Conventional Monte Carlo Simulation Research Articles

Exchange Monte Carlo Dynamics in the SK Model

The equilibration process of the Sherrington–Kirkpatrick spin-glass model is investigated using the exchange Monte Carlo (MC) method. It is found that a MC time scale for equilibration, τ eq , is nearly proportional to the system size and weakly depends on temperature, in contrast to the thermal activation behavior observed in a conventional MC simulation. Fluctuation of the overlap distribution over samples is confirmed to follow the non trivial probability law predicted by the replica-symmetry-breaking theory.

Adatom dynamics and diffusion in a model of O/W(110)

We consider adatom dynamics and diffusion in a lattice-gas model of the O/W(110) system under conditions where the adatom interaction effects are important. In particular, we study the behavior of the tracer and collective diffusion coefficients as a function of temperature when crossing over from the high-temperature disordered phase to a low-temperature symmetry broken phase. To this end, we utilize a combined analytical and numerical approach based on the recently developed dynamical mean field theory (DMF) in addition to conventional Monte Carlo simulations. In the case studied here, the origin of the strong temperature dependence of the effective activation barrier ${E}_{A}^{D}$ close to an order-disorder transition, i.e., the non-Arrhenius behavior of the diffusion coefficients, can be traced back to that of the average microscopic jump rate $\ensuremath{\Gamma}$ appearing within the DMF. This is in contrast to the usual assumption that thermodynamics controls diffusion near phase transitions. The behavior of $\ensuremath{\Gamma}$, in turn, is found to arise predominantly from critical effects in the short-time behavior of the waiting-time distribution of single-particle jumps, $W(t)$, which is an experimentally accessible quantity. The long-time decay of $W(t)$ is then used to define another effective barrier ${E}_{A}^{W}$, which shows no anomalous effects near the transition.

Color-dielectric gauge theory on a transverse lattice

We investigate in some detail consequences of the effective colour-dielectric formulation of lattice gauge theory using the light-cone Hamiltonian formalism with a transverse lattice. As a quantitative test of this approach, we have performed extensive analytic and numerical calculations for 2+1-dimensional pure gauge theory in the large N limit. Because of Eguchi-Kawai reduction, one effectively studies a 1+1-dimensional gauge theory coupled to matter in the adjoint representation. We study the structure of coupling constant space for our effective potential by comparing with the physical results available from conventional Euclidean lattice Monte Carlo simulations of this system. In particular, we calculate and measure the scaling behaviour of the entire low-lying glueball spectrum, glueball wavefunctions, string tension, asymptotic density of states, and deconfining temperature. We employ a new hybrid DLCQ/wavefunction basis in our calculations of the light-cone Hamiltonian matrix elements, along with extrapolation in Tamm-Dancoff truncation, significantly reducing numerical errors. Finally we discuss, in light of our results, what further measurements and calculations could be made in order to systematically remove lattice spacing dependence from our effective potential a priori.

Calculating Prices and Sensitivities for Path-Independent Derivatives Securities in Multifactor Models

Pricing derivatives written on assets whose prices depend on multiple factors can be difficult, even if payoffs depend only on prices at maturity. In complicated problems, Monte Carlo simulation is a popular technique, but it can be time-consuming when accuracy is needed. Conditional Monte Carlo is a known variance reduction method, but its use has been limited by a presumption in the literature that it can be used only when factors are instantaneously uncorrelated. I show that this presumption is not necessarily true. For example, we can express the prices and sensitivities of an European call option written on an asset that has stochastic volatility as an expectation over its Black-Scholes counterparts, even when the volatility and price processes exhibit instantaneous correlation. Combining this conditional Monte Carlo technique with low-discrepancy ("quasi-random") techniques provides price and delta estimators that require a fraction of a second to compute and yet have exact errors that are accurate to a few one-hundredths of a percent of the option's price. By comparison, conventional Monte Carlo simulation requires at least several minutes of computation time to achieve the same accuracy.

