Discrete lattices

Discrete lattice of cells the system substrate consists of a one-, two- or three-dimensional lattice of cells. 

We shall extensively employ the notation of graph theory it provides a powerful and elegant formalism for the description of both the structure of the discrete lattices on which the CA live, and the complete dynamics (he. the global state transitions) induced by those structures. Graph theory also allows the correspondence between CA configurations and the words of a regular language to be made in a very natural fashion. 

Discrete Cellular State Space the discrete lattice of cells or sites upon which CA live , and their dynamics unfolds. C can be one-dimensional, two-dimensional... 

Producing a reasonably good accuracy for analytically defined surfaces, this scheme of calculation is very inaccurate when the field is specified by the discrete set of values (the lattice scalar field). The surface in this case is located between the lattice sites of different signs. The first, second, and mixed derivatives can be evaluated numerically by using some finite difference schemes, which normally results in poor accuracy for discrete lattices. In addition, the triangulation of the surface is necessary in order to compute the integral in Eq. (8) or calculate the total surface area S. That makes this method very inefficient on a lattice in comparison to the other methods. 

In solution theory the specialized distribution functions of this kind should appear in the theory of ion pairs in ionic solutions, and a form of the Bjerrum-Fuoss ionic association theory adapted to a discrete lattice is generally used for the treatment of the complexes in ionic crystals mentioned above. In fact, the above equation is not used in this treatment. Comparison of the two procedures is made in Section VI-D. 

Although the theory of solutions has been widely used in formulating problems of defects in solids the problems encountered differ in certain respects. The most obvious point is that defects are restricted to discrete lattice sites, whereas the ions in a solution can occupy any position in the fluid. Sometimes no allowance is made for this fact. For example, it has not been demonstrated that at very low concentrations, in the absence of ion-pair effects, the activity coefficients are identical with those of the Debye-Hiickel theory. It can be plausibly argued51 that at sufficiently low concentrations the effect of discreteness is likely to be negligible, but clearly in developing a theory for any but the lowest concentrations the effect should be investigated. A second point... 

Since in this problem not only the limit but also the character of convergence matters we conclude that consistent homogenization of the micromodel should lead to a description in a broader functional space than is currently accepted. One interesting observation is that the concave part of the energy is relevant only in the region with zero