On the Sampling Method for Grand Canonical Monte Carlo Simulations

Abstract

A bulk Lennard-Jones fluid was simulated using the grand canonical Monte Carlo method. Three different sampling methods were used in the transition matrix, namely the Metropolis, Barker and a third novel method. While it can be shown that the Metropolis method will give the most accurate ensemble averages in the limit of an infinitely long run, the new method termed “Modified Barker Sampling” (MBS), is shown to be superior for the runs of practical length for the particular system studied.     

Taboo-based Monte Carlo Search as a Method to Improve Sampling Efficiency

Abstract

Taboo-based Monte Carlo search which restricts the sampling of the region near an old configuration, is developed. In this procedure, Monte Carlo simulation and random search method are combined to improve the sampling efficiency. The feasibility of this method is tested on global optimization of a continuous model function, melting of the 256 Lennard-Jones particles at T? = 0.680 and ρ? = 0.850 and polypeptides (alanine dipeptide and Metenkephalin). From the comparison of results for the model function between our method and other methods, we find the increase of convergence rate and the high possibility of escaping from the local energy minima. The results of the Lennard-Jones solids and polypeptides show that the convergence property to reach the equilibrium state is better than that of others. It is also found that no significant bias in ensemble distribution is detected, though taboo-based Monte Carlo search does not sample the correct ensemble distribution owing to the restriction of the sampling of the region near an old configuration.     

Grand Canonical Ensemble Monte Carlo Simulation on a Transputer Array

Five methods are described for the distribution over a 3D Transputer array of the calculation of the pair interaction component of particle energy. The most efficient method, expressed in terms of the time to complete a simulation, depends on the size of the simulation and the Transputer array. This dependence is quantified, with emphasis on Grand Canonical Ensemble Monte Carlo simulation, and yields criteria for the optimum strategy for parallel implementation of GCEMC algorithms. The equations derived are generally applicable, and have implications for the programming of Molecular Dynamics simulations.     

Simulations on the Primitive Electrolyte Environment of a High Charge-density Polyelectrolyte. A Sampling Problem and its Solution

Abstract

We show that the classical Metropolis Monte Carlo (MMC) algorithm converges very slowly when applied to the primitive electrolyte environment for a high charge-density polyelectrolyte. This slowness of convergence, which is due to the large density inhomogeneity around the polyelectrolyte, produces noticeable errors in the ion distribution functions for MMC runs of 1.3 × 10⁶ trial steps started from nonequilibrium distributions. We report that an algorithm which we call DSMC (for density-scaled Monte Carlo) overcomes this problem and provides relatively rapid convergence in this application. We suggest that DSMC should be well-suited for other Monte Carlo simulations on physical systems where large density inhomogeneities occur.     

Computer Simulation of Two-Dimensional Continuum Flows by the Direct Simulation Monte Carlo Method

Abstract

Computer simulations using particles are an attractive method to extract microscopic information of flow phenomena [1]. The molecular dynamics (MD) method, in which Newton's equations of motions are integrated, gives the temporal development of the system. In the MD simulation of fluid flows, the computational region is limited to atomistic scales [2]. On the other hand, the direct simulation Monte Carlo (DSMC) method, in which collisions of particles are made on a probabilistic basis, has a potential of treating a realistic system with a macroscopic scale length retaining the atomistic details. The DSMC method provides an efficient way to integrate the Boltzmann equation from the rarefied gas to the near-continuum region. Bird clarified the validity of the DSMC method in the near-continuum flow region [3]. However, the DSMC method has not been applied to the continuum region and compared with the continuum hydrodynamics.     

