Discrete lattice approach

In order to study theoretically defect aggregation, several methods of physical and chemical kinetics were developed in recent years. Irrespective of the particular method used, the two basic approaches - a continuous and discrete-lattice ones - are used. In the former model intrinsic defect volume is ignored and thus a number of similar defects in any volume element is unlimited. In its turn, in the latter model any lattice site could be occupied by no more than a single particle (v or i) [15]. 

We now consider a more quantitative model of the vibrational density of states which makes a remarkable linkage between continuum and discrete lattice descriptions. In particular, we undertake the Debye model in which the vibrational density of states is built in terms of an isotropic linear elastic reckoning of the phonon dispersions. Recall from above that in order to effect an accurate calculation of the true phonon dispersion relation, one must consider the dynamical matrix. Our approach here, on the other hand, is to produce a model representation of the phonon dispersions which is valid for long wavelengths and breaks down at... 

This is possible within the framework of the self-consistent field (SCF) approach to polymer configurations, described more completely elsewhere [18, 19, 51, 52]. Implementation of this method in its full form invariably requires numerical computations which are done in one of two equivalent ways (1) as solutions to diffusion- or Schrodinger-type equations for the polymer configuration subject to the SCF (in which solutions to the continuous-space formulation of the equations are obtained by discretization) or (2) as solutions to matrix equations resulting from a discrete-space formulation of the problem on a lattice. 

To represent the elasticity and dispersion forces of the surface, an approach similar to that of Eqs. (3) and (4) can be taken. The waU molecules can be assumed to be smeared out. And after performing the necessary integration over the surface and over layers of molecules within the surface, a 10-4 or 9-3 version of the potential can be obtained [54,55], Discrete representation of a hexagonal lattice of wall molecules is also possible by the Steele potential [56], The potential is essentially one dimensional, depending on the distance from the wall, but with periodic variations according to lateral displacement from the lattice molecules. Such a representation, however, has not been developed in the cylindrical pore... 

Such an approach is conceptually different from the continuum description of momentum transport in a fluid in terms of the NS equations. It can be demonstrated, however, that, with a proper choice of the lattice (viz. its symmetry properties), with the collision rules, and with the proper redistribution of particle mass over the (discrete) velocity directions, the NS equations are obeyed at least in the incompressible limit. It is all about translating the above characteristic LB features into the physical concepts momentum, density, and viscosity. The collision rules can be translated into the common variable viscosity, since colliding particles lead to viscous behavior indeed. The reader interested in more details is referred to Succi (2001). 

Following the approach discussed in Section 2.2.2, let us divide the whole reaction volume V of the spatially extended system into N equivalent cells (domains) [81]. However, there is an essential difference with the mesoscopic level of treatment in Section 2.2.2 a number of particles in cells were expected to be much greater than unity. Note that this restriction is not imposed on the microscopic level of system s treatment. Their volumes are chosen to be so small that each cell can be occupied by a single particle only. (There is an analogy with the lattice gas model in the theory of phase transitions [76].) Despite the finiteness of vq coming from atomistic reasons or lattice discreteness, at the very end we make the limiting transition vo - 0, iV - oo, v0N = V, to the continuous pattern of point dimensionless particles. 

The transition to the continuum fluid may be mimicked by a discretization of the model choosing > 1. To this end, Panagiotopoulos and Kumar [292] performed simulations for several integer ratios 1 < < 5. For — 2 the tricritical point is shifted to very high density and was not exactly located. The absence of a liquid-vapor transition for = 1 and 2 appears to follow from solidification, before a liquid is formed. For > 3, ordinary liquid-vapor critical points were observed which were consistent with Ising-like behavior. Obviously, for finely discretisized lattice models the behavior approaches that of the continuum RPM. Already at = 4 the critical parameters of the lattice and continuum RPM agree closely. From the computational point of view, the exploitation of these discretization effects may open many possibilities for methodological improvements of simulations [292], From the fundamental point of view these discretization effects need to be explored in detail. 

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