Showing papers on "Monte Carlo method published in 1979"

Monte Carlo methods in statistical physics

Abstract: This book provides an introduction to Monte Carlo simulations in classical statistical physics and is aimed both at students beginning work in the field and at more experienced researchers who wish to learn more about Monte Carlo methods. The material covered includes methods for both equilibrium and out of equilibrium systems, and common algorithms like the Metropolis and heat-bath algorithms are discussed in detail, as well as more sophisticated ones such as continuous time Monte Carlo, cluster algorithms, multigrid methods, entropic sampling and simulated tempering. Data analysis techniques are also explained starting with straightforward measurement and error-estimation techniques and progressing to topics such as the single and multiple histogram methods and finite size scaling. The last few chapters of the book are devoted to implementation issues, including discussions of such topics as lattice representations, efficient implementation of data structures, multispin coding, parallelization of Monte Carlo algorithms, and random number generation. At the end of the book the authors give a number of example programmes demonstrating the applications of these techniques to a variety of well-known models.

On optimal and data based histograms

Abstract: SUMMARY In this paper the formula for the optimal histogram bin width is derived which asymptotically minimizes the integrated mean squared error. Monte Carlo methods are used to verify the usefulness of this formula for small samples. A data-based procedure for choosing the bin width parameter is proposed, which assumes a Gaussian reference standard and requires only the sample size and an estimate of the standard deviation. The sensitivity of the procedure is investigated using several probability models which violate the Gaussian assumption.

Probability weighted moments compared with some traditional techniques in estimating Gumbel Parameters and quantiles

Abstract: Estimates of the parameters and quantiles of the Gumbel distribution by the methods of probability weighted moments, (conventional) moments, and maximum likelihood were compared. Results were derived from Monte Carlo experiments by using both independent and serially correlated Gumbel numbers. The method of probability weighted moments was seen to compare favorably with the other two techniques.

A quantum-statistical Monte Carlo method; path integrals with boundary conditions

Abstract: A new Monte Carlo method for problems in quantum‐statistical mechanics is described. The method is based on the use of iterated short‐time Green’s functions, for which ’’image’’ approximations are used. It is similar to the use of Feynman or Wiener path integrals but with a modification to take account of hard‐core boundary conditions. It is applied to two one‐dimensional test problems: that of a single particle in a hard‐walled box and that of two hard particles in a hard‐walled box. For these test problems, the results are in excellent agreement with exact quantum‐mechanical results both at high temperatures (near the classical limit) and at very low temperatures such that essentially only the ground state is occupied. Generalizations to three‐dimensional systems, to many‐body systems, and to more realistic potentials are discussed briefly.

A Monte Carlo simulation of the hydrophobic interaction

Abstract: The hydrophobic interaction between two apolar (Lennard–Jones) spheres dissolved in a model of liquid water (ST 2 water) is simulated using the force‐bias Monte Carlo technique recently devised by the authors. Importance sampling techniques are devised and used to give a relatively accurate determination of the potential of mean force of the two apolar spheres as a function of their separation. This determination shows that there are two relatively stable configurations for the spheres. In one configuration each member of the pair sits in its own water cage with one water molecule fitting between them. There is a free energy barrier separating this from the other stable configuration which is such that no water molecule sits between the spheres. This conclusion is shown to be quantitatively consistent with the recent semiempirical theory of Pratt and Chandler and is in disagreement with some previous Monte Carlo studies.

Experiments with a Gauge-Invariant Ising System

Abstract: Using Monte Carlo techniques, we evaluate the path integral for the four-dimensional lattice gauge theory with a Z/sub 2/ gauge group. The system exhibits a first-order transition. This is contrary to the implications of the approximate Migdal recursion relations but consistant with mean-field-theory arguments. Our ''data'' agree well with a low-temperature expansion and the exact duality between the high- and low-temperature phases.

Monte Carlo Renormalization Group

Abstract: In 1976, Ma1 made the suggestion of combining Monte Carlo (MC) computer simulations with a real-space renormalization-group (RG) analysis to calculate critical exponents at second-order phase transitions. Since then, numerous authors2–14 have presented various ways of implementing Ma's idea to produce a useful theoretical tool. In these lectures, I will discuss a particular Monte Carlo renormalization-group (MCRG) method that I and several coworkers have been using.7–14 The method is still in the early stages of development, but it has a number of advantages over older methods, and has already produced excellent results for some systems of interest.

A Comparison of Procedures for Multiple Comparisons of Means with Unequal Variances

Abstract: Nine procedures for multiple comparisons of means with unequal variances are reviewed. Modifications in some procedures are proposed either for improvement in their performance or easier implementation. A Monte Carlo sampling study is carried out for pair-wise differences as well as a few selected contrasts and the procedures are compared based on the results of this study. Recommendations for the choice of the procedures are given. Robustness of two procedures designed for homogeneous variances under violation of that assumption is also examined in the Monte Carlo study.

A Monte Carlo Analysis of the hydrologic effects of spatial variability of infiltration

Abstract: Monte Carlo simulation is employed to analyze the effects of random distribution of soil infiltration properties on the hydrologic performance of catchment areas. Soil hydraulic parameters are distributed with log normal distribution as reported from previous field measurements. Although independent sampling units are assumed, a discussion is presented on the nature and effect of spatial dependence. The composite infiltration rate of an area is simply the average, at an instant in time, of the infiltration rate of all points within that area. Rigorous treatment of the topic is a forbidding problem in stochastic differential equations. Monte Carlo simulation is used here to study the effects of random distribution of soil properties on distribution of ponding time, composite areal rate, and bias due to variability in space. More accurate treatment for watershed studies must also consider interaction of various points on the surface by way of overland flow paths. A distributed watershed simulation model is used for this purpose to demonstrate hydrograph bias due to deterministic spatial variability, and from random variation, plus the relation of bias to rainfall rate. An analysis is also made of the partial area contribution in early parts of storm runoff caused solely by random variation in surface soil properties.