Conformational free energy of alkylsilanes by nonequilibrium-pulling Monte Carlo simulation

The stationary phase in supercritical fluid chromatography includes alkylsilanes, bearing typically 18-carbon alkane chains, bonded to silica. The silanes are in contact with supercritical carbon dioxide. Interaction of the stationary phase with analytes from the mobile phase depends on conformation of the silanes, whether they form a collapsed layer between the silica and the carbon dioxide or are extended into the carbon dioxide. Although equilibrium conformation of alkylsilanes can be determined by equilibrium Monte Carlo (MC) simulation, that is hampered by slow relaxation of the chains. An alternative is to pull alkylsilanes from collapsed to extended conformations, then calculate free energy change from the Jarzynski equality. This work compares conformational results from equilibrium MC simulation to free energies from nonequilibrium pulling simulations. Because both equilibrium and nonequilibrium simulations are faster for shorter silanes, this work also compares results from 8-carbon and 18-carbon silanes. Free energies from nonequilibrium pulling predict that alkylsilanes tend to bend over and form a layer between silica and carbon dioxide. Results from equilibrium simulations are qualitatively consistent with results from nonequilibrium pulling. Longer-chain silanes have greater tendency to extend slightly into the carbon dioxide.     

Development of Dual Ensemble Monte Carlo Program and its Application to the CO2/N2 Separation

Abstract

A novel Monte Carlo simulation named as Dual Ensemble Monte Carlo (DEMC) method is developed for the investigation of the membrane separation process. In this method the spatial combination of Grand Canonical MC and Canonical MC techniques is employed. The DEMC method can be used to calculate the separation factor at a specific chemical potential gradient. At first, a check on the accuracy of the DEMC method is made by generating gas density gradient between two reservoir regions. Thereafter, we applied this method to CO₂/N₂ gas separation by inorganic membranes and calculated the separation factor dependence on the size of micropore in membranes.     

Monte Carlo Simulations of the Chemical Potential and Free Energy for Trimer and Hexamer Rings

Abstract

The chemical potential of a trimer and hexamer model ring system was determined by computer simulation over a range of temperatures and densities. Such ring molecules are important as model aromatic and naphthenic hydrocarbons. Thermodynamic integration of the pressure along a reversible path, Widom's ghost particle insertion method and Kirkwood's charging parameter method were used over a molecular density range of 0.05 to 0.30. Data were obtained by Monte Carlo simulation of a 96 molecule system that was modelled with a Lennard-Jones 6-12 truncated potential. The original insertion method, which does not take into account the orientation of the molecule when it is inserted, gives results for the chemical potential which deviate from that obtained using the thermodynamic pressure integration. At high density or temperature the deviation is significant. We have modified the Widom insertion technique to account for this short range orientation and find good agreement between this technique and the thermodynamic integration method for the chemical potential. We also calculated the free energy difference between our model ring molecules and ring molecules made up of hard spheres.     

A Monte Carlo Simulation of Nematic and Discotic Ordering in a Polymeric Liquid Crystal

A cyclic polymeric liquid crystal system is simulated using the Metropolis Monte Carlo method in the NVT ensemble. The polymeric system consists of mesogenic moieties attached by alkyl chain spacers to siloxane ring polymers. In the model, the mesogenic moieties are represented individually by an anisotropic Lennard-Jones potential and the polymer ring is represented solely as a constraint on the relative motions of the attached mesogens. A transition from calamitic ordering to discotic ordering is observed as the ring-mesogen bond is varied from full flexible to rigid.     

Transition Path Sampling Study of the Local Molecular Structure in the Aqueous Solvation of Sodium Chloride

Abstract

The aqueous solvation of sodium chloride has been investigated using the recently introduced technique of the transition path sampling. We performed a series of Monte Carlo simulations for each element of an ensemble of chains of states. The evolution of the local solvent structure during the dissociation process has been observed. The incoming of a couple of waters to the first coordination shell is responsible for the structural changes which allow the dissociation occur: waters which leave the second coordination shell produce voids and a local molecular reorganization in order to allocate the dissociated ion pair.     

