Showing papers on "Dynamic Monte Carlo method published in 1968"
TL;DR: In this article, the Monte Carlo method was adapted to the calculation of the equation of state and radial distribution function for hard disks, and the results obtained for a small system (12 hard disks), as an example of the method, were presented.
Abstract: The N p T ensemble for hard disks is formulated as an equivalent N V T ensemble for a pseudopotential interaction in a reduced configuration space, with exact relations for small periodic systems emphasized. The Monte Carlo method originally devised by Metropolis et al. is then adapted to the calculation of the equation of state and radial distribution function. The present paper then describes some results obtained for a small system (12 hard disks), as an example of the method, and with emphasis upon the statistical reliability of the results and on the effect of different pseudorandom‐number‐generating procedures. At least for such small systems it is possible to obtain, as a byproduct, the N V T ‐ensemble equation of state over a range of densities from an N p T ‐ensemble calculation at a single pressure. Subsequent papers will describe results already in hand for larger systems, including reasonably reliable extrapolations to the “thermodynamic” limit.
198 citations
168 citations
TL;DR: The first section of this paper is a mathematical construction of a certain Monte Carlo procedure for sampling from the distribution by defining a particular random variable.
Abstract: The first section of this paper is a mathematical construction of a certain Monte Carlo procedure for sampling from the distributionThe construction begins by defining a particular random variable ...
120 citations
TL;DR: The proper justification of Monte Carlo integration must be based not on the randomness of the procedure, which is spurious, but on equidistribution properties of the sets of points at which the integrand values are computed as discussed by the authors.
Abstract: The proper justification of the normal practice of Monte Carlo integration must be based not on the randomness of the procedure, which is spurious, but on equidistribution properties of the sets of points at which the integrand values are computed. Besides the discrepancy, which it is proposed to call henceforth extreme discrepancy, another concept, that of mean square discrepancy, can be regarded as a measure of the lack of equidistribution of a sequence of points in a multidimensional cube. Determinate upper bounds can be obtained, in terms of either discrepancy, for the absolute value of the error in the computation of the integral. There exist sequences of points yielding, for sufficiently smooth functions, errors of a much smaller order of magnitude than that which is claimed by the Monte Carlo method. In the case of two dimensions, sequences with optimum properties can be generated with the help of Fibonacci numbers. The previous arguments do not apply to domains of integration which cannot be reduced to multidimensional intervals. Difficult questions arising in this connection still await an answer.
96 citations
TL;DR: In this paper, the electron distribution function in the (000 and 100) valleys of gallium arsenide and the resulting velocity-field relationship have been calculated using a Monte Carlo method.
76 citations
TL;DR: In this article, an analysis is presented of the dynamical properties of a model meant to represent the electron jump mechanism that has been proposed to describe the alkali-metal-halogen-molecule reactions.
Abstract: An analysis is presented of the dynamical properties of a model meant to represent the “harpooning,” or electron‐jump mechanism that has been proposed to describe the alkali‐metal–halogen‐molecule reactions (M + X2→MX + X). Individual trajectories are computed from classical equations of motion with the starting conditions chosen by Monte Carlo procedures. With 500 or so trajectories for each trial, a comparison can be made with available observations from molecular‐beam experiments; the way in which the reaction energy is distributed between kinetic energy of translation and the internal modes of the product, along with a measure of the differential cross section, are of particular concern. The trajectories include a sudden crossing from an initial homopolar surface to a final surface with the long‐range forces required of an M+X− bond to simulate electron transfer. Two different potential functions are used as the final surface: one has a phenomenological form to include various types of X–X forces after transition, and the other has a simple induced‐dipole term as an interaction of the departing X with the resulting charges (M+,X−). Except for an extreme trial closely approximating pure stripping, nine trials of the first potential failed to agree with experiment. The last potential gave good results for trials of K + Br2,K + I2,Rb + Br2,Rb + I2, and Cs + Br2.
