Showing papers on "Dynamic Monte Carlo method published in 2007"
01 Jan 2007
TL;DR: The purpose of this chapter is to provide an introduction to this KMC method, by taking the reader through the basic concepts underpinning KMC and how it is typically implemented, assuming no prior knowledge of these kinds of simulations.
Abstract: Monte Carlo refers to a broad class of algorithms that solve problems through the use of random numbers. They first emerged in the late 1940’s and 1950’s as electronic computers came into use [1], and the name means just what it sounds like, whimsically referring to the random nature of the gambling at Monte Carlo, Monaco. The most famous of the Monte Carlo methods is the Metropolis algorithm [2], invented just over 50 years ago at Los Alamos National Laboratory. Metropolis Monte Carlo (which is not the subject of this chapter) offers an elegant and powerful way to generate a sampling of geometries appropriate for a desired physical ensemble, such as a thermal ensemble. This is accomplished through surprisingly simple rules, involving almost nothing more than moving one atom at a time by a small random displacement. The Metropolis algorithm and the numerous methods built on it are at the heart of many, if not most, of the simulations studies of equilibrium properties of physical systems. In the 1960’s researchers began to develop a different kind of Monte Carlo algorithm for evolving systems dynamically from state to state. The earliest application of this approach for an atomistic system may have been Beeler’s 1966 simulation of radiation damage annealing [3]. Over the next 20 years, there were developments and applications in this area (e.g., see [3, 4, 5, 6, 7]), as well as in surface adsorption, diffusion and growth (e.g., see [8, 9, 10, 11, 12, 13, 14, 15, 16, 17]), in statistical physics (e.g., see [18, 19, 20]), and likely other areas, too. In the 1990’s the terminology for this approach settled in as kinetic Monte Carlo, though the early papers typically don’t use this term [21]. The popularity and range of applications of kinetic Monte Carlo (KMC) has continued to grow and KMC is now a common tool for studying materials subject to irradiation, the topic of this book. The purpose of this chapter is to provide an introduction to this KMC method, by taking the reader through the basic concepts underpinning KMC and how it is typically implemented, assuming no prior knowledge of these kinds of simulations. An appealing property of KMC is that it can, in principle, give the exact dynamical evolution of a system. Although this ideal is virtually never achieved, and usually not even attempted, the KMC method is presented here from this point of view because it makes a good framework for
549 citations
TL;DR: In this article, a method for optimizing all parameters of many-body wave functions in quantum Monte Carlo calculations, applicable to continuum systems and lattice models, is presented, based on a strong zero-variance principle.
Abstract: We present a simple, robust, and highly efficient method for optimizing all parameters of many-body wave functions in quantum Monte Carlo calculations, applicable to continuum systems and lattice models. Based on a strong zero-variance principle, diagonalization of the Hamiltonian matrix in the space spanned by the wave function and its derivatives determines the optimal parameters. It systematically reduces the fixed-node error, as demonstrated by the calculation of the binding energy of the small but challenging C(2) molecule to the experimental accuracy of 0.02 eV.
454 citations
365 citations
TL;DR: It is observed for the C2 molecule studied here, and for other systems the authors have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improvesMonotonically, implying that the nodal hypersurface also improves monotonically.
Abstract: We study three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear, and perturbative methods. In the Newton method, the parameter variations are calculated from the energy gradient and Hessian, using a reduced variance statistical estimator for the latter. In the linear method, the parameter variations are found by diagonalizing a nonsymmetric estimator of the Hamiltonian matrix in the space spanned by the wave function and its derivatives with respect to the parameters, making use of a strong zero-variance principle. In the less computationally expensive perturbative method, the parameter variations are calculated by approximately solving the generalized eigenvalue equation of the linear method by a nonorthogonal perturbation theory. These general methods are illustrated here by the optimization of wave functions consisting of a Jastrow factor multiplied by an expansion in configuration state functions (CSFs) for the C2 molecule, including both valence and core electrons in the calculation. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF, and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters. Although the optimization is performed at the variational Monte Carlo level, we observe for the C2 molecule studied here, and for other systems we have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves monotonically.
261 citations
TL;DR: This letter presents a new solution to this problem, where the fermionic determinant is represented using n pseudofermion fields, each with an nth root kernel, within the framework of the rational hybrid Monte Carlo algorithm.
Abstract: There has been much recent progress in the understanding and reduction of the computational cost of the hybrid Monte Carlo algorithm for lattice QCD as the quark mass parameter is reduced. In this letter we present a new solution to this problem, where we represent the fermionic determinant using n pseudofermion fields, each with an nth root kernel. We implement this within the framework of the rational hybrid Monte Carlo algorithm. We compare this algorithm with other recent methods in this area and find it is competitive with them.
