. It is shown that either disconnected or connected (iterated) kernels can be obtained by various choices of W. One particular realization of the coupled equations is seen to be similar but not identical to the Lovelace form of the Faddeev equations. Since the matrix form of the coupled equations is similar to the one-body Lippmann-Schwinger equation, the introduction of Møller wave operators is straightforward, and these are used to derive coupled integral equations for the channel state vectors.     

Phase transitions in hadronic matter

We formulate an equation of state for strongly interacting matter, which leads to a phase transition from massive resonance excitation to ideal gas behaviour. The structural similarity to the Van der Waals equation is discussed, as are extensions to describe hadron to quark matter transitions.     

Various velocity correlations functions in a Lorentz gas - simulation and mode coupling theory

We present computer simulation results for several types of velocity correlation function in the two dimensional, overlapping Lorentz gas. Only the normal velocity autocorrelation function, whose integral gives the diffusion constant, shows obvious anomalous behaviour at the percolation transition. The other functions are fairly well approximated by the Lorentz-Boltzmann equation, even for densities at which the travelling particle is trapped. We do, however, at a sub-percolation density, examine the long time behaviour of the autocorrelation function corresponding to the second rank, irreducible tensor of the velocity, and find an algebraic decay with an exponent of 3.0 ± 0.1, consistent with the theoretically expected value of 3. With these observations in mind we re-examine the mode coupling theory of Götze, Leutheusser and Yip (Phys. Rev. A 23 (1981) 2634,) replacing their one (frequency dependent) relaxation time approximation to a kinetic operator by a two (frequency dependent) relaxation time model. We find that this leads to a significantly better estimate of the diffusions constant at low density. Furthermore the theory correctly predicts no striking anomalous behaviour in the types of velocity correlation function that are unrelated to diffusion as the percolation threshold is crossed.