A Monte Carlo scheme based on mid-density in a hysteresis loop to determine equilibrium phase transition

A new and simple method to determine equilibrium phase transition in adsorption systems exhibiting a hysteresis loop is presented as an alternative to methods such as multiple histogram reweighting, gauge cell method and thermodynamic integration. This method is based on the NVT-grand canonical Monte Carlo mid-density scheme to determine the coexistence chemical potential and coexistence densities of an adsorption system. We illustrate this new scheme with argon and methane adsorption in a number of model solids having slit and cylindrical pores. This method does not have a strong basis on thermodynamic ground, but it does provide a simple heuristic approach that is simpler to understand physically.     

Vapour-Liquid Phase Equilibria of n-alkanes by Direct Monte Carlo Simulations

Abstract

We report results of direct Monte Carlo simulations of n-pentane and n-decane at the liquidvapour interface for a number of temperatures. The intermolecular interactions are modeled using the last version of the anisotropic united atom model (AUA4). We have used the local long range correction energy and an algorithm allowing to select randomly with equal probability two different displacements. The liquid and vapour densities are in excellent agreement with experimental data and with those previously calculated using the GEMC method.     

Monte Carlo Simulation of Nonionic Surfactants at the Oil-Water Interface

Abstract

A Monte Carlo simulation method has been developed for modelling amphiphiles at an oil-water interface. Properties are calculated for the mixture water, benzene and tetraoxyethylene glycol dodecyl ether.     

Direct Determination of Fluid Phase Equilibria by Simulation in the Gibbs Ensemble: A Review

This paper provides an extensive review of the literature on the Gibbs ensemble Monte Carlo method for direct determination of phase coexistence in fluids. The Gibbs ensemble technique is based on performing a simulation in two distinct regions in a way that ensures that the conditions of phase coexistence are satisfied in a statistical sense. Contrary to most other available techniques for this purpose, such as thermodynamic integration, grand canonical Monte Carlo or Widom test particle insertions, the Gibbs ensemble technique involves only a single simulation per coexistence point. A significant body of literature now exists on the method, its theoretical foundations, and proposed modifications for efficient determination of equilibria involving dense fluids and complex intermolecular potentials. Some practical aspects of Gibbs ensemble simulation are also discussed in this review. Applications of the technique to date range from studies of simple model potentials (for example Lennard–Jones, square-well or Yukawa fluids) to calculations of equilibria in mixtures with components described by realistic potentials. We conclude by discussing the limitations of the technique and potential future applications.     

On the existence of a hysteresis loop in open and closed end pores

We have studied the adsorption of argon at 87 K in slit pores of finite length with a smooth graphitic potential, open at both ends or closed at one end. Simulations were carried out using conventional GCMC (grand canonical Monte Carlo) or kMC (kinetic Monte Carlo) in the canonical ensemble with extremely long Markov chain, of at least 2 × 10⁸ configurations; selected simulations with much longer Markov chains do not show any change in the results. When the pore width is in the micropore range (0.65 nm), type I isotherms are obtained for both pore models and for both simulation methods. However, wider pores (1, 2 and 3 nm in width) all exhibit hysteresis loops in the GCMC simulations, while in the canonical ensemble simulations, the isotherms pass through a sigmoid van der Waals type loop in the transition region. This loop locates the true equilibrium transition. For the pores with one closed end, this transition is close to, or coincides with, the adsorption branch of the GCMC hysteresis loop, but for the open-ended pores, it is more closely associated with the desorption branch. In a separate study of adsorption hysteresis in an infinitely long slit pore, using both simulation techniques, the van der Waals loop follows the adsorption branch of the GCMC isotherm to the transition, then reverts to a long vertical section that falls midway between the two hysteresis branches and finally moves to the desorption transition close to the evaporation pressure. An examination of molecular distributions inside the pores reveals two coexisting phases in the canonical simulations, whereas in the grand canonical simulations, the molecules are uniformly distributed along the length of the pores.     