239 citations
Journal Article•
TL;DR: In this article, the authors studied three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear, and perturbative methods, and showed that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves.
Abstract: We study three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear, and perturbative methods. In the Newton method, the parameter variations are calculated from the energy gradient and Hessian, using a reduced variance statistical estimator for the latter. In the linear method, the parameter variations are found by diagonalizing a nonsymmetric estimator of the Hamiltonian matrix in the space spanned by the wave function and its derivatives with respect to the parameters, making use of a strong zero-variance principle. In the less computationally expensive perturbative method, the parameter variations are calculated by approximately solving the generalized eigenvalue equation of the linear method by a nonorthogonal perturbation theory. These general methods are illustrated here by the optimization of wave functions consisting of a Jastrow factor multiplied by an expansion in configuration state functions (CSFs) for the C2 molecule, including both valence and core electrons in the calculation. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF, and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters. Although the optimization is performed at the variational Monte Carlo level, we observe for the C2 molecule studied here, and for other systems we have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves monotonically.
191 citations
TL;DR: A new open system Monte Carlo procedure designed to overcome difficulties with insertion and deletion of molecules is introduced, and is shown to yield correct results for the volumetric properties of the Lennard-Jones fluid and water as well as the phase behavior of the CO2-ethanol binary system.
Abstract: A new open system Monte Carlo procedure designed to overcome difficulties with insertion and deletion of molecules is introduced. The method utilizes gradual insertions and deletions of molecules through the use of a continuous coupling parameter and an adaptive bias potential. The method draws upon concepts from previous open system molecular dynamics and expanded ensemble Monte Carlo techniques and is applied to both the grand canonical and osmotic ensembles. It is shown to yield correct results for the volumetric properties of the Lennard-Jones fluid and water as well as the phase behavior of the CO2-ethanol binary system.
183 citations
TL;DR: A "virtual-move" Monte Carlo algorithm for systems of pairwise-interacting particles that employs a size- and shape-dependent damping of cluster movements, motivated by collective hydrodynamic effects neglected in simple implementations of Brownian dynamics is introduced.
Abstract: We introduce a “virtual-move” Monte Carlo algorithm for systems of pairwise-interacting particles. This algorithm facilitates the simulation of particles possessing attractions of short range and arbitrary strength and geometry, an important realization being self-assembling particles endowed with strong, short-ranged, and angularly specific (“patchy”) attractions. Standard Monte Carlo techniques employ sequential updates of particles and can suffer from low acceptance rates when attractions are strong. In this event, collective motion can be strongly suppressed. Our algorithm avoids this problem by proposing simultaneous moves of collections (clusters) of particles according to gradients of interaction energies. One particle first executes a “virtual” trial move. We determine which of its neighbors move in a similar fashion by calculating individual bond energies before and after the proposed move. We iterate this procedure and update simultaneously the positions of all affected particles. Particles move according to an approximation of realistic dynamics without requiring the explicit computation of forces and without the step size restrictions required when integrating equations of motion. We employ a size- and shape-dependent damping of cluster movements, motivated by collective hydrodynamic effects neglected in simple implementations of Brownian dynamics. We discuss the virtual-move algorithm in the context of other Monte Carlo cluster-move schemes and demonstrate its utility by applying it to a model of biological self-assembly.
181 citations
14 Mar 2007
162 citations
TL;DR: Finite size corrections are proposed, which allow us to estimate approximately the free energy of the solid phase in the thermodynamic limit from the known value of the freeEnergy of theSolid phase with N molecules.
Abstract: In this paper a new method to evaluate the free energy of solids is proposed. The method can be regarded as a variant of the method proposed by Frenkel and Ladd [J. Chem. Phys. 81, 3188 (1984)]. The main equations of the method can be derived in a simple way. The method can be easily implemented within a Monte Carlo program. We have applied the method to determine the free energy of hard spheres in the solid phase for several system sizes. The obtained free energies agree within the numerical uncertainty with those obtained by Polson et al. [J. Chem. Phys. 112, 5339 (2000)]. The fluid-solid equilibria has been determined for several system sizes and compared to the values published previously by Wilding and Bruce [Phys. Rev. Lett. 85, 5138 (2000)] using the phase switch methodology. It is shown that both the free energies and the coexistence pressures present a strong size dependence and that the results obtained from free energy calculations agree with those obtained using the phase switch method, which constitutes a cross-check of both methodologies. From the results of this work we estimate the coexistence pressure of the fluid-solid transition of hard spheres in the thermodynamic limit to be p*=11.54(4), which is slightly lower than the classical value of Hoover and Ree (p*=11.70) [J. Chem. Phys. 49, 3609 (1968)]. Taking into account the strong size dependence of the free energy of the solid phase, we propose to introduce finite size corrections, which allow us to estimate approximately the free energy of the solid phase in the thermodynamic limit from the known value of the free energy of the solid phase with N molecules. We have also determined the free energy of a Lennard-Jones solid by using both the methodology of this work and the finite size correction. It is shown how a relatively good estimate of the free energy of the system in the thermodynamic limit is obtained even from the free energy of a relatively small system.
