Critical exponents for the self-avoiding random walk in three dimensions

We compute by direct Monte Carlo simulation the main critical exponents, , ₄, andv and the effective coordination number for the self-avoiding random walk in three dimensions on a cubic lattice. We find both hyperscaling relationsdv=2– anddv– 2 ₄+=0 satisfied ind = 3.     

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.     

Molecular trajectory algorithm for random walks on percolation systems at criticality in two and three dimensions

Random walk simulations based on a molecular trajectory algorithm are performed on critical percolation clusters. The analysis of corrections to scaling is carried out. It has been found that the fractal dimension of the random walk on the incipient infinite cluster is d_w=2.873±0.008 in two dimensions and 3.78 ± 0.02 in three dimensions. If instead the diffusion is averaged over all clusters at the threshold not subject to the infinite restriction, the corresponding critical exponent k is found to be k=0.3307±0.0014 for two-dimensional space and 0.199 ± 0.002 for three-dimensional space. Moreover, in our simulations the asymptotic behaviors of local critical exponents are reached much earlier than in other numerical methods.     

Random alloy diffusion kinetics for the application to multicomponent alloy systems

In this paper, extensive Monte Carlo simulation results are reported on tracer and collective diffusion correlation effects in the random ternary alloy, as an example of a multicomponent alloy system. The problem of analytically describing both collective and tracer diffusion kinetics is also addressed for the random multicomponent alloy by application of a combination of the Manning theory and Holdsworth and Elliott theory. It is found that the overall results from the combined theory agree reasonably well with Monte Carlo results. This combined approach is much more accurate than Manning’s approach itself and much more manageable than the almost exact, but unfortunately difficult to use, self-consistent theory of Moleko, Allnatt and Allnatt. Some relations between the Onsager phenomenological coefficients and tracer diffusion coefficients are derived and are tested with our Monte Carlo data. Good agreement is found.     

Critical behavior of the specific heat for pure and site-diluted simple cubic Ising systems

The exponent of the specific heatC is determined for the pure and the site-diluted simple cubic Ising model (concentrationx=0, 0.2, 0.4 of nonmagnetic sites) by a finite-size scaling analysis of the peak value C_max(L) for systems of linear dimensionsL=8, 16, 32, and 64. The C_max values are obtained by the Ferrenberg-Swendsen algorithm, using Monte Carlo data from a fully-vectorized multi-spin coding program. We obtain =0.11 for x=0 and a crossover to a negative value upon dilution, with =–0.029(4) both forx=0.2 andx=0.4.     

Application of a radioactive tracer method for diffusion study in some liquids

In this paper a radioactive tracer technique based on sliding cell method developed in our laboratory for the study of diffusion in liquids is presented in detail. This method consists of radioactive and non-radioactive liquid columns of equal length and the radiation detector is placed in a vertical geometry over the diffusion column. Some liquid metals and aqueous electrolyte solutions were studied by this method. The data of liquid metals like mercury and gallium have been analyzed from the point of view of hard sphere model and those of electrolyte solutions from phenomenological theories. Onsager’s coefficientsL ₁₁,L ₁₂,L ₂₂ have been calculated from the experimental data and the variation of diffusion with solute concentration has been explained from ion-ion interaction.     

Monte Carlo evidence for a cusp in the uniform susceptibility of the site-diluted simple cubic Ising antiferromagnet

Monte Carlo simulations are performed for pure and site-diluted Ising antiferromagnets on a simple cubic lattice with up to 40³ sites and impurity concentrationx=0, 0.2. and 0.5. Forx=0.5 a cusp emerges for the temperature dependence of the uniform susceptibility at the critical temperature which is contrasted with the smooth behavior for a pure antiferromagnet, in agreement with the theoretical prediction of Fishman and Aharony.     

Monte Carlo simulation for temperature dependence of Ga diffusion length on GaAs(0 0 1)

Diffusion length of Ga on the GaAs(0 0 1)-(2×4)β2 is investigated by a newly developed Monte Carlo-based computational method. The new computational method incorporates chemical potential of Ga in the vapor phase and Ga migration potential on the reconstructed surface obtained by ab initio calculations; therefore we can investigate the adsorption, diffusion and desorption kinetics of adsorbate atoms on the surface. The calculated results imply that Ga diffusion length before desorption decreases exponentially with temperature because Ga surface lifetime decreases exponentially. Furthermore, Ga diffusion length L along and [1 1 0] on the GaAs(0 0 1)-(2×4)β2 are estimated to be and L_[110]200 nm, respectively, at the incorporation–desorption transition temperature (T860 K).     

