Size effect numerical results

Steward (S3) proposed an algorithm based on tearing a variable from only one equation at a time and evaluating each tear on the basis of the size of the resulting subsystems of simultaneous equations in the torn system and numerical considerations of the particular equations. Each variable is torn successively from each equation in which it appears and the effectiveness of the tear evaluated. 

The numerical efficiency of the viscosity lower bound method has allowed calculations on considerably longer chains. The long chain Hmit results for 12-arm stars without intramolecular interactions and with EV (up to 325 beads) and in the theta region (up to 145 beads) [164] are close to the previous estimates with shorter chains (the extrapolated ratio g obtained in this study is also included in Table 4). The lower bound method has also served to characterize globule-coil transitions of 12-arm star chains from intrinsic viscosity calculations [143], though finite size effects are considerably more important than in the characterization of this transition from the radius of gyration data (see Fig. 17). This is due to the noticeable increase in the solvent permeabiHty associated with the chain expansion in better solvent conditions. However, the permeability effects are smaller in the more compact star chains than in their linear counterparts. 

In the past, the equivalence between the size distribution generated by the Smoluchowski equation and simple statistical methods [9, 12, 40-42] was a source of some confusion. The Spouge proof and the numerical results obtained for the kinetics models with more complex aggregation physics, e.g., with a presence of substitution effects [43,44], revealed the non-equivalence of kinetics and statistical models of polymerization processes. More elaborated statistical models, however, with the complete analysis made repeatedly at small time intervals have been shown to produce polymer size distributions equivalent to those generated kinetically [45]. Recently, Faliagas [46] has demonstrated that the kinetics and statistical models which are both the mean-field models can be considered as special cases of a general stochastic Markov process. 

Lanthanides doped into nanocrystalline semiconductors have been the subject of numerous investigations in the past decades. If the size of a semiconductor particle is smaller than the Bohr radius of the excitons, the so-called quantum confinement occurs. As a result, the band gap of the semiconductor increases and discrete energy levels occur at the edges of the valence and conduction bands (Bol et al., 2002 Bras, 1986). These quantum size effects have stimulated extensive interest in both basic and applied research. 

At larger particle size, several experimental results suggest the existence of a moderate coercivity increase as the particle size is decreased [54], This unexpected behavior was examined by numerical modeling. When more than two particles are involved in reversal [116], a competition may exist between the various local simultaneous processes, thus accounting for the observed effect. 

The observed differences among the data obtained at similar JSC and LFJSC operating conditions suggest that comparisons of data from the various combustors must allow for possible effects of combustor type in the results. Extensions of the results to larger, commercial size gas turbine combustors or to similar laboratory combustors should also be made with extreme caution. Qualitative trends such as the dependence of fuel nitrogen conversion on HC concentrations should be valid within a given combustor however, quantitative numerical results may depend strongly on specific combustor characteristics. For example,... 

Breadth of Phase Transition. The abruptness or width of the coexistence region of the order-disorder transition has also been estimated theoretically. For a monodisperse solution of a relatively high molecular weight polymer, an intrinsic coexistence region, ATc, exists because of standard flnite-size fluctuation effects. Numerical calculations yield the result (22)... 

Abstract In a solid with orbital degree of freedom, an orbital configuration does not minimize simultaneously bond energies in equivalent directions. This is a kind of frustration effect which exists intrinsically in orbital degenerate system. We review in this paper the intrinsic orbital frustration effects in Mott insulating systems. We introduce recent our theoretical studies in three orbital models, i.e. the cubic lattice orbital model, the two-dimensional orbital compass model and the honeycomb lattice orbital model. We show numerical results obtained by the Monte-Carlo simulations in finite size systems, and introduce some non-trivial orbital states due to the orbital frustration effect. 

Numerical results obtained by Han and Lawler (25) for the effects of hydrodynamic interactions on particle transport by fluid shear are summarized graphically in Figure 9. These results are based on the work of Adler (26). The effects of particle size, velocity gradient, and van der Waals interaction are characterized by a dimensionless group, HA, defined as follows ... 

The standard way to overcome the surface effects which result from the finite size of systems studied in numerical simulations is the use of periodic boundary conditions. With this requirement, systems of 1000-10000 particles interacting by potentials having a range of the order of a few partide diameters are sufficient to perform simulations where the bias on the computation data, induced by the finite size effects is at an acceptable level of 1%. However, when the interactions between the particles are of Coulombic and/or dipolar types, their long range cannot be neglected because it is at the origin... 

There are, however, important limitations in the application of MD First, practical considerations limit the model system to relatively small sizes of 10 to 10 atoms. This makes it difficult to extrapolate the results of MD, and to compare them with those of macroscopic systems consisting of 10 atoms. Second, our knowledge of the interaction potentials among various kinds of atoms is seriously deficient. It is therefore difficult to compare MD results directly with real experiments. Moreover, chemical reactions are basically controlled by quantum mechanics, not classical mechanics. This imposes additional difficulties in MD simulations of chemical detonation. Third, only numerical solutions are available, and from these it is difficult to develop a broad theoretical understanding of the particular effect under study. Great care must be exercised in the interpretation of the numerical results, and in making certain that they are physically realistic, and are not simply an idiosyncrasy of the numerical model or the assumed boundary conditions. 

In more recent theoretical work Hubbard Colonomos and Wolynes (81) developed a molecular theory of the dynamics of the two depolarization processes which predicts specific effects of ion size reducing to the Hubbard-Onsager result for slip boundary conditions in the limit of large ion radii. Numerical results for methanol were later tested by measurements of Winsor and Cole (82) for five salts which... 

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