Simulation, Monte Carlo

Monte Carlo simulation (MCS) is a well-known method for evaluating the integral in Equation (2.125) [225]. It is robust since it does not depend on the type of problems or the number of random variables, i.e., the dimension of the integral. The key of MCS is to simulate random samples of 0 according to the distribution p 0) 0, 02, , 6n- Then, the integral in Equation (2.125) can be evaluated directly by statistical averaging  

In evaluating the type of updated robust integral in Equation (2.126) by MCS, it requires the parameter samples distributed according to the updated PDF p 0 T , C). Therefore, generation 

Bayesian Methods for Structural Dynamics and Civil Engineering 

The procedure for Monte Carlo simulation of the cosine law is simple. It can be derived by following the general scheme for continuous distributions [34], This way 

From the contents of Chapter 5 it follows that the original evaluation of the VTC data in terms of the energy of desorption presented in Refs. [29, 31, 35] should be revised. The value of ed was deduced implying the model of mobile adsorption. Meanwhile, if the model were valid, the unhindered movement of the adsorbed molecules across the surface would result in the column, independently of 

Reproduced from Radiochimica Acta, 92(8), Hohn A, Eichler R, Eichler B, Investigations on adsorption and transport behavior of carrier free Ag, Au and Pt in quartz columns under vacuum conditions, 513-516, 2004, with permission from Oldenbourg Wissenschaftsverlag. 

Jonsson JA (1987) Dispersion and peak shapes in chromatography. In Jonsson JA (ed) Chromatographic science series vol 38, Chromatographic theory and basic principles. Dekker, New York 

Giddings JC (1965) Dynamics of chromatography, Part I, Principles and theory. Dekker, New York 

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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... 

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. 

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... 

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

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]. 

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)... 

Binder K and Heermann D W 1997 Monte Carlo Simulation in Statistical Physics 3rd edn, vol 80 Solid State Sciences (Berlin Springer)... 

Butler B D, Ayton C, Jepps C G and Evans D J 1998 Configurational temperature verification of Monte Carlo simulations J. Chem. Phys. fOS 6519-22... 

Gil-Villegas A, McGrother S C and Jackson G 1997 Reaction-field and Ewald summation methods in Monte Carlo simulations of dipolar liquid crystals Mol. Phys. 92 723-34... 

Manousiouthakis V I and Deem M W 1999 Strict detailed balance is unnecessary in Monte Carlo simulation J. Chem. Phys. 1102752-Q... 

Kofke D A and Glandt E D 1988 Monte Carlo simulation of multicomponent equilibria in a semigrand canonical ensemble/Wo/. Phys. 64 1105-31... 

Mon K K and Griffiths R B 1985 Chemical potential by gradual insertion of a particle in Monte Carlo simulation Phys. Rev. A 31 956-9... 

Nezbeda I and Kolafa J 1991 A new version of the insertion particle method for determining the chemical potential by Monte Carlo simulation Mol. SImul. 5 391-403... 

Harris J and Rice S A 1988 A lattice model of a supported monolayer of amphiphile molecules—Monte Carlo simulations J. Ohem. Phys. 88 1298-306... 

Panagiotopoulos A Z 1987 Direot determination of phase ooexistenoe properties of fluids by Monte Carlo simulation in a new ensemble Mol. Phys. 61 813-26... 

Panagiotopoulos A Z 1987 Adsorption and oapillary oondensation of fluids in oylindrioal pores by Monte Carlo simulation in the Gibbs ensemble Mol. Phys. 62 701-19... 

Panagiotopoulos A Z 1989 Exaot oaloulations of fluid-phase equilibria by Monte Carlo simulation in a new statistioal ensemble Int. J. Thermophys. 10 447-57... 

Esoobedo F A and de Pablo J J 1996 Expanded grand oanonioal and Gibbs ensemble Monte Carlo simulation of polymers J. Chem. Phys. 105 4391-4... 

R N. The exponent v = 0.588 has been calculated using renonnalization group teclmiques [9, 10], enumeration teclmiques for short chain lengths and Monte Carlo simulations [13]. 

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

Monte Carlo simulations, which include fluctuations, then yields Simulations of a coarse-grained polymer blend by Wemer et al find = 1 [49] in the strong segregation limit, in rather good... 

A multitude of different variants of this model has been investigated using Monte Carlo simulations (see, for example [M])- The studies aim at correlating the phase behaviour with the molecular architecture and revealing the local structure of the aggregates. This type of model has also proven useful for studying rather complex structures (e.g., vesicles or pores in bilayers). 

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

By virtue of their simple stnicture, some properties of continuum models can be solved analytically in a mean field approxunation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is hrvestigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-parameter model are described in the review by Gompper and Schick [76]. A very interesting feature of tiiese models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a fiinctional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

Kremer K and Binder K 1988 Monte Carlo simulations of lattice models for macromolecules Comp. Phys. Rep. 7 259... 

Deutsoh H-P and Binder K 1993 Mean-field to Ising orossover in the oritioal behavior of polymer mixtures—a finite size sealing analysis of Monte Carlo simulations J. Physique II 3 1049... 

Larson R G 1996 Monte Carlo simulations of the phase behavior of surfaotant solutions J. Physique 116 1441... 

Liverpool T B and Bernardes A T 1995 Monte Carlo simulation of the formation of layered struotures and membranes by amphiphiles J. Physique II 5 1003... 

The most important molecular interactions of all are those that take place in liquid water. For many years, chemists have worked to model liquid water, using molecular dynamics and Monte Carlo simulations. Until relatively recently, however, all such work was done using effective potentials [4T], designed to reproduce the condensed-phase properties but with no serious claim to represent the tme interactions between a pair of water molecules. 

Monte Carlo simulations generate a large number of confonnations of tire microscopic model under study that confonn to tire probability distribution dictated by macroscopic constrains imposed on tire systems. For example, a Monte Carlo simulation of a melt at a given temperature T produces an ensemble of confonnations in which confonnation with energy E. occurs witli a probability proportional to exp (- Ej / kT). An advantage of tire Monte Carlo metliod is tliat, by judicious choice of tire elementary moves, one can circumvent tire limitations of molecular dynamics techniques and effect rapid equilibration of multiple chain systems [65]. Flowever, Monte Carlo... 

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