Application of a new Gibbs ensemble Monte Carlo method to site-site interaction model fluids

A new Gibbs ensemble Monte Carlo simulation method has been tested for rigid site–site interaction model fluids. In this method the total system is considered at constant enthalpy, pressure and number of particles. Vapour-liquid equilibrium data were determined for a potential model of liquid carbon disulphide and a potential model of liquid hydrogen sulphide. Conventional Gibbs ensemble Monte Carlo simulations were also carried out for comparison. The two methods yielded essentially the same results, but the new scheme showed some advantages.

Trajectory Weighting: a Fast Monte Carlo Method for Calculating Large Numbers of XPS/AES Spectra

A new, efficient, hybrid Monte Carlo method (‘trajectory-weighting’ method) for the calculation of XPS and AES spectra is presented. The method produces, in one single calculation, XPS or AES spectra for a large number of sample compositions. The method consists of a conventional Monte Carlo calculation for a single standard composition plus transformations for many other compositions. The transformations are based upon proper weighting of the electron trajectories constructed for the standard composition. The weighting factors depend on the elastic and inelastic scattering cross-sections. The trajectory-weighting method has been tested against a conventional Monte Carlo simulation and an analytical model. Furthermore, it has been applied to establish the XPS spectrum dependence on thickness, depth and composition of a buried layer in an otherwise homogeneous sample. © 1997 by John Wiley & Sons, Ltd.

Modeling of protein conformational fluctuations in p Ka predictions

A method is presented to account for conformational fluctuations of a protein in predicting the p K a values of its titrating groups. Conformations of the protein are generated by conventional molecular dynamics or Monte Carlo simulations, in which the protonations of the titrating groups are fixed. For each protein conformation, the electrostatic free energies required to add a proton charge to a titrating group while other groups are either unprotonated or protonated are calculated within a dielectric continuum model. These are used to determine the mean protonations of the titrating groups in the conformation at a series of pH values. The mean protonations are then used to determine the relative weight of the particular conformation with the titrating groups having all possible protonations. A conformationally averaged mean protonation for each titrating group is finally obtained by the weighted sum of the group’s mean protonations in all the conformations. This method is applied to yeast iso-1-ferricytochrome c. The predicted p K a values are in general agreement with experimental results.

Application of a new Gibbs ensemble Monte Carlo method to site-site interaction model fluids

A new Gibbs ensemble Monte Carlo simulation method has been tested for rigid site–site interaction model fluids. In this method the total system is considered at constant enthalpy, pressure and number of particles. Vapour-liquid equilibrium data were determined for a potential model of liquid carbon disulphide and a potential model of liquid hydrogen sulphide. Conventional Gibbs ensemble Monte Carlo simulations were also carried out for comparison. The two methods yielded essentially the same results, but the new scheme showed some advantages.

On repeated parameter sampling in Monte Carlo simulations

In a realistic model of a biological system there is usually uncertainty in some of the model parameters. A conventional Monte Carlo simulation handles this situation by randomly choosing parameters at the start of the simulation (from specified probability distributions) and then keeping them fixed for the duration of the simulation. This is certainly appropriate in situations where the parameters are thought to be constant through time but the precise values are not known with certainty. If any of the parameters vary through time in the real system, however, it is intuitively appealing to allow the model parameters to vary by resampling at intervals during the simulation. Previous work on a simple model of algal competition had suggested that the second of these two approaches gave qualitatively different (and biologically more reasonable) results than the first. By analyzing the behavior of the model more thoroughly we explain the reasons for these results, and show that in fact when the procedure is carried out correctly, there is little or no difference in model output between the two types of simulation.