52 citations
38 citations
TL;DR: The functional integration techniques of Kac are shown to be applicable to an evaluation of Anderson's relaxation function in this paper, and the evaluation is worked through for the rotational diffusion model, and the essential features of the Monte Carlo calculation by Saunders and Johnson are confirmed.
Abstract: The functional integration techniques of Kac are shown to be applicable to an evaluation of Anderson's relaxation function. The evaluation is worked through for the rotational diffusion model, and the essential features of the Monte Carlo calculation by Saunders and Johnson are confirmed.
26 citations
TL;DR: In this paper, a Monte Carlo simulation technique is described that can be used to determine the cumulative distribution functions of stochastic variables such as ultimate strength, factor of safety and region of safety.
Abstract: Probabilistic concepts of structural safety are reviewed. A Monte Carlo simulation technique is described that can be used to determine the cumulative distribution functions of stochastic variables such as ultimate strength, factor of safety and region of safety. In a worked example, the variability in strength of an axially loaded short reinforced concrete column is investigated using the Monte Carlo technique and the results are compared with a closed form solution. Short cut methods of selective sampling are also described and demonstrated.
25 citations
23 citations
TL;DR: In this article, a Monte Carlo calculation of the nucleonic cascade in the atmosphere has been carried out and preliminary results are presented and compared with measurements of the secondary proton spectrum at sea level.
Abstract: A Monte Carlo calculation of the nucleonic cascade in the atmosphere has been carried out. Preliminary results are presented and compared with measurements of the secondary proton spectrum at sea level.
DOI•
02 Dec 1968
TL;DR: A brief survey of the existing methods for determining when to stop sampling in Monte Carlo simulations is presented and the distinction is made between stopping rules for simulations using independent samples and those using correlated samples.
Abstract: A brief survey of the existing methods for determining when to stop sampling in Monte Carlo simulations is presented. The distinction is made between stopping rules for simulations using independent samples and those using correlated samples. Possible avenues for further research are mentioned.
TL;DR: In this article, the feasibility of using Monte Carlo methods to compute the criticality of thermal reactors was investigated by analyzing three simple critical assemblies with the 05R Monte Carlo neutron transport (MCT) method.
Abstract: The feasibility of using Monte Carlo methods to compute the criticality of thermal reactors is investigated by analyzing three simple critical assemblies with the 05R Monte Carlo neutron transport ...
TL;DR: In this paper, the authors performed the calculation of time independent spin correlation function for a Ising ferromagnet with a Monte Carlo method in the case of a simple cubic lattice, S = ∞ and interactions only between nearest neighbours.
TL;DR: A Monte Carlo model of γ-ray diffusion is presented with a discussion of its application to dosimetry problems and the Klein-Nishina free electron approximation as a model for τ-ray scattering is shown to be an adequate approximation.
Abstract: A Monte Carlo model of γ-ray diffusion is presented with a discussion of its application to dosimetry problems. Two methods of using photon history data to calculate γ-ray dose are reviewed: summing the energy losses from individual events, and calculating the energy fluence. Approximations in the γ-ray history simulation model are analysed and the relative importance of the various assumptions established. Studies of the spectra of scattered radiation show that results are not affected by use of a finite energy cut-off to terminate histories. The Klein-Nishina free electron approximation as a model for γ-ray scattering is shown to be an adequate approximation for the purposes of γ-ray dosimetry.
TL;DR: In this paper, a Monte Carlo study of the configurational and statistical thermodynamic properties of ring systems generated on the tetrahedral lattice is presented, and it is shown that the above parameters obtained are in close agreement with those found for open walks.
Abstract: A Monte Carlo study of the configurational and statistical thermodynamic properties of ring systems generated on the tetrahedral lattice is presented. It is shown that the above parameters obtained are in close agreement with those found for open walks. It is suggested that a number of ratios of configurational parameters may be fruitful quantities for studying a variety of systems.