149 citations
TL;DR: The resulting method can capture arbitrarily small deviations from equilibrium at a computational cost that is independent of the magnitude of this deviation, and results in a significant computational efficiency advantage for low-signal flows.
TL;DR: Grand Canonical Monte Carlo simulations have explained the breathing of a metal-organic framework upon CO(2) adsorption, first suggested by microcalorimetry.
TL;DR: In this paper, the authors used a standard Monte Carlo algorithm to study the slow dynamics of a binary Lennard-Jones glass-forming mixture at low temperature and found that the Monte Carlo approach is by far the most efficient way to simulate a stochastic dynamics since the relaxation is about 10 times faster than in Brownian dynamics and about 30 times faster in stochastically dynamic dynamics.
Abstract: We use a standard Monte Carlo algorithm to study the slow dynamics of a binary Lennard-Jones glass-forming mixture at low temperature. We find that the Monte Carlo approach is by far the most efficient way to simulate a stochastic dynamics since the relaxation is about 10 times faster than in Brownian dynamics and about 30 times faster than in stochastic dynamics. Moreover, the average dynamical behaviour of the system is in quantitative agreement with that obtained using Newtonian dynamics, apart from at very short times where thermal vibrations are suppressed. We show, however, that dynamic fluctuations quantified by four-point dynamic susceptibilities do retain a dependence on the microscopic dynamics, as recently predicted theoretically.
TL;DR: The formalism of tensor-network states, such as the matrix-product states (MPS), can be used as a basis for variational quantum Monte Carlo simulations by explicit MPS calculations for the transverse Ising chain with up to N=256 spins at criticality.
Abstract: We show that the formalism of tensor-network states, such as the matrix-product states (MPS), can be used as a basis for variational quantum Monte Carlo simulations. Using a stochastic optimization method, we demonstrate the potential of this approach by explicit MPS calculations for the transverse Ising chain with up to N=256 spins at criticality, using periodic boundary conditions and D x D matrices with D up to 48. The computational cost of our scheme formally scales as ND3, whereas standard MPS approaches and the related density matrix renormalization group method scale as ND5 and ND6, respectively, for periodic systems.
TL;DR: It is shown that down to very small cluster sizes, classical nucleation theory built on the liquid drop model can be used very accurately to describe the work required to add a monomer to the cluster.
Abstract: We carry out molecular Monte Carlo simulations of clusters in an imperfect vapor. We show that down to very small cluster sizes, classical nucleation theory built on the liquid drop model can be used very accurately to describe the work required to add a monomer to the cluster. However, the error made in modeling the smallest of clusters as liquid drops results in an erroneous absolute value for the cluster work of formation throughout the size range. We calculate factors needed to correct the cluster formation work given by the liquid drop model. The corrected work of formation results in nucleation rates in good agreement with recent nucleation experiments on argon and water.
TL;DR: In this article, a method to treat low-energy scattering problems in few-nucleon systems was described, and applied to the five-body case of neutron-alpha scattering.
Abstract: We describe a new method to treat low-energy scattering problems in few-nucleon systems, and we apply it to the five-body case of neutron-alpha scattering. The method allows precise calculations of low-lying resonances and their widths. We find that a good three-nucleon interaction is crucial to obtain an accurate description of neutron-alpha scattering.
TL;DR: In this paper, the authors compared four Monte Carlo (MC) methods for the numerical solution of the general dynamic equation (GDE) in particulate systems and found that when run with a comparable number of particles, all methods compute the size distribution within comparable levels of error.
TL;DR: In this paper, a Monte Carlo simulation of lattice models is used to determine the Casimir forces acting on the confining surfaces of soft media for the Ising and the XY universality classes.
Abstract: The confinement of critical fluctuations in soft media induces critical Casimir forces acting on the confining surfaces. The temperature and geometry dependences of such forces are characterized by universal scaling functions. A novel approach is presented to determine them for films via Monte Carlo simulations of lattice models. The method is based on an integration scheme of free energy differences. Our results for the Ising and the XY universality class agree well with corresponding experimental results for wetting layers of classical binary liquid mixtures and of 4He, respectively.
