Monte Carlo Work Research Articles

Monte Carlo simulations of Ising models

Choosing from the plenitude of recent Monte Carlo simulations of Ising models, results on dilute systems and metamagnets are presented. In general, good Monte Carlo work may require highly accurate data, careful analyses of a variety of aspects, including finite size effects, and comparison to suitable analytic theories and experiments. Depending on the type of problem, it is shown that one or the other of these points may deserve special care.

Testing for monotonicity in the intensity of a nonhomogeneous Poisson process

Based on the times of occurrences of a nonhomogeneous Poisson process in a fixed time interval, a pseudo-likelihood ratio test of the null hypothesis that the intensity is nondecreasing with the alternative that it is not nondecreasing is developed. The modifications needed to test for a nonincreasing intensity are also presented. For these statistics, the collection of constant intensities is shown to be least favorable within the collection of nondecreasing intensities. For constant intensities, a suitable approximation to the conditional distribution of the likelihood ratio statistics, given the number of occurrences, is obtained and the approximate conditional tests are studied. Through Monte Carlo work certain types of alternatives are identified which require a large number of occurrences for the likelihood ratio test to detect with reasonable probability. Some inferences about nonhomogeneous Poisson processes are based on a researcher's a priori belief that the intensity is nondecreasing (nonincreasing). These tests provide a statistical method for assessing the validity of the researcher's belief.

New series expansion data for surface and bulk resistivity and conductivity in two-dimensional percolation

Low-density series expansions are obtained for random resistor networks. The expansions are obtained on the square lattice to order 18 and on the triangular lattice to order 14. Bulk and surface expansions are given for both resistive and conductive susceptibilities. The balance of the evidence obtained from analysing these series is in favour of the value for the critical exponent of the resistance scale and supports the existence of only a single such scale for bulk and surface susceptibilities. This value is in agreement with earlier Monte Carlo work.

An efficient algorithm for choosing scattering directions in Monte Carlo work with arbitrary phase functions

This paper describes an efficient Monte Carlo algorithm for choosing a new direction of a photon after a scattering interaction. The algorithm chooses a scattering angle by linear interpolation in a table of the inverse cumulative scattering probability. A Legendre expansion of the phase function makes it easy to apply Clenshaw's algorithm to build the interpolation table. The points in the table are close enough together that linear interpolation is accurate. With a table of 100,000 entries, we can keep the absolute and relative errors in matching the probability distribution below 10 −5.

Multiple-histogram Monte Carlo study of the Ising antiferromagnet on a stacked triangular lattice.

The nearest-neighbor Ising antiferromagnet on a stacked triangular lattice is a frustrated cooperative system in which it is known that at least two long-range ordered states exist at low temperature. This model has also been of considerable interest as it is known to be a reasonable description of two antiferromagnetic insulators, ${\mathrm{CsCoBr}}_{3}$ and ${\mathrm{CsCoCl}}_{3}$. It has also been the subject of previous theoretical and simulation studies which have yielded conflicting results for the critical phenomena displayed near the transition from the paramagnetic to the high-temperature ordered phase. We have carried out a detailed Monte Carlo study of this system using the recently developed multiple-histogram technique and finite-size scaling analysis, with the purpose of extracting estimates for the critical exponents relevant to this continuous transition. Our results give \ensuremath{\beta}=0.311(4), \ensuremath{\gamma}=1.43(3), \ensuremath{\alpha}=-0.05(3), and \ensuremath{\nu}=0.685(3) which are not in agreement with previous Monte Carlo work. In addition, although they are close to the expectations from previous symmetry arguments, there are systematic differences between our results and these theoretical predictions. A possible interpretation of these Monte Carlo exponent estimates is that they do not correspond to those calculated for any known universality class, and add to the growing number of simple models of interacting spins, in which geometrical frustration is relevant, which appear to exhibit novel critical behavior. Finally, we have examined the evolution of real-space spin configurations and have seen that a buildup of correlations between anti-phase-domain walls, or solitons, along the stacking direction precedes the transition, an observation which is consistent with recent neutron-scattering measurements on ${\mathrm{CsCoBr}}_{3}$.

Testing regression disturbances for normality with stable alternatives: further monte carlo evidence

This paper extends previous Monte Carlo work on testing for normality of ordinary least squares regression residuals. In addition to considering variation in sample size, this study also considers ...