Sampling protein conformations using segment libraries and a genetic algorithm

We present a new simulation algorithm for minimizing empirical contact potentials for a simplified model of protein structure. The model consists of backbone atoms only (including Cβ) with the φ and ψ dihedral angles as the only degrees of freedom. In addition, φ and ψ are restricted to a finite set of 532 discrete pairs of values, and the secondary structural elements are held fixed in ideal geometries. The potential function consists of a look-up table based on discretized inter-residue atomic distances. The minimization consists of two principal elements: the use of preselected lists of trial moves and the use of a genetic algorithm. The trial moves consist of substitutions of one or two complete loop regions, and the lists are in turn built up using preselected lists of randomly-generated three-residue segments. The genetic algorithm consists of mutation steps (namely, the loop replacements), as well as a hybridization step in which new structures are created by combining parts of two “parents’’ and a selection step in which hybrid structures are introduced into the population. These methods are combined into a Monte Carlo simulated annealing algorithm which has the overall structure of a random walk on a restricted set of preselected conformations. The algorithm is tested using two types of simple model potential. The first uses global information derived from the radius of gyration and the rms deviation to drive the folding, whereas the second is based exclusively on distance-geometry constraints. The hierarchical algorithm significantly outperforms conventional Monte Carlo simulation for a set of test proteins in both cases, with the greatest advantage being for the largest molecule having 193 residues. When tested on a realistic potential function, the method consistently generates structures ranked lower than the crystal structure. The results also show that the improved efficiency of the hierarchical algorithm exceeds that which would be anticipated from tests on either of the two main elements used independently.

AN OPTIMAL CONTROL APPROACH TO EXTREME LOCAL MAXIMA FOR STOCHASTIC DUFFING-TYPE OSCILLATORS

A new way is presented to calculate extreme exceedance probabilities associated with the height of local maxima on stationary sample paths of a non-linear oscillator driven by narrow-band Gaussian excitation. Deterministic “least effort” optimal control functions are used to construct a linear filter which is applied to the excitation process. By defining suitable form of conditioning on the filter output local maxima, the most difficult problem can be switched around, so that use can be made of the conventional theory for a normal process. With subsequent simplifications, the required probabilities can be expressed just in terms of an unknown conditional mean and variance. An efficient numerical scheme is developed to estimate these two moments via non-linear regression using a small set of simulated oscillator sample paths. In testing the method on a Duffing-type oscillator with strongly non-linear stiffness and damping, comparison with conventional Monte Carlo simulation is concentrated in a probability range 10 −2−10 −5but using only about 1% of the local maxima normally required. The method offers a practical tool for use in the analysis of extreme non-linear random vibration.

A decoupled photon-electron transport method for calculation of energy deposition in gamma-ray detectors

Abstract A computationally efficient method is presented to facilitate the calculation of energy deposition by photons and electrons in gamma-ray detectors for the purpose of determining the response function. The method is based on decoupling photon from electron transport simulation using electron-energy-leakage functions. A variety of detector response functions are calculated for various gamma-ray source arrangements. The results are validated against those obtained using conventional Monte Carlo simulation of coupled photon-electron transport. Analytical functions for the detection efficiency and photopeak efficiency are formulated using data generated by the decoupled photon-electron transport method.

Harmonic wave propagation through viscoelastic heterogeneous media exhibiting mild stochasticity — II. Applications

The results obtained in Part I ( Soil Dynam. Earth. Engng, 1996, 15, 119-27) of this work for modelling harmonic wave propagation through viscoelastic heterogeneous media that exhibit a small random fluctuation of their material properties about mean values are now used here to investigate SH wave propagation in two naturally occurring media, namely sandstone and topsoil. These results are in the form of depth dependent, deterministic mean values and non-isotropic covariances for both the wave speed profile in the medium and for the fundamental solution in terms of displacement due to a unit point source. The results are also compared against conventional Monte Carlo simulations.

Generalized Kawasaki dynamics of the Heisenberg model.

Kawasaki spin-exchange dynamics is generalized to study ordering dynamics in the three-dimensional ferromagnetic Heisenberg model with a conserved vector order parameter. It is found, by using conventional temperature-quenching Monte Carlo simulations, that the generalized Kawasaki dynamics enables the system to reach thermodynamic equilibrium faster than the conventional Kawasaki dynamics does at lower temperatures, while both are similar at higher temperatures. With the generalized Kawasaki dynamics the domain size grows with time as ${\mathit{t}}^{1/4}$, in agreement with recent studies using Langevin-type dynamics. Evidence for the existence of spin waves is observed in the Monte Carlo simulations for the generalized Kawasaki dynamics. Its relation to domain growth is discussed.