TL;DR: The method of Monte Carlo simulations demonstrates the competition between subdiffusion and Lévy flights in the framework of the fractional Fokker-Planck dynamics on thelevel of realizations as well as on the level of probability density functions of the anomalous diffusion process.
Abstract: In this paper we answer positively a question raised by Metzler and Klafter [Phys. Rep. 339, 1 (2000)]: can one see a competition between subdiffusion and L\'evy flights in the framework of the fractional Fokker-Planck dynamics? Our method of Monte Carlo simulations demonstrates the competition on the level of realizations as well as on the level of probability density functions of the anomalous diffusion process. The simulation algorithm is based on a stochastic representation of the above dynamics.
TL;DR: The effects of beam energy, mass transport versus reaction-rate-limited growth, and the effects of surface diffusion on the EBID process are compared.
Abstract: A computer simulation was developed to simulate electron-beam-induced deposition (EBID). Simulated growth produced high-aspect-ratio, nanoscale pillar structures by simulating a stationary Gaussian electron beam. The simulator stores in memory the spatial and temporal coordinates of deposited atoms in addition to the type of electron, either primary (PE), back-scattered (BSE), or secondary (SE), that induced its deposition. The results provided in this paper apply to tungsten pillar growth by EBID on a tungsten substrate from WF6 precursor, although the simulation may be applied to any substrate–precursor set. The details of the simulation are described including the Monte Carlo electron–solid interaction simulation used to generate scattered electron trajectories and SE generation, the probability of molecular dissociation of the precursor gas when an electron traverses the surface, and the gas dynamics which control the surface coverage of the WF6 precursor on the substrate and pillar surface. In this paper, three specific studies are compared: the effects of beam energy, mass transport versus reaction-rate-limited growth, and the effects of surface diffusion on the EBID process.
TL;DR: Previously developed DDMC techniques are extended in several ways that improve the accuracy and utility of DDMC for nonlinear, time-dependent, radiative-transfer calculations and treats the interface between optically thick and optically thin regions with an improved method that can produce accurate results regardless of the angular distribution of the incident Monte Carlo particles.
TL;DR: In this article, a three-dimensional Monte Carlo code for modeling radiation transport in Type Ia supernovae is described, where a scheme involving volume-based Monte Carlo estimators is used to allow properties of the emergent radiation field to be extracted for specific viewing angles in a multidimensional structure.
Abstract: A three-dimensional Monte Carlo code for modelling radiation transport in Type Ia supernovae is described. In addition to tracking Monte Carlo quanta to follow the emission, scattering and deposition of radiative energy, a scheme involving volume-based Monte Carlo estimators is used to allow properties of the emergent radiation field to be extracted for specific viewing angles in a multidimensional structure. This eliminates the need to compute spectra or light curves by angular binning of emergent quanta. The code is applied to two test problems to illustrate consequences of multidimensional structure on the modelling of light curves. First, elliptical models are used to quantify how large-scale asphericity can introduce angular dependence to light curves. Secondly, a model which incorporates complex structural inhomogeneity, as predicted by modern explosion models, is used to investigate how such structure may affect light-curve properties.
TL;DR: In the authors' pseudopotential DMC calculations, the total energies of the water monomer and dimer obtained using the locality approximation are compared with those from the variational scheme recently proposed by Casula, and the errors cancel when energy differences are taken.
Abstract: We report a study of the electronic dissociation energy of the water dimer using quantum Monte Carlo techniques. We have performed variational quantum Monte Carlo and diffusion quantum Monte Carlo (DMC) calculations of the electronic ground state of the water monomer and dimer using all-electron and pseudopotential approaches. We have used Slater-Jastrow trial wave functions with B3LYP type single-particle orbitals, into which we have incorporated backflow correlations. When backflow correlations are introduced, the total energy of the water monomer decreases by about 4–5mhartree, yielding a DMC energy of −76.42830(5)hartree, which is only 10mhartree above the experimental value. In our pseudopotential DMC calculations, we have compared the total energies of the water monomer and dimer obtained using the locality approximation with those from the variational scheme recently proposed by Casula [Phys. Rev. B 74, 161102–R (2006)]. The time step errors in the Casula scheme are larger, and the extrapolation of the energy to zero time step always lies above the result obtained with the locality approximation. However, the errors cancel when energy differences are taken, yielding electronic dissociation energies within error bars of each other. The dissociation energies obtained in our various all-electron and pseudopotential calculations range between 5.03(7) and 5.47(9)kcal∕mol and are in good agreement with experiment. Our calculations give monomer dipole moments which range between 1.897(2) and 1.909(4)D and dimer dipole moments which range between 2.628(6) and 2.672(5)D.