TL;DR: In this article, the effect of the size and length-to-radius ratio of the orifice and the effects of different condensation coefficients on the flux ratio of a given cell was analyzed.
Abstract: Data are presented for effusion probabilities and cell‐wall flux gradients calculated dynamically by means of Monte Carlo computer studies. Typical effusion cell dimensions were programmed for both knife‐edged and Clausing orifices; all calculations apply to a flat sample. The influence of the size and the length‐to‐radius ratio of the orifice and the effects of different condensation coefficients are described. An analytical expression relating the equilibrium sample flux to the flux at steady state has been derived. It is shown that for a given cell design, the ratio of these fluxes is constant and independent of the nature of the vaporizing species. Because the Monte Carlo method requires no mathematical approximations or simplifying assumptions, results analogous to real cell behavior are to be expected. Comparison of these results with the treatment derived by Motzfeldt shows that his formula is applicable and quite accurate over a wide range of experimentally encountered conditions.
TL;DR: In this paper, an extensive series of numerical studies directed toward understanding and evaluating the various errors in the Boltzmann collision integral solutions of the Nordsieck equation are presented.
TL;DR: In this article, the authors present a method for conducting a Monte Carlo simulation of a space mission when nonlinear effects must be considered and accounted for, which can be used in the analysis of a multiphase mission.
Abstract: This paper presents a method for conducting a Monte Carlo simulation of a space mission when nonlinear effects must be considered and accounted for. The simulation uses precomputed relationships between variables to generate the Monte Carlo samples from which the appropriate statistical information is computed. The method reduces the computing time required for the Monte Carlo simulation to just about that required to perform a standard root-sum-square analysis. The results, however, are considerably greater in information content, allow definitive probabilistic statements to be made about the performance of the vehicle, and can be used in the analysis of a multiphase mission. Thus, a very realistic assessment of the over-all performance of a space vehicle can be made at a reasonable computing cost.
TL;DR: In this article, the authors give a general description of a class of formulas which combine the Monte Carlo and classical approaches to get a lower bound on quadrature errors, and show that for the class D * the error of any non-random (e.g. Newton-Cotes, Gaussian) method is Q(N-) ; for random methods the best he could show was (r = Ö(i\\M) and he showed that there in fact exist methods for which
Abstract: /Ytf, X, ' • • , X) (Ox)*\" • • • (da?)* is continuous on G8 whenever mf ^2, • • • , n8^n. N. S. Bahvalov [ l ] , in a study of lower bounds on quadrature errors showed that for the class D\" the error of any nonrandom (e.g. Newton-Cotes, Gaussian) quadrature method is Q(N-) ; for random methods the best he could show was (r = Ö(i\\M) and he showed that for the set of periodic functions in D* there in fact exist methods for which
TL;DR: In this paper, a recent Monte Carlo study of a hydrogenous plasma near the ionization temperature showed that distribution functions obtained are unusually sensitive to two parameters: the cutoff imposed at small radii on the Coulomb potential between unlike particles, and the maximum step length through which the particles are allowed to move in the Monte Carlo procedure.
Abstract: Results of a recent Monte Carlo study of a hydrogenous plasma near the ionization temperature show that distribution functions obtained are unusually sensitive to two parameters. The first is the cutoff imposed at small radii on the Coulomb potential between unlike particles, and it becomes necessary to consider quantum-mechanical effects at these radii. The second is the maximum step length $\ensuremath{\Delta}$ through which the particles are allowed to move in the Monte Carlo procedure. It appears that near the ionization temperature the plasma behaves as a mixture of two phases, one ionized, the other un-ionized, and the magnitude chosen for $\ensuremath{\Delta}$ influences which phase dominates.
TL;DR: In this paper, algebraic Monte Carlo method was used to determine simultaneous effect of several input statistical variables on the output variable, and the authors constructed a histogram based on the simultaneous effect.
Abstract: Histogram construction via algebraic Monte Carlo method, determining simultaneous effect of several input statistical variables on output variable