On a transient transport theory applied to a GaAs/Al xGa 1− xAs heterojunction

A linear quantum theory of transient transport is applied to a GaAs/AlGaAs heterojunction at 77 K. The electron energy subband structure is included and transient transport is investigated in the electric quantum limit with electron interactions in the presence of impurities and an electric field. The further inclusion of polar optical phonons as well as ionized impurity scattering is made. The results of the present theory are compared with a Monte Carlo work and to a more sophisticated non-linear approach for an electric field of E = 1 kV cm −1.

Linearity of ability-performance relationships: A reconfirmation.

Previous research on the nature of ability-performance relationships has found the relationship to be linear. Recent Monte Carlo work suggests that the test of the significance of the difference between the Pearson correlation and the correlation ratio (eta), used in previous research, has low power to detect nonlinear relationships. Using 174 studies of the relationship between 9 scales of the General Aptitude Test Battery and job performance, with a mean sample size of 210, and using a power polynomial approach, which has been shown to have higher power than the r vs. eta test, the issue of linearity was reexamined

Orbital partitioning errors in transition metal relativistic effective potentials

The large errors in atomic excitation energies of the first transition elements (seen in SCF calculations using Ar-core relativistic effective potentials) can be rationalized in terms of the partitioning of the 4s orbital as required to form nodeless pseudoorbitals. We are able to show that with the partitioning revised to better represent the 4s electron density (and therefore critical coulombic interactions) in the region dominated by the 3s subshell, errors can be dramatically reduced. Such potentials would make large transition metal cluster calculations simpler and might be appropriate for quantum Monte Carlo work.

Three-way metric unfolding via alternating weighted least squares

Three-way unfolding was developed by DeSarbo (1978) and reported in DeSarbo and Carroll (1980, 1981) as a new model to accommodate the analysis of two-mode three-way data (e.g., nonsymmetric proximities for stimulus objects collected over time) and three-mode, three-way data (e.g., subjects rendering preference judgments for various stimuli in different usage occasions or situations). This paper presents a revised objective function and new algorithm which attempt to prevent the common type of degenerate solutions encountered in typical unfolding analysis. We begin with an introduction of the problem and a review of three-way unfolding. The three-way unfolding model, weighted objective function, and new algorithm are presented. Monte Carlo work via a fractional factorial experimental design is described investigating the effect of several data and model factors on overall algorithm performance. Finally, three applications of the methodology are reported illustrating the flexibility and robustness of the procedure.

Orthogonal triplet probes: an efficient method for unbiased estimation of length and surface of objects with unknown orientation in space.

If the axis of anisotropy of oriented, tubular or lamellar objects is unknown, the unbiased stereological estimation of length density and surface density (Lv and Sv) requires counts on sections with isotropic uniform random (IUR) orientation. It is shown theoretically that in homogeneous, anisotropic specimens the precision of Lv and Sv estimation is considerably augmented if IUR-oriented sets of three mutually perpendicular sections (orthogonal triplet probes = ortrips) are used instead of three directionally independent IUR sections. The mechanism of variance reduction results from a negative covariance between sections within 'ortrips' and corresponds to the antithetic variate principle of Monte Carlo work. Heterogeneity decreases the efficiency of the ortrip method, but this effect can often be counteracted by systematic sampling of ortrips within specimens. Practical estimation of length and surface area of the highly anisotropic, tubular myocardial capillaries per tissue volume in the left ventricles of eight normal, adult, male perfusion-fixed Wistar rats provided estimates of excellent precision with CEs of 3.3% (Lv) and 2.1% (Sv) of the group mean. The method will hopefully allow stereologists dealing with arbitrary anisotropic structures to apply the same simple, efficient, and unbiased sampling designs that have long been used in the study of liver, lung, and kidney tissue.

A Monte Carlo primer for health physicists.

The basic ideas and principles of Monte Carlo calculations are presented in the form of a "primer" for health physicists. A simple integral with a known answer is evaluated by two different Monte Carlo approaches. Random numbers, which underlie Monte Carlo work, are discussed, and a sample table of random numbers generated by a hand calculator is presented. Monte Carlo calculations of dose and linear energy transfer (LET) from 100-keV neutrons incident on a tissue slab are discussed. The random-number table is used in a hand calculation of the initial sequence of events for a 100-keV neutron entering the slab. Some pitfalls in Monte Carlo work are described. While this primer addresses mainly the "bare bones" of Monte Carlo, a final section briefly describes some of the more sophisticated techniques used in practice to reduce variance and computing time.

