Monte-Carlo simulation boundary conditions

Monte Carlo simulations have been done on the TV x x cubic lattice (TV = 27) with the lattice spacing h = 0.8 [47,49] for a bulk system. The usual temperature factor k T is set to 1, since it only sets the energy scale. The following periodic boundary conditions are used = [Pg.714]

The temperature, pore width and average pore densities were the same as those used by Snook and van Megen In their Monte Carlo simulations, which were performed for a constant chemical potential (12.). Periodic boundary conditions were used In the y and z directions. The periodic length was chosen to be twice r. Newton s equations of motion were solved using the predictor-corrector method developed by Beeman (14). The local fluid density was computed form... 

Monte Carlo simulations were carried out to determine the free energy curve for the reaction in solution. The simulations were executed for the solute surrounded by 250 water molecules (or 180 DMF molecules) in the isothermal-isobaric ensemble at 25 °C and 1 atm, including periodic boundary conditions. As a consequence, the Gibbs free energy is obtained in this case. There is sufficient solvent to adequately represent the bulk participation in the chemical reaction. 

After this computer experiment, a great number of papers followed. Some of them attempted to simulate with the ab-initio data the properties of the ion in solution at room temperature [76,77], others [78] attempted to determine, via Monte Carlo simulations, the free energy, enthalpy and entropy for the reaction (24). The discrepancy between experimental and simulated data was rationalized in terms of the inadequacy of a two-body potential to represent correctly the n-body system. In addition, the radial distribution function for the Li+(H20)6 cluster showed [78] only one maximum, pointing out that the six water molecules are in the first hydration shell of the ion. The Monte Carlo simulation [77] for the system Li+(H20)2oo predicted five water molecules in the first hydration shell. A subsequent MD simulation [79] of a system composed of one Li+ ion and 343 water molecules at T=298 K, with periodic boundary conditions, yielded... 

In order to describe the fluorescence radiation profile of scattering samples in total, Eqs. (8.3) and (8.4) have to be coupled. This system of differential equations is not soluble exactly, and even if simple boundary conditions are introduced the solution is possible only by numerical approximation. The most flexible procedure to overcome all analytical difficulties is to use a Monte Carlo simulation. However, this method is little elegant, gives noisy results, and allows resimulation only according to the method of trial and error which can be very time consuming, even in the age of fast computers. Therefore different steps of simplifications have been introduced that allow closed analytical approximations of sufficient accuracy for most practical purposes. In a first... 

In figure 1 a) we address the comparison between the analytical adsorption isotherm in one dimension and Monte Carlo simulation. The simulations have been performed for monomers, dimers and 10>mers adsorbed on chains of M/k =1000 sites with periodic boundary conditions. Different values of the parameter c have been considered. In all cases, the computational data fully agree with the theoretical predictions, which reinforce the robustness of the two methodologies employed here. 

Further, Imdakm and Matsuura [61] have developed a Monte Carlo simulation model to smdy vapor permeation through membrane pores in association with DCMD, where a three-dimensional network of interconnected cylindrical pores with a pore size distribution represents the porous membrane. The network has 12 nodes (sites) in every direction plus boundary condition sites (feed and permeate). The pore length / is assumed to be of constant length (1.0 p,m), however, it could have any value evaluated experimentally or theoretically [62]. 

Fig. 6. Visulization of the Monte Carlo simulation periodic boundary conditions are used where the mirror image of a particle enters through the opposite face when a particle leaves (Ref. 24).

Fig. 47. Log-log plot of the surface layer magnetization mi vs. reduced temperature, for various ratios of the exchange 7S in the surface planes to the exchange J in the bulk. Slopes of the straight lines yield effective exponents jS (indicated by the number). Data are from Monte Carlo simulation of 50 x 50 x 40 lattices with two free 50 x 50 surfaces and otherwise periodic boundary conditions. From Binder and Landau (1984).

Fig. 50. Magnetization profiles across thin Ising films [eq. (1) with H — H — 0], upper part, and near the surface of semi-infinite Heisenberg ferromagnels, lower part (where bulk behavior in the Monte Carlo simulation is enforced by an effective field boundary condition at z — 16). Note that in the Ising case (where three film thicknesses L = 5, 10, and 20 are shown) the surface layer magnetization m- — m(z — 0) is independent of L, and for L > 10 already the bulk value of the order parameter is reached in the center of the film. For the Heisenberg model, on the other hand, at a comparable temperature distance from % the free surface produces a Long-range perturbation of the local magnetization m(z). From Binder and Hohenbcrg (1974).

Diffusion is intricately linked with all aspects of the radical-pair mechanism. The CIDNP kinetics for the reaction of a sensitizer with a large spherical molecule that has only a small reactive spot on its surface were studied theoretically. This situation is t)qjical for protein CIDNP, where only three amino acids are readily polarizable, and where such a polarizable amino acid must be exposed to the bulk solution to be able to react with a photoexcited dye. Goez and Heun carried out Monte Carlo simulations of diffusion for radical ion pairs both in homogeneous phase and in micelles. The advantage of this approach compared to numerical solutions of the diffusion equation is that it can easily accommodate arbitrary boundary conditions, such as non-spherical symmetry, as opposed to the commonly used "model of the microreactor" ° where a diffusive excursion starts at the micelle centre and one radical is kept fixed there. 

Like MD, Monte Carlo simulations are typically performed on ensembles containing several thousand particles, to which periodic boundary conditions are applied in the case of the simulation of solids. The simulation again starts... 

The two extreme conditions discussed above are more amenable to analysis since, in both cases, the sheath can be described as a DC sheath actually a series of DC sheaths at the different moments in time during the RF cycle when wt+ 1, and a DC sheath at the time-average voltage when o t+ 1. The most difficult situation to analyze is when cut+ 1. Monte Carlo simulations have been performed in this intermediate regime [179, 180] but the boundary conditions for ion injection are not well understood, and comparison with experimental data is lacking. A fluid model developed by Miller and Riley [32] attempts the bridge the gap between the low and high frequency sheaths. 

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