TL;DR: The single component adsorption of alkanes in carbon slit pores was studied using configurational-biased grand canonical Monte Carlo simulations and the behavior of long alkanes at high temperatures was found to be similar to short alkane at lower temperatures.
Abstract: The single component adsorption of alkanes in carbon slit pores was studied using configurational-biased grand canonical Monte Carlo simulations. Wide ranges of temperature, pressure, alkane chain length, and slit height were studied to evaluate their effects on adsorption. Adsorption isotherms and density and orientation profiles were calculated. The behavior of long alkanes at high temperatures was found to be similar to short alkanes at lower temperatures. This suggests that the isotherms may be related through the Polanyi potential theory.
TL;DR: The average dynamical behavior of the system is in quantitative agreement with results obtained from molecular dynamics simulations, at least in the long-time regime corresponding to the alpha -relaxation, and the emergence of dynamic heterogeneity is discussed.
Abstract: We implement a standard Monte Carlo algorithm to study the slow, equilibrium dynamics of a silica melt in a wide temperature regime, from 6100K down to 2750K . We find that the average dynamical behavior of the system is in quantitative agreement with results obtained from molecular dynamics simulations, at least in the long-time regime corresponding to the alpha -relaxation. By contrast, the strong thermal vibrations related to the boson peak present at short times in molecular dynamics are efficiently suppressed by the Monte Carlo algorithm. This allows us to reconsider silica dynamics in the context of mode-coupling theory, because several shortcomings of the theory were previously attributed to thermal vibrations. A mode-coupling theory analysis of our data is qualitatively correct, but quantitative tests of the theory fail, raising doubts about the very existence of an avoided singularity in this system. We discuss the emergence of dynamic heterogeneity and report detailed measurements of a decoupling between translational diffusion and structural relaxation, and of a growing four-point dynamic susceptibility. Dynamic heterogeneity appears to be less pronounced than in more fragile glass-forming models, but not of a qualitatively different nature.
TL;DR: A novel, parallelised approach to Monte Carlo simulations for the computation of full molecular weight distributions arising from complex polymerisation reactions is presented and it seems viable to fuse parallel Monte Carlo methods with those based on the h-p Galerkin methods to achieve an optimum of information depths for the modelling of complex macromolecular kinetics and the resulting microstructural information.
Abstract: A novel, parallelised approach to Monte Carlo simulations for the computation of full molecular weight distributions (MWDs) arising from complex polymerisation reactions is presented. The parallel Monte Carlo method constitutes perhaps the most comprehensive route to the simulation of full MWDs of multiple chain length polymer entities and can also provide detailed microstructural information. New fundamental insights have been developed with regard to the Monte Carlo process in at least three key areas: (i) an insufficient system size is demonstrated to create inaccuracies via poor representation of the most improbable events and least numerous species,- (ii) advanced algorithmic principles and compiler technology known to computer science have been used to provide speed improvements and (iii) the parallelisability of the algorithm has been explored and excellent scalability demonstrated. At present, the parallel Monte Carlo method presented herein compares very favourably in speed with the latest developments in the h-p Galerkin methodbased PREDICI software package while providing significantly more detailed microstructural information. It seems viable to fuse parallel Monte Carlo methods with those based on the h-p Galerkin methods to achieve an optimum of information depths for the modelling of complex macromolecular kinetics and the resulting microstructural information. © 2007 WILEY-VCH Verlag GmbH & Co. KCaA.
TL;DR: It is shown that sequential updating is the key to improving efficiency in parallel simulations through domain decomposition in Markov chain theory and its role in accelerating Monte Carlo simulations via cluster computing is explored.
Abstract: With strict detailed balance, parallel Monte Carlo simulation through domain decomposition cannot be validated with conventional Markov chain theory, which describes an intrinsically serial stochastic process. In this work, the parallel version of Markov chain theory and its role in accelerating Monte Carlo simulations via cluster computing is explored. It is shown that sequential updating is the key to improving efficiency in parallel simulations through domain decomposition. A parallel scheme is proposed to reduce interprocessor communication or synchronization, which slows down parallel simulation with increasing number of processors. Parallel simulation results for the two-dimensional lattice gas model show substantial reduction of simulation time for systems of moderate and large size.
TL;DR: In this paper, a method for the prediction of extreme response statistics of floating offshore structures subjected to random seas by Monte Carlo simulation is described, taking into account both the first order, wave frequency and the second order, slow-drift motions.