Validation of practical diffusion approximation for virtual near infrared spectroscopy using a digital head phantom

Light propagation in the digital head phantom for virtual near infrared spectroscopy and imaging is calculated by diffusion theory. In theory, diffusion approximation is not valid in a low-scattering cerebrospinal fluid (CSF) layer around the brain. The optical path length and spatial sensitivity profile predicted by the finite element method based upon the diffusion theory are compared with those predicted by the Monte Carlo method to validate a practical implementation of diffusion approximation to light propagation in an adult head. The transport scattering coefficient of the CSF layer is varied from 0.01 to 1.0 mm⁻¹ to evaluate the influence of that layer on the error caused by diffusion approximation. The error is practically ignored and the geometry of the brain surface such as the sulcus structure in the digital head phantom scarcely affects the error when the transport scattering coefficient of the CSF layer is greater than 0.3 mm⁻¹.     

Surface diffusion of Pb clusters on Cu(1 1 1) influenced by size dependent cluster-substrate misfit: Monte Carlo tight binding simulation model

The migration of two-dimensional Pb clusters on Cu(1 1 1) surface is studied in the framework of three-dimensional (3D) continuum space tight binding (TB) computational model. Monte Carlo simulations based on many-body TB interactions reveal significant influence of cluster-substrate misfit on diffusion behavior of two-dimensional (2D) clusters. The analysis of pair distribution function (PDF) demonstrates that cluster lattice constant depends on the number of atoms N for 1 < N < 10. The observed effect is more pronounced for heteroepitaxial than homoepitaxial systems. It can be explained in the framework of model accounting for enhanced charge transfer in heteroepitaxy and strong influence of surface potential on cluster atomic arrangement. The variation of lattice constant leads to variable misfit which affects the island migration behavior. The results are discussed in a physical model that implies cluster diffusion with size dependent cluster-substrate misfit.     

Analysis of Light Propagation in a Realistic Head Model by a Hybrid Method for Optical Brain Function Measurement

Near infrared spectroscopy (NIRS) is used to measure the change in blood volume and oxygenation in the brain cortex induced by functional brain activation. The development of an adequate model to calculate light propagation in the head is very important because the light is strongly scattered in the tissue and this causes ambiguity in the volume of tissue interrogated with a source–detector pair of the NIRS instrument. In this study, a two-dimensional realistic head model is generated from a MRI scan of a human adult head. The light propagation in the head model is calculated by the hybrid Monte Carlo–diffusion method to obtain the change in detected intensity caused by a focal absorption change in the grey matter or in the white matter to discuss the relationship between the depth of the activated region and the sensitivity of the NIRS signal. The sensitivity to the activated region in the white matter steeply decreases with an increase of the depth of the activated region because the spatial sensitivity profile is mainly confined to the grey matter. The contribution of the focal brain activity to the NIRS signal is determined by not only the depth of the activated region from the head surface but also the depth of the activated region from the brain surface.     

Lifshitz' law for the volume of a two-dimensional droplet at zero temperature

We study a simple model of the zero-temperature stochastic dynamics for interfaces in two dimensions-essentially Glauber dynamics of the two-dimensional Ising model atT=0. Using elementary geometric considerations, we show that the (rescaled) volume of an initially square droplet decreases linearly to zero as a function of (rescaled) time.     

Monte Carlo simulation of many-arm star polymers in two-dimensional good solvents in the bulk and at a surface

A Monte Carlo technique is proposed for the simulation of statistical properties of many-arm star polymers on lattices. In this vectorizing algorithm, the length of each arml is increased by one, step by step, from a starting configuration withl=1 orl=2 which is generated directly. This procedure is carried out for a large sample (e.g., 100,000 configurations). As an application, we have studied self-avoiding stars on the square lattice with arm lengths up tol _max=125 and up tof=20 arms, both in the bulk and in the geometry where the center of the star is adsorbed on a repulsive surface. The total number of configurations, which behaves asNl ^{G–1 fl} , where=2.6386 is the usual effective coordination number for self-avoiding walks on the square lattice, is analyzed, and the resulting exponents G=(f) and _s (f) for the bulk and surface geometries are found to be compatible with predictions of Duplantier and Saleur based on conformai invariance methods. We also obtain distribution functions for the monomer density and the distance of the end of an arm from its center. The results are consistent with a scaling theory developed by us.     

Diffusion of interacting particles on a percolating lattice

The diffusion has been simulated by the Monte Carlo method on a random lattice. As in the ant in the labyrinth problem the particles move by stepping to allowed, randomly chosen neighboring fields. The particle interaction has been defined by the constraint that only one particle can occupy a site at a time. Biased diffusion means that one of the directions will be chosen with a greater probability than the others. It was shown that, with an increasing number of walkers, the displacement of the particles first of all increases to a maximum value and then decreases. This filling-up effect will not occur with small bias fields and on lattices with a high concentration of allowed sites.