Connectivity and percolation in simulated grain-boundary networks

Random percolation theory is a common basis for modelling intergranular phenomena such as cracking, corrosion or diffusion. However, crystallographic constraints in real microstructures dictate that grain boundaries are not assembled at random. In this work a Monte Carlo method is used to construct physically realistic networks composed of high-angle grain boundaries that are susceptible to intergranular attack, as well as twin-variant boundaries that are damage resistant. When crystallographic constraints are enforced, the simulated networks exhibit triple-junction distributions that agree with experiment and reveal the non-random nature of grain-boundary connectivity. The percolation threshold has been determined for several constrained boundary networks and is substantially different from the classical result of percolation theory; compared with a randomly assembled network, about 50-75% more resistant boundaries are required to break up the network of susceptible boundaries. Triple-junction distributions are also shown to capture many details of the correlated percolation problem and to provide a simple means of ranking microstructures.     

Universal behavior of optimal paths in weighted networks with general disorder

We study the statistics of the optimal path in both random and scale-free networks, where weights are taken from a general distribution P(w). We find that different types of disorder lead to the same universal behavior. Specifically, we find that a single parameter (S defined as AL(-1/v) for d-dimensional lattices, and S defined as AN(-1/3) for random networks) determines the distributions of the optimal path length, including both strong and weak disorder regimes. Here v is the percolation connectivity exponent, and A depends on the percolation threshold and P(w). We show that for a uniform P(w), Poisson or Gaussian, the crossover from weak to strong does not occur, and only weak disorder exists.     

Comparison between fixed and Gaussian steplength in Monte Carlo simulations for diffusion processes

We analyze the different degrees of accuracy of two Monte Carlo methods for the simulation of one-dimensional diffusion processes with homogeneous or spatial dependent diffusion coefficient that we assume correctly described by a differential equation. The methods analyzed correspond to fixed and Gaussian steplengths. For a homogeneous diffusion coefficient it is known that the Gaussian steplength generates exact results at fixed time steps Δt. For spatial dependent diffusion coefficients the symmetric character of the Gaussian distribution introduces an error that increases with time. As an example, we consider a diffusion coefficient with constant gradient and show that the error is not present for fixed steplength with appropriate asymmetric jump probabilities.     

Influence of a uniform driving force on tracer diffusion in a one-dimensional hard-core lattice gas

The influence of a uniform driving force on tracer diffusion is investigated for a one-dimensional lattice gas where particles jump stochastically to unoccupied neighboring sites. A new, simple calculation is presented for the diffusion coefficient of a tracer particle with respect to its average drift, obtained recently by rigorous methods by De Masi and Ferrari. A theoretical expression describing the tracer particle mean square displacement approximately for all times is derived and found to be in excellent agreement with the results of Monte Carlo simulations.     

Conductivity exponent in three-dimensional percolation by diffusion based on molecular trajectory algorithm and blind-ant rules

Diffusion on random systems above and at their percolation threshold in three dimensions is carried out by a molecular trajectory method and a simple lattice random walk method, respectively. The classical regimes of diffusion on percolation near the threshold are observed in our simulations by both methods. Our Monte Carlo simulations by the simple lattice random walk method give the conductivity exponent μ/ν=2.32±0.02 for diffusion on the incipient infinite clusters and μ/ν=2.21±0.03 for diffusion on a percolating lattice above the threshold. However, while diffusion is performed by the molecular trajectory algorithm either on the incipient infinite clusters or on a percolating lattice above the threshold, the result is found to be μ/ν=2.26±0.02. In addition, it takes less time step for diffusion based on the molecular trajectory algorithm to reach the asymptotic limit comparing with the simple lattice random walk.     

Impurity transport in percolation media

An equation describing the impurity transport in a percolation medium is obtained and the inferences drawn from this equation are analyzed based on the scale invariance concept. A determining part in this analysis is allowance for the sinks inherent in such media. At distances shorter than the correlation length, the particles are transferred in the regime of subdiffusion; at large distances, the concentration asymptotics exhibits a characteristic “tail” shape. In the medium occurring in the state above the percolation threshold, the impurity transport over time periods longer than the characteristic time related to the correlation length is well described by the classical equation with a renormalized diffusion coefficient. In this case, the concentration tail has a Gaussian shape at moderate distances and tends to subdiffusion asymptotics at very long distances. A relation is established between the factor determining renormalization of the diffusion coefficient and the factor determining a decrease in the number of active impurity particles at large times.     

