Ghost particles

The excess chemiccil potential is thus determined from the average of exp[—lT (r )/fe In ensembles other than the canonical ensemble the expressions for the excess chem potential are slightly different. The ghost particle does not remain in the system and the system is unaffected by the procedure. To achieve statistically significant results m Widom insertion moves may be required. However, practical difficulties are encounte when applying the Widom insertion method to dense fluids and/or to systems contain molecules, because the proportion of insertions that give rise to low values of y f, dramatically. This is because it is difficult to find a hole of the appropriate size and sha... 

MD simulations, complete with ghost particle insertions (160, 161), may be used to obtain static and dynamic information. (These particle insertions were performed after the MD runs and do not affect the calculations they merely probe the insertion of particles into the system.) The MD simulations performed by Snurr et al. (155) were slightly more expensive than the GC-MC calculations, but they produced similar isotherms and also yielded important information about the structure of the adsorbed fluid. The methane molecules appeared to behave like an ordered fluid at all concentrations, although the structure does change. This change reflects the changing importance of sorbate-sorbate and zeolite-sorbate interactions as a function of loading. 

Several factors can affect the accuracy of Fraunhofer diffraction (i) particles smaller than the lower limit of Fraunhofer theory (ii) nonexistent ghost particles in particle size distribution obtained by Fraunhofer diffraction applied to systems containing particles with edges, or a large fraction of small particles (below 10 pm) (iii) computer algorithms that are unknown to the user and vary with the manufacturer s software version (iv) the composition-dependent optical properties of the particles and dispersion medium and (v) if the density of all particles is not the same, the result may be inaccurate. 

For simple fluids at low to moderate densities, Eq. (4.1) provides one of the most reliable means for estimating the chemical potential. For long, articulated molecules, however, random insertions of a ghost particle lead to frequent overlaps with the host system that hamper collection of reliable statistics for Eq. (4.1). For molecules of intermediate size, this sampling problem can be alleviated by resorting to configurational bias ideas... 

At liquid-like densities, the configurational-bias ghost particle approach is reliable only for chain molecules of intermediate length [71]. This limitation can be partially overcome by applying other techniques, such as the method of expanded ensembles. 

To simulate an infinite system as closely as possible, a small number of particles was enclosed in a cell of volume V and periodic boundary conditions were applied to the system. These overcome the problem intrinsic to the use of a small number of particles which is the escape of particles from the cell. The periodic boundary conditions are implemented as follows. Each particle inside the cell has a ghost particle outside that mimics its position and motion exactly. As a result when one particle moves out of the cell, another compensatory particle enters from the other side (Fig. 13.8). 

FIGURE 7.7 Snapshots of the A, = 1/3 mixture model which phase separates. The SPC/E water molecules are shown as red spheres (oxygen site) with two white spheres (hydrogen sites), while weak water is shown with the oxygen as a cyan sphere. Here, x is the mole fraction of weak water. For x = 0.2 and x = 0.8, the majority species is shown as ghost particles. (See color insert.)... 

Figure 2. Order parameter (P versus r (the distance from the simulation box center, measured in lattice units a) in a sample containing a single cylindrical fiber with R = 5a. Planar anchoring along the 2-axis (a) nematic (T = 1.0) and (b) isotropic phase (T = 1.2). In the plots each of the curves corresponds to a different degree of ordering in the ghost particle system P2)g 1.0, 0.75, 0.50, 0.25, and 0 (top to bottom).

Figure 9. Array of several distorted fibers (sample C) ghost particles representing the fixed polymer fiber network P2)g 0.28.

Choose a uniform random number 0 < <1 for each ghost particle and assign it to the state n for which Pn-l[Pg.464]

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