Monte Carlo Simulation of the β-Sheet–Random Coil Transition of a Homopolypeptide. I. Equilibrium Study

Abstract

Monte Carlo simulation of the β-sheet – random coil conversion of a homopolypeptide chain was carried out on the basis of a model where successive two amino acid residues were assumed to change their states simultaneously and hence constituted a basic unit. Only three states were considered for each unit: extended, turn and coil. The conversion exhibited a transition between two states, random coil (C) and the β-sheet (B). In the transition region, two population maxima were always found, each corresponded to the local minimum of the free energy and there was an energy barrier between them. This behavior is characteristic of the all-or-none type transition. We have found that the nature of the first-order transition is retained in the case of a small system consisting of 100 units. The size of the cooperative unit was evaluated. According to the analytical theory of Kanô, a transition curve was obtained which was very close to the present one. This consistent result has suggested that equilibrium properties of the β-sheet-random coil transition are well evaluated with the mean field approximation. The matrix method of Mattice is also discussed.     

Monte Carlo Comparison of Multiple Comparison Procedures

Seven procedures of multiple comparisons: Tukey, Scheffé, Bonferroni, Studentized Maximum Modulus, Duncan, Newman-Keuls and F are compared with respect to the probability of the correct decision. Monte Carlo simulation shows that there is no the best procedure. AMS 1985 Subject Classification: 62 J 15.     

Monte Carlo estimates of natural variation in HIV infection

We describe a Monte Carlo simulation of the within-host dynamics of human immunodeficiency virus 1 (HIV-1). The simulation proceeds at the level of individual T-cells and virions in a small volume of plasma, thus capturing the inherent stochasticity in viral replication, mutation and T-cell infection. When cell lifetimes are distributed exponentially in the Monte Carlo approach, our simulation results are in perfect agreement with the predictions of the corresponding systems of differential equations from the literature. The Monte Carlo model, however, uniquely allows us to estimate the natural variability in important parameters such as the T-cell count, viral load, and the basic reproductive ratio, in both the presence and absence of drug therapy. The simulation also yields the probability that an infection will not become established after exposure to a viral inoculum of a given size. Finally, we extend the Monte Carlo approach to include distributions of cell lifetimes that are less-dispersed than exponential.     

A semi-analytical model for calculating the penetration depth of a high energy electron beam in a water phantom with a magnetic field

PurposeAs an electron beam is incident on a uniform water phantom in the presence of a lateral magnetic field, the depth-dose distribution of the electron beam changes significantly and forms the well-known ‘Bragg peak’, with a depth-dose distribution similar to that of heavy ions. This phenomenon has pioneered a new field in the clinical application of electron beams. For such clinical applications, evaluating the penetration depth of electron beams quickly and accurately is the critical problem.MethodsThis paper describes a model for calculating the penetration depth of an electron beam rapidly and correctly in a water phantom under the influence of a magnetic field. The model was used to calculate the penetration depths under different conditions: the energies of electron beams of 6, 8, 12 and 15 MeV and the magnetic induction intensities of 0.75, 1.0, 1.5, 2.0 and 3.0 T. In addition, the calculation results were compared with the results of a Monte Carlo simulation.ResultsThe comparison results indicate that the difference between the two calculation methods was less than 0.5 cm. Moreover, the computing time of the calculation model was less than a second.ConclusionsThe semi-analytical model proposed in the present study enables the penetration depth of the electron beam in the presence of a magnetic field to be obtained with a computational efficiency higher than that of the Monte Carlo approach; thus, the proposed model has high potential for application.     

A Modified Gibbs Ensemble Method for Calculating Fluid Phase Equilibria

Abstract

A modification of the Gibbs ensemble Monte Carlo computer simulation method for fluid phase equilibrium is described. The modification, which is based on the assumption of a thermodynamic model for the vapor phase, reduces the computational time for the simulation as compared to the original Gibbs ensemble methods. Since the computational time is largely proportional to the number of particle-particle interactions, avoiding the direct simulation of the vapor phase typically leads to a thirty to forty percent reduction in computational time. For a pure Leonard-Jones-(12,6) fluid the results obtained at moderate reduced temperatures, T/T_c < 0.8, are in good agreement with the full Gibbs ensemble method.