MMC-a high-performance Monte Carlo code for electron beam treatment planning

The macro Monte Carlo (MMC) method has been developed to improve the speed of traditional Monte Carlo (MC) high-energy electron transport calculations without loss in accuracy. The MMC algorithm uses results derived from conventional MC simulations of electron transport through macroscopic spheres of various radii and consisting of a variety of media. Based on these results, electrons are transported in macroscopic steps through the absorber. The absorber geometry is represented by a three-dimensional (3D) density matrix, typically derived from computer tomographic (CT) data. Energy lost by the electrons along their paths through the absorber is scored in a 3D dose matrix. Transport of secondary electrons and bremsstrahlung photons is taken into account. Major modifications of the original implementation of the MMC algorithm have resulted in an improved version of the code, resolving earlier problems with electron transport across interfaces of different materials, and running at a substantially higher speed. Furthermore, the code has been integrated into a clinical 3D treatment planning system. MMC results are in good agreement with results from conventional MC codes and are obtained with a speed gain of about one order of magnitude for clinically relevant irradiation situations. Calculation times to obtain a relative statistical accuracy of 2% per dose grid voxel for small electron field sizes are short enough to be routinely useful in radiotherapy clinics on present day affordable workstation computers. Considering speed, accuracy and memory requirements, MMC is a promising alternative to currently available electron dose planning algorithms.

INCORPORATION OF POLARIZATION EFFECTS IN MONTE CARLO SIMULATIONS OF RADIATIVE HEAT TRANSFER

The electric field vector of individual photons has been incorporated into Monte Carlo simulations of radiative heat transfer to examine the effects of polarisation on the optical properties of arrays of fixed discrete surfaces. Simulations are performed on arrays that have specular surfaces with high and low reflectivity. Two different arrays are illuminated by polarized and unpolarized light and compared with conventional Monte Carlo simulations. The results show that if the initial illumination is either partially or fully polarized, polarization effects are substantial, especially for low-reflectivity surfaces and for arrays that favor a large number of grazing-angte reflections.

Monte Carlo studies of engineering system reliability

The Monte Carlo simulation technique, which uses well-developed neutron transport theory, is applied to study the reliability of engineering systems with either Markovian or non-Markovian properties. In order to extend the application of this Monte Carlo simulation technique to the study of the perturbed system behaviour, for example, after design changes have been made to the system, the correlated sampling technique is incorporated with conventional Monte Carlo simulation to achieve this purpose. We conclude that Monte Carlo simulation function becomes an attractive method for reliability studies, especially after the correlated sampling technique has been incorporated.

Monte Carlo simulation of ultrafast processes in photoexcited semiconductors: Coherent and incoherent dynamics.

The ultrafast dynamics of photoexcited carriers in a semiconductor is investigated by using a Monte Carlo simulation. In addition to a ``conventional'' Monte Carlo simulation, the coherence of the external light field and the resulting coherence in the carrier system are fully taken into account. This allows us to treat the correct time dependence of the generation process showing a time-dependent linewidth associated with a recombination from states off resonance due to stimulated emission. The subsequent dephasing of the carriers due to scattering processes is analyzed. In addition, the simulation contains the carrier-carrier interaction in Hartree-Fock approximation giving rise to a band-gap renormalization and excitonic effects which cannot be treated in a conventional Monte Carlo simulation where polarization effects are neglected. Thus the approach presents a unified numerical method for the investigation of phenomena occurring close to the band gap and those typical for the energy relaxation of hot carriers.

Monte Carlo Simulation of Diffusion Process on Quasi-Lattice

The characteristics of the diffusion process of quasicrystals are studied for one-dimensional and two-dimensional models by the conventional Monte Carlo simulation. For the one-dimensional model, the tracer diffusion coefficients of an isolated particle are calculated for the three types of lattice models, i.e., single crystal, binary solid solution and quasicrystal. The characteristics of diffusivity of the one-dimensional quasicrystal model are derived from comparisons among the three types of models. For the two-dimensional model, the characteristics of diffusivity of a quasicrystal are discussed in comparison with those of the honeycomb lattice. In particular, the concentration dependence of diffusivity on the two-dimensional Penrose lattice is calculated and the effects of interatomic potential are discussed.

Computing the performance of Fano and stack decoders using importance sampling

An exact analysis method for the evaluation of the bit error rate of coded communication systems which employ the Fano and stack decoders is presented. The proposed method is an extremely efficient importance sampling scheme which substantially reduces the required number of conventional Monte Carlo simulation runs.