Monte Carlo,

For the study of systems at equilibrium, Monte Carlo (MC) simulation offers an alternative to MD. MC relies on a Markov chain of configurations created by sampling the phase space using a set of specified moves that are accepted or rejected according to a criterion that ensures convergence of the chain of configurations to their equilibrium distribution. The evolution of a system simulated by the MC method obeys the Master Equation  

The distribution of configurations at equilibrium is then specified in the usual manner by choosing 

Fand W,(f) in turn are related through the fluctuation-dissipation theorem, v, is the particle velocity. The hydrodynamic interaction between particles associated with motion of the intervening solvent may also be included, but doing so compromises considerably the computational benefits of the implicit solvent treatment, and has not been considered in the study of polymer crystallization to date. As with the use of the thermostat in MD, the solvent is presumed small enough 

In writing Equation (6.6), the probability of generating a new trial configuration j is presumed to be a constant, and the kinetic energy is assumed to be the same, on average, for any two configurations i and j. 

A real strength of MC simulation is that one is not constrained to transition from one configuration to another through a sequence of physically realistic steps such as those prescribed by Newton s Law. Instead, very unphysical moves can be designed, so long as they pre- 

This algorithm will generate the ensemble of conformations at thermodynamic equilibrium, if the conformational variations introduced in step 1 (the so-called moves ) are sufficient to cover all possible regions of conformational space and are locally reversible. It is easy to understand that feature of the Metropolis procedure the transition probability into the higher energy state is given by the Boltzmann factor, the transition probability into the lower energy state is one. The relative population of the two states is then the ratio of the transition probabilities, thus the Boltzmann factor this is the outcome expected from thermodynamics. 

The universal algorithm of MC methods was provided early after computers came into use by Metropolis et al. (1953). The name MC stems from a random number generator in the method, similar to that used in casinos. 

In the MC method, molecules are moved randomly from an initial configuration, so that only the immediately previous configuration affects the current position. Using the individual potential (e.g., SPC or TIP4P) between 

As in the molecular dynamic calculations, MC calculations for water structures were first tested against experimental values. Beveridge and coworkers (Swaminathan et al., 1978) and Owicki and Scheraga (1977) obtained acceptable comparison of their calculations against experimental values for the oxygen-oxygen radial distribution function for both water and methane dissolved in water. 

There are substantially fewer MC studies of hydrates than there areMD studies. The initial MC study of hydrates was by Tester et al. (1972), followed by Tse and Davidson (1982), who checked the Lennard-Jones-Devonshire spherical cell approximation for interaction of guest with the cavity. Lund (1990) and Kvamme et al. (1993) studied guest-guest interactions within the lattice. More recently Natarajan and Bishnoi (1995) have studied the technique for calculation of the Langmuir coefficients. 

The first term in each equation represents the contribution of kinetic energy, which is analytically integrable. In the harmonic (low-temperature) limit, E given by Eq. [2] will be a linear function of temperature and Cy from Eq. [3] will be constant, in accordance with the Equipartition Theorem.Eor a small cluster of, say, 6 atoms, the integrals implicit in the calculation of Eqs. [1] 

To calculate the desired thermodynamic averages, it is necessary to have some method available for computation of the potential energy, either explicitly (in the form of a function representing the interaction potential as in molecular mechanics) or implicitly (in the form of direct quantum-mechanical calculations). Throughout this chapter we shall assume that 1/ is known or can be computed as needed, although this computation is typically the most computationally expensive part of the procedure (because 1/ may need to be computed many, many times). For this reason, all possible measures should be taken to assure the maximum efficiency of the method used in the computation of 17. 

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Keywords deterministic methods, STOllP, GllP, reserves, ultimate recovery, net oil sands, area-depth and area-thickness methods, gross rock volume, expectation curves, probability of excedence curves, uncertainty, probability of success, annual reporting requirements, Monte-Carlo simulation, parametric method... 

Figure 6.10 Probability distributions for two variables input for Monte Carlo...

A Monte Carlo simulation is fast to perform on a computer, and the presentation of the results is attractive. However, one cannot guarantee that the outcome of a Monte Carlo simulation run twice with the same input variables will yield exactly the same output, making the result less auditable. The more simulation runs performed, the less of a problem this becomes. The simulation as described does not indicate which of the input variables the result is most sensitive to, but one of the routines in Crystal Ball and Risk does allow a sensitivity analysis to be performed as the simulation is run.This is done by calculating the correlation coefficient of each input variable with the outcome (for example between area and UR). The higher the coefficient, the stronger the dependence between the input variable and the outcome. 

Figure 6.11 Schematic of Monte Carlo simulation 6.2.5 The parametric method...

The parametric method is an established statistical technique used for combining variables containing uncertainties, and has been advocated for use within the oil and gas industry as an alternative to Monte Carlo simulation. The main advantages of the method are its simplicity and its ability to identify the sensitivity of the result to the input variables. This allows a ranking of the variables in terms of their impact on the uncertainty of the result, and hence indicates where effort should be directed to better understand or manage the key variables in order to intervene to mitigate downside and/or take advantage of upside in the outcome. 

From the probability distributions for each of the variables on the right hand side, the values of K, p, o can be calculated. Assuming that the variables are independent, they can now be combined using the above rules to calculate K, p, o for ultimate recovery. Assuming the distribution for UR is Log-Normal, the value of UR for any confidence level can be calculated. This whole process can be performed on paper, or quickly written on a spreadsheet. The results are often within 10% of those generated by Monte Carlo simulation. 

Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. 

In Fig. III-7 we show a molecular dynamics computation for the density profile and pressure difference P - p across the interface of an argonlike system [66] (see also Refs. 67, 68 and citations therein). Similar calculations have been made of 5 in Eq. III-20 [69, 70]. Monte Carlo calculations of the density profile of the vapor-liquid interface of magnesium how stratification penetrating about three atomic diameters into the liquid [71]. Experimental measurement of the transverse structure of the vapor-liquid interface of mercury and gallium showed structures that were indistinguishable from that of the bulk fluids [72, 73]. 

Both the Monte Carlo and the molecular dynamics methods (see Section III-2B) have been used to obtain theoretical density-versus-depth profiles for a hypothetical liquid-vapor interface. Rice and co-workers (see Refs. 72 and 121) have found that density along the normal to the surface tends to be a... 

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

Anderson J B, Traynor C A and Boghosian B M 1993 An exact quantum Monte-Carlo calculation of the helium-helium intermolecular potential J. Chem. Phys. 99 345... 

The solutions to this approximation are obtained numerically. Fast Fourier transfonn methods and a refomuilation of the FINC (and other integral equation approximations) in tenns of the screened Coulomb potential by Allnatt [M are especially useful in the numerical solution. Figure A2.3.12 compares the osmotic coefficient of a 1-1 RPM electrolyte at 25°C with each of the available Monte Carlo calculations of Card and Valleau [ ]. 

Figure A2.3.12 The osmotic coefficient of a 1-1 RPM electrolyte compared with the Monte Carlo results of...

The agreement is excellent up to a 1 molar concentration. The excess energies for 1-1, 2-1, 2-2 and 3-1 charge types calculated from the MS and HNC approximations are shown in figure A2.3.13. The Monte Carlo... 

Figure A2.3.13 The excess energy of 1-1, 2-1, 3-1 and 2-2 RPM electrolytes in water at 25°C. The frill and dashed curves are from the HNC and MS approximations, respectively. The Monte Carlo results of Card and Valleau [63] for the 1-3 and 2-2 charge types are also shown.

Orkoulas G and Panagiotopoulos A Z 1999 Phase behavior of the restricted primitive model and square-well fluids from Monte Carlo simulations in the grand canonical ensemble J. Chem. Phys. 110 1581... 

Card D N and Valleau J 1970 Monte Carlo study of the thermodynamics of electrolyte solutions J. Chem. Phys. 52 6232... 

Valleau J P and Cohen L K 1980 Primitive model electrolytes. I. Grand canonical Monte Carlo computations J. Chem. Phys. 72 5932... 

Jorgenson W L and Ravimohan C 1985 Monte Carlo simulation of the differences in free energy of hydration J. Chem. Phys. 83 3050... 

Alavi A 1996 Path integrals and ab initio molecular dynamics Monte Carlo and Molecular Dynamics of Condensed Matter Systems ed K Binder and G Ciccotti (Bologna SIF)... 

The alternative simulation approaches are based on molecular dynamics calculations. This is conceptually simpler that the Monte Carlo method the equations of motion are solved for a system of A molecules, and periodic boundary conditions are again imposed. This method pennits both the equilibrium and transport properties of the system to be evaluated, essentially by numerically solvmg the equations of motion... 

In principle, simulation teclmiques can be used, and Monte Carlo simulations of the primitive model of electrolyte solutions have appeared since the 1960s. Results for the osmotic coefficients are given for comparison in table A2.4.4 together with results from the MSA, PY and HNC approaches. The primitive model is clearly deficient for values of r. close to the closest distance of approach of the ions. Many years ago, Gurney [H] noted that when two ions are close enough together for their solvation sheaths to overlap, some solvent molecules become freed from ionic attraction and are effectively returned to the bulk [12]. 

Progress in the theoretical description of reaction rates in solution of course correlates strongly with that in other theoretical disciplines, in particular those which have profited most from the enonnous advances in computing power such as quantum chemistry and equilibrium as well as non-equilibrium statistical mechanics of liquid solutions where Monte Carlo and molecular dynamics simulations in many cases have taken on the traditional role of experunents, as they allow the detailed investigation of the influence of intra- and intemiolecular potential parameters on the microscopic dynamics not accessible to measurements in the laboratory. No attempt, however, will be made here to address these areas in more than a cursory way, and the interested reader is referred to the corresponding chapters of the encyclopedia. 

Specific solute-solvent interactions involving the first solvation shell only can be treated in detail by discrete solvent models. The various approaches like point charge models, siipennoleciilar calculations, quantum theories of reactions in solution, and their implementations in Monte Carlo methods and molecular dynamics simulations like the Car-Parrinello method are discussed elsewhere in this encyclopedia. Here only some points will be briefly mentioned that seem of relevance for later sections. 

Bunker D L and Davidson B S 1972 Photolytic cage effect. Monte Carlo experiments J. Am. Chem. Soc. 94 1843... 

Berne B J 1985 Molecular dynamics and Monte Carlo simulations of rare events Multiple Timescales ed J V Brackbill and B I Cohen (New York Academic Press)... 

Bunker D L 1964 Monte Carlo calculations. IV. Further studies of unimolecular dissociation J. Chem. 

Bunker D L 1962 Monte Carlo calculation of triatomic dissociation rates. I. N2O and O3 J. Chem. Rhys. 37 393-403... 

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