Domain growth in the clock model.

The development of order in the clock (${Z}_{N}$) model is studied following a quench from the disordered phase to a low-temperature unstable state below the ferromagnetic critical point. Growth laws for the average size of domains are obtained by computer simulation for N=3, 4, 8, 16, and 26 degenerate states. In disagreement with recent Monte Carlo work, no pinning by vortices is observed.

Monte Carlo and series study of corrections to scaling in two-dimensional percolation

Corrections to scaling for percolation cluster numbers in two dimensions are studied by Monte Carlo simulations of very large systems (up to 17*109 lattice sites) and by series analysis. Both series and Monte Carlo work suggests that the value of the correction-to-scaling exponent is slightly lower at the percolation threshold than away from it. Moreover, the corrections to scaling observed at pc ( Omega equivalent to 0.64) might be due to the mixing of scaling fields rather than to the irrelevant scaling fields. The Monte Carlo results are compatible with finite-size scaling, and finite-size scaling corrections are estimated. Technical problems associated with Monte Carlo simulation of very large systems are discussed in an appendix.

Monte Carlo calculation of the intermolecular dipolar spin relaxation in a liquid solution

The time correlation function characteristic of the intermolecular dipolar spin relaxation in a liquid solution is calculated by a Monte Carlo diffusive simulation. The validity of the Monte Carlo method for modeling molecular dynamics is examined referring to the careful analysis by Hilhorst and Deutch of the Monte Carlo work on polymer kinetics of Verdier et al. The Monte Carlo estimates of the time correlation function are compared with those obtained by analytical solutions of translational diffusion equations. The normalized solutions of the Smoluchowski diffusion equation, including the effects of the nonuniform radial molecular distribution, are appropriate for describing the relative random walks of two given molecules in the liquid.

A Review of Simultaneous Painvise Multiple Comparisons

Abstract A Review of Simultaneous Pairwise Multiple Comparisons. Simultaneous pairwise comparisons can be accomplished with numerous multiple comparison procedures. The methods differ in two essential ways: the choice of critical value and the specification of the estimated standard error of the mean difference. Those methods that assume homogeneous variances are not robust to violations of this assumption. The methods are contrasted via a numerical example. Results of recent Monte Carlo work are described. A choice between the Games‐Howell, Dunnett, and Cochran procedures is recommended.

Antithetic Sampling with Multivariate Inputs

SYNOPTIC ABSTRACTThis paper extends the “antithetic-variates theorem” to include the case of two or more random variables, each of which is generated by a sampling scheme requiring a fixed-dimension input vector of independent random numbers. Three examples illustrate the application of this result in Monte Carlo work.

Covariate Analysis of Survival Data: A Small-Sample Study of Cox's Model

Cox's proportional-hazards model is frequently used to adjust for covariate effects in survival-data analysis. The small-sample performances of the maximum partial likelihood estimators of the regression parameters in a two-covariate hazard function model are evaluated with respect to bias, variance, and power in hypothesis tests. Previous Monte Carlo work on the two-sample problem is reviewed.

Essential singularities in dilute magnets

Harris argued in favor of an essential singularity at zero magnetic field in the equation of state for randomly dilute low-temperature ferromagnets. His assumptions and conclusions are reexamined and criticized with the help of earlier Monte Carlo data on the number ${W}_{n}$ of clusters with $n$ spins each. These data suggest for large clusters $log{W}_{n}\ensuremath{\propto}\ensuremath{-} n$ on the paramagnetic side, whereas $log{W}_{n}\ensuremath{\propto}\ensuremath{-} {n}^{0.4}$ on the ferromagnetic side. More Monte Carlo work is suggested, if $n\ensuremath{\sim}100$, for square site percolation near $p=0.40$ or square bond percolation near $p=0.35$.

Scattering of 2-20 keV electrons in aluminium

An investigation of the major scattering processes involved when an incident electron suffers an appreciable change of momentum in a metal has been undertaken. The results of this study have been incorporated within a Monte Carlo framework, allowing calculations of energy and angular distributions of electrons, both transmitted and backscattered from films, to be made. The advantage of this treatment of the problem over earlier Monte Carlo work is that fundamental scattering processes are used rather than invoking mean energy-loss approximations, thus allowing an investigation to be made of the shortcomings of the theoretical treatments of the individual collision types.