Testing the adequacy of a time series model

Abstract: SUMMARY The problem of testing the adequacy of an autoregressive-moving average time series model is considered. A new approach is proposed, and some Monte Carlo results on the finite sample size and power of the test procedures of this paper are reported. The test statistics discussed by Box & Pierce (1970) and Ljung & Box (1978) are also examined.

Monte Carlo simulation of the classical two-dimensional one-component plasma

Abstract: Monte Carlo simulation, lattice dynamics in the harmonic approximation, and solution of the hypernetted chain equation were used to study the classical two-dimensional one component plasma. The system consists of a single species of charged particles immersed in a uniform neutralizing background. The particles interact via a l/r potential, where r is the two dimensional separation. Equations of state were calculated for both the liquid and solid phases. Results of calculation of the thermodynamic functions and one and two particle correlation functions are presented.

On the force bias Monte Carlo simulation of water: methodology, optimization and comparison with molecular dynamics

Abstract: An optimized version of the force bias scheme is presented for the Monte Carlo simulation of water in which the random walk of the individual molecule is biased in the direction of the force and torque acting on the molecule. A new criterion is developed to judge the efficiency of sampling the configuration space by studying the translational and rotational diffusion of individual molecules during the random walk. The force bias method is compared with the uniform sampling method using ST2 water as an example, and with molecular dynamics. Similarities and differences between molecular dynamics and Monte Carlo are discussed with respect to pair correlations and energy distributions.

Estimation for the multiple factor model when data are missing

Abstract: A maximum likelihood method of estimating the parameters of the multiple factor model when data are missing from the sample is presented. A Monte Carlo study compares the method with 5 heuristic methods of dealing with the problem. The present method shows some advantage in accuracy of estimation over the heuristic methods but is considerably more costly computationally.

Properties of liquid and solid He 4

Abstract: A Monte Carlo method is used to compute the properties of the fluid and crystal phases of the Lennard-Jones model of $^{4}\mathrm{He}$ at absolute zero. The method yields exact results subject only to statistical sampling errors. The energy, structure factor, and momentum distribution are calculated at several densitites in both phases. In addition, in the crystal phase we have carried out a detailed study of the single-particle distribution function. The densities at which melting and freezing occurs are determined. In both phases perturbative estimates of the three-body Axilrod-Teller potential are computed. Overall the agreement with experiment is good to excellent. However there are significant discrepancies between the computed and experimental equations of state. We believe this is due to the inadequacy of the Lennard-Jones potential.

An investigation of the accuracy of numerical solutions of Boltzmann's equation for electron swarms in gases with large inelastic cross sections

Abstract: A Monte Carlo simulation technique has been used to test the accuracy of electron energy distribution functions and transport coefficients calculated using conventional numerical solutions of Boltzmann's equation based on a two-term approximation. The tests have been applied to a number of model gases, some of which have characteristics close to those of real gases, and include cases where the scattering is anisotropic. The results show that, in general, previous application of the numerical solution to real gases has been valid.

Evidence Against Spin-Glass Order in the Two-Dimensional Random-Bond Ising Model

Abstract: By recursive methods, numerically exact free energies are calculated for L×L Ising lattices with bonds of randomly chosen sign, with 6<~L<~18. Ground states of these systems are identified, and the response to ordering fields is studied. By performing Monte Carlo simulations for precisely the same systems we are able to unambiguously distinguish nonequilibrium phenomena from equilibrium properties. The L dependence of our results suggests that there is no nonzero spin-glass order parameter for L→∞.

Monte Carlo studies on the freely jointed polymer chain with excluded volume interaction

Abstract: We have carried out computer simulations of the freely jointed polymer chain with an excluded volume interaction using a dynamic Monte Carlo method for chain lengths N between N=8 and N=100. The equilibrium values of end‐to‐end distances and radius of gyration approach the asymptotic form ∼Nν for N≳70 (scaling limit) and excluded volume parameter d/l=1.0. The scaling limit decreases to lower N for lower d/l. Here l is the length of a chain unit and d is the excluded volume distance. The exponent is ν?0.6 for d/l≠0 and ν=0.5 for d/l=0. The structure function varies as S (k) ∼q−1/ν (q=kNν) over a wide range of q. This asymptotic behavior for large N is found for N≳16 and d/l≠0. Our results are carefully compared to previous studies on the same model where other types of Monte Carlo methods are used.

Monte Carlo Renormalization Group and Ising Models with n>=2

Abstract: We suggest that the Monte Carlo renormalization group, when combined with the type of cell-spin transformation introduced by van Leeuwen, should be a powerful tool in the study of Ising models with n > or = 2. Numerical results are presented for the Baxter model and the Ising model with nearest-- and next-nearest--neighbor interactions on a square lattice.

A new Monte Carlo method for phase determination

Abstract: An extended version of the multi-solution method has been devised by introducing a Monte Carlo technique. This Monte Carlo direct method differs from the ordinary multi-solution procedure in two respects: (1) The starting set usually consists of as many as 10-50 phases; (2) Tentative phase values assigned to the members of the starting set are derived from successively generated random numbers. The application of the new method to several unknown structures has shown that it can be used as an effective means of phase determination.