Alan M. Ferrenberg

Monte Carlo study of phase transitions in ferromagnetic bilayers

We have used Monte Carlo computer simulations to study the behavior of an Ising model consisting of two ferromagnetic layers with different interaction constants coupled weakly together. For the range of lattice sizes studied it appears as though the system undergoes a single transition at the transition temperature of an isolated layer with the stronger coupling, but substantial changes in the thermodynamic properties also occur near the transition temperature of an isolated layer with the weaker of the two couplings.

Statistical and systematic errors in Monte Carlo sampling

We have studied the statistical and systematic errors which arise in Monte Carlo simulations and how the magnitude of these errors depends on the size of the system being examined when a fixed amount of computer time is used. We find that, depending on the degree of self-averaging exhibited by the quantities measured, the statistical errors can increase, decrease, or stay the same as the system size is increased. The systematic underestimation of response functions due to the finite number of measurements made is also studied. We develop a scaling formalism to describe the size dependence of these errors, as well as their dependence on the “bin length” (size of the statistical sample), both at and away from a phase transition. The formalism is tested using simulations of thed=3 Ising model at the infinite-lattice transition temperature. We show that for a 96×96×96 system noticeable systematic errors (systematic underestimation of response functions) are still present for total run lengths of 106 Monte Carlo steps/site (MCS) with measurements taken at regular intervals of 10 MCS.

New monte carlo methods for improved efficiency of computer simulations in statistical mechanics

In this paper, we have reviewed two complementary new approaches to improve the accuracy and efficiency of Monte Carlo computer simulations in the study of phase transitions. Substantial gains have already been achieved, and since these methods represent approaches to the problem of simulation that are still being developed, they promise further improvements in the future. In both cases, a key feature is the lifting of the restriction of thinking about computer simulations as simply “computer experiments,” and recognising that there are certain aspects of simulations that can be exploited to greatly increase their value.

Cluster hybrid Monte Carlo simulation algorithms

We show that addition of Metropolis single spin flips to the Wolff cluster-flipping Monte Carlo procedure leads to a dramatic increase in performance for the spin-1/2 Ising model. We also show that adding Wolff cluster flipping to the Metropolis or heat bath algorithms in systems where just cluster flipping is not immediately obvious (such as the spin-3/2 Ising model) can substantially reduce the statistical errors of the simulations. A further advantage of these methods is that systematic errors introduced by the use of imperfect random-number generation may be largely healed by hybridizing single spin flips with cluster flipping.

Monte Carlo study of critical behavior in the three-dimensional classical Heisenberg ferromagnet

The static critical properties of classical, isotropic Heisenberg ferromagnets have been studied numerically. Using an improved cluster algorithm we performed extensive Monte Carlo simulations on L×L×L simple cubic and body‐centered cubic systems with L≤40 over a temperature range covering the critical region. Temperature dependences of thermodynamic quantities were determined with an optimized multiple‐histogram method, and both critical temperatures and critical exponents were extracted from finite‐size scaling of several different thermodynamic quantities. It was found that within their respective error bars the critical exponents for both ferromagnets agree with each other and with field theoretical predictions.

Detection of deleterious genotypes in multigenerational studies. III. Estimation of selection components in highly selfing populations

New paradigms in genetics have increased the chance of finding genes that appear redundant but in fact may have been preserved due to a small level of positive selection potential acting during each generation. Monitoring changes in genotypic frequencies within and between generations allows the dissection of the fertility, viability and meiotic drive selection components acting on such genes in natural and experimental populations. Here, a formal maximum likelihood procedure is developed to identify and estimate these selection components in highly selfing populations by fitting the time-dependent solutions for genotypic frequencies to observed multigenerational counts. With adult census alone, we can not simultaneously estimate all three selection components considered. In such cases, we instead consider a hierarchy of 11 models with either fewer selection components, complete dominance, or multiplicative meiotic drive with a single parameter. We identify the best-fitting of these models by applying likelihood ratio tests to nested models and Akaikes Information Criterion (AIC) and the Bayesian Information Criterion (BIC) to non-nested models. With seed census, fertility and viability selection are not distinguishable and thus can only be estimated jointly. A combination of joint seed and adult census data allows us to estimate all three selection components simultaneously. Simulated data validate the estimation procedure and provide some practical guidelines for experimental design. An application to Arabidopsis data establishes that viability selection is the major selective force acting on the ACT2 actin gene in laboratory-grown Arabidopsis populations.

Container medium characterization by scanning electron microscopy and comparison with a Monte Carlo computer simulated medium

Abstract A 1:1 by volume pine bark and sand containermedium was embedded with Spurrs epoxy resin and cross‐sectioned for SEM characterization of particle distribution. A Monte Carlo computer simulated medium was built with spheres of the same particle size distribution as the pine bark and sand grains used in the experiment. Cross‐sections from the simulated systems were compared with actual experimental data and good agreement was obtained. SEM makes it possible to analyze particle and pore distribution of container media. Computer simulation provides a good approximation of container medium structure and may prove useful in estimating properties of container medium mixtures.

Monte Carlo studies of magnetic critical phenomena using histogram techniques (invited)

Recent developments in the use of histograms (measured discrete probability distributions) now make it possible to determine critical properties with high accuracy using Monte Carlo (MC) simulations combined with finite‐size scaling analyses. The advantages and limitations of the use of histograms for extracting the maximum information from MC simulation data are described. Results from extensive MC studies of the critical properties of two important magnetic systems, the d=3 ferromagnetic Ising and Heisenberg models, are presented. It is shown that the careful use of histogram techniques can provide results comparable or superior to these obtained with other numerical methods while maintaining all the advantages of standard MC techniques.

Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model

While the three-dimensional Ising model has defied analytic solution, various numerical methods like Monte Carlo, Monte Carlo renormalization group, and series expansion have provided precise information about the phase transition. Using Monte Carlo simulation that employs the Wolff cluster flipping algorithm with both 32-bit and 53-bit random number generators and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising Model, with lattice sizes ranging from 16^{3} to 1024^{3}. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, e.g., logarithmic derivatives of magnetization and derivatives of magnetization cumulants, we have obtained the critical inverse temperature K_{c}=0.221654626(5) and the critical exponent of the correlation length ν=0.629912(86) with precision that exceeds all previous Monte Carlo estimates.

92 Years of the Ising Model: A High Resolution Monte Carlo Study

Using extensive Monte Carlo simulations that employ the Wolff cluster flipping and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising model with lattice sizes ranging from 163 to 10243. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, we obtained the critical inverse temperature K c = 0.221 654 626(5) and the critical exponent of the correlation length ν = 0.629 912(86) with precision that improves upon previous Monte Carlo estimates.