Target-Searching on Percolation

We study target-searching processes on a percolation, on which a hunter tracks a target by smelling odors it emits. The odor intensity is supposed to be inversely proportional to the distance it propagates. The Monte Carlo simulation is performed on a 2-dimensional bond-percolation above the threshold. Having no idea of the location of the target, the hunter determines its moves only by random attempts in each direction. For lager percolation connectivity p 〉 0.90, it reveals a scaling law for the searching time versus the distance to the position of the target. The scaling exponent is dependent on the sensitivity of the hunter. For smaller p, the scaling law is broken and the probability of finding out the target significantly reduces. The hunter seems trapped in the cluster of the percolation and can hardly reach the goal.     

Percolation thresholds and percolation conductivities of octagonal and dodecagonal quasicrystalline lattices

The octagonal and dodecagonal quaislattices were generated by means of the grid method. Monte Carlo simulation and cluster counting procedure were used for numerical determination of the site and bond percolation thresholds. Two types of connectivity called ferromagnetic and chemical were studied. The estimated site percolation thresholds are 0.5435… and 0.585… for octagonal lattice and 0.617… and 0.628… for dodecagonal lattice respectively. The obtained spanning fraction curves (for site percolation) seem to approach the 50% value at the percolation threshold. The site percolation conductivity for these lattices was studied by means of a transfer-matrix approach. The critical behavior was found to be the same as for the periodic lattices.     

Hybrid neural nets with poisson and gaussian connectivities

The dynamic behavior of neural nets with different patterns of interneuronal synaptic connectivity is investigated. Our method is based on probabilistic neural nets for the net structure and dynamics. Each net is divided into several different subsystems, which are characterized by different distribution laws for the number of connections that the neurons make. We start from the binomial distribution, which, under appropriate conditions, reduces to the Poisson and Gaussian distributions. The overall net now acquires a hybrid character. The expression for the neural activity is generalized to include this effect, and new expressions are derived, based on the isolated single-net equations. The dynamics of nets with sustained external inputs is also studied. The results obtained by this approach also show multiple stability and multiple hysteresis effects, as in the case of single nets. The differences between pure Poisson, Gaussian, and hybrid nets are explained in terms of the structural properties of the model. As expected, the hybrid case falls in between the two other distributions. Finally, we performed Monte Carlo computer calculations for the hybrid nets. For the range of parameters examined we find very good agreement with the developed formalism     

Gas diffusion in a pulmonary acinus model: experiments with hyperpolarized helium-3

Diffusion of hyperpolarized helium-3 in epoxy phantoms was experimentally studied by pulsed-gradient nuclear magnetic resonance (NMR). One phantom with a dichotomic branching structure densely filling a cubic volume was built using the Kitaoka algorithm to model a healthy human acinus. Two other phantoms, one with a different size and the other one with a partial destruction of the branched structure, were built to simulate changes occurring at the early stages of emphysema. Gas pressure and composition (mixture with nitrogen) were varied, thus exploring different diffusion regimes. Preliminary measurements in a cylindrical glass cell allowed us to calibrate the gradient intensity with 1% accuracy. Measurements of NMR signal attenuation due to gas diffusion were compared to a classical Gaussian model and to Monte Carlo simulations. In the slow diffusion regime, the Gaussian model was in reasonable agreement with experiments for low gradient intensity, but there was a significant systematic deviation at larger gradient intensity. An apparent diffusion coefficient Dapp was deduced, and in agreement with previous findings, a linear decrease of Dapp/D0 with D0(1/2) was observed, where D0 is the free diffusion coefficient. In the regime of intermediate diffusion, experimental data could be described by the Gaussian model for very small gradient intensities only. The corresponding Dapp/D0 values seemed to reach a constant value. Monte Carlo simulations were generally in fair agreement with the measurements in both regimes. Our results suggest that, for diffusion times typical of medical magnetic resonance imaging, an increase in alveolar size has more impact on signal attenuation than a partial destruction of the branched structure at equivalent surface-to-volume ratio.     

A dynamical mean field theory for the study of surface diffusion constants

We present a combined analytical and numerical approach based on the Mori projection operator formalism and Monte Carlo simulations to study surface diffusion within the lattice-gas model. In the present theory, the average jump rate and the susceptibility factor appearing are evaluated through Monte Carlo simulations, while the memory functions are approximated by the known results for a Langmuir gas model. This leads to a dynamical mean field theory (DMF) for collective diffusion, while approximate correlation effects beyond DMF are included for tracer diffusion. We apply our formalism to three very different strongly interacting systems and compare the results of the new approach with those of usual Monte Carlo simulations. We find that the combined approach works very well for collective diffusion, whereas for tracer diffusion the influence of interactions on the memory effects is more prominent.     

Monte Carlo simulation of the percolation process caused by the random sequential adsorption of k-mers on heterogeneous triangular lattices

Mixed site-bond percolation is studied for a random sequential adsorption process of k-mers on heterogeneous lattices with variable connectivity by means of Monte Carlo simulation. The percolation phase diagrams are built to render evidence about complex structures. Critical exponents are also calculated to show that the universality class corresponding to ordinary percolation in two dimensions is preserved.     

Static correlations of higher order and diffusion in an interacting lattice gas

For a lattice gas with extended hard core interaction on a square lattice the static correlation functions of higher order, which determine the average jump rate in the diffusion process, are calculated both by the Monte Carlo method and by analytic approximations. It is found that the superposition approximation is very inaccurate for the correlation functions of third and fourth order, but gives better results for the average jump rate. Up to concentrations ofc = 0.3 better consistency with the Monte Carlo data for both quantities is obtained by treating the site occupation numbers as Gaussian random variables and accordingly expressing the correlation functions of higher order by products of averages of two particle correlations. For concentrationsc > 0.3, however, a Bethe-Peierls cluster approximation is superior to the superposition approximation.The results of this paper were presented at the I.L.L, workshop Beyond Radial Distribution, Grenoble, July 15–16, 1985.     

Diffusion controlled growth of hydrogen clusters in metals

We show that a simple diffusion controlled growth, with subsequent aggregation, mean field theory accounts numerically for the growth of hydrogen clusters in certain metals (e.g. V, Nb, Zr, etc.). We also calculate the percolation threshold for the formation of an “infinite” aggregate both analytically and by a Monte Carlo simulation in a cubic lattice. The results are comparable and seem to correspond to the observed physical behavior.     

Depletion-induced percolation in networks of nanorods

Above a certain density threshold, suspensions of rodlike colloidal particles form system-spanning networks. Using Monte Carlo simulations, we investigate how the depletion forces caused by spherical particles affect these networks in isotropic suspensions of rods. Although the depletion forces are strongly anisotropic and favor alignment of the rods, the percolation threshold of the rods decreases significantly. The relative size of the effect increases with the aspect ratio of the rods. The structural changes induced in the suspension by the depletant are characterized in detail and the system is compared to an ideal fluid of freely interpenetrable rods.     

Size of largest and second largest cluster in random percolation

The number of sites in the second largest cluster is found to have a maximum at the percolation threshold where it is about one third and one sixth of the size of the largest cluster, in the simple cubic and triangular lattice, respectively. The correlation exponent η is slightly negative in three dimensions. These results are obtained from Monte Carlo simulations, extrapolated to infinite lattice sizes.     

Two-state random walk model of lattice diffusion. 1. Self-correlation function

Diffusion with interruptions (arising from localized oscillations, or traps, or mixing between jump diffusion and fluid-like diffusion, etc.) is a very general phenomenon. Its manifestations range from superionic conductance to the behaviour of hydrogen in metals. Based on a continuous-time random walk approach, we present a comprehensive two-state random walk model for the diffusion of a particle on a lattice, incorporating arbitrary holding-time distributions for both localized residence at the sites and inter-site flights, and also the correct first-waiting-time distributions. A synthesis is thus achieved of the two extremes of jump diffusion (zero flight time) and fluid-like diffusion (zero residence time). Various earlier models emerge as special cases of our theory. Among the noteworthy results obtained are: closed-form solutions (ind dimensions, and with arbitrary directional bias) for temporally uncorrelated jump diffusion and for the ‘fluid diffusion’ counterpart; a compact, general formula for the mean square displacement; the effects of a continuous spectrum of time scales in the holding-time distributions, etc. The dynamic mobility and the structure factor for ‘oscillatory diffusion’ are taken up in part 2.     

Shadowing function with single reflection from anisotropic Gaussian rough surface. Application to Gaussian, Lorentzian and sea correlations

In this paper, the monostatic (transmitter and receiver are located at the same place) and bistatic (transmitter and receiver are distinct) statistical shadowing functions from an anisotropic two-dimensional randomly rough surface are presented. This parameter is especially important in the case of grazing angles for computing the bistatic scattering coefficient in optical and microwave frequencies. The objective of this paper is to extend the previous work (Bourlier C, Berginc G and Saillard J 2002 Waves Random Media 12 145-74), valid for a one-dimensional surface, to a two-dimensional anistropic surface by considering a joint Gaussian process of surface slopes and heights separating two points of the surface. The monostatic average (statistical shadowing function average over the statistical variables) shadowing function is then performed in polar coordinates with respect to the incidence angle, the azimuthal direction and the surface height two-dimensional autocorrelation function. In addition, for a bistatic configuration, it depends on the incidence angle and azimuthal direction of the receiver. For Gaussian and Lorentzian correlation profiles and practically important power-type spectra such as the Pierson-Moskowitz sea roughness spectrum, the numerical solution, obtained from generating the surface Gaussian elevations (Monte Carlo method), is compared with the uncorrelated and correlated models. The results show that the correlation underestimates the shadow slightly, whereas the uncorrelated results weakly overpredict the shadow and are close to the numerical solution.     

Corrections to scaling for diffusion exponents on three-dimensional percolation systems at criticality

Recent results of Monte Carlo simulations of the ant-in-the-labyrinth method in three-dimensional percolation lattices are reanalyzed in the light of more accurate corrections to scaling ansatz, motivated by inconsistent results that have appeared in the literature. The results are observed to be sensitive to the form of the scaling correction terms. Using a single correction term, we estimate the valuek=0.197±0.004 for the anomalous diffusion exponent at criticality. When two correction terms are included,k=0.200±0.002 is obtained. These new estimates are consistent with known theoretical bounds, with recent series expansion results, and with numerical calculations of the conductance of random resistor networks above criticality.     

Diffusion correlation effects of molybdenum and silicon in molybdenum disilicide

Diffusion data for both principal directions of silicon and molybdenum as well as germanium are briefly summarised. Analysis is performed of the defect formation energies (available from previous ab initio calculations and experimental measurements) for diffusion mechanisms via home and foreign sublattices. The home sublattice mechanism is shown to be the preferred one for both silicon and molybdenum. Tracer correlation factors for silicon and molybdenum diffusion via sublattice vacancies in the respective sublattices of the tetragonal C11_b structure of molybdenum disilicide are calculated by a direct Monte Carlo simulation technique. Correlation factors for Si diffusion on its sublattice are compared with literature values that were calculated using a more complicated Monte Carlo method based on the matrix approach. It is shown that there is no need for this complicated approach and that the direct Monte Carlo simulation technique gives highly accurate correlation factors. Correlation factors and anisotropy ratios of vacancy-mediated diffusion in both sublattices are deduced and compared with experimental data. Tracer correlation in the tetragonal direction is shown to contribute 0.40?eV (i.e. over 55%) of the migration energy of the corresponding Si diffusivity. Two possible jump rates for Si diffusion are separately estimated. Mo diffusion correlation factors are calculated using the direct Monte Carlo technique. A comparison with experiment is made and the ratio of two possible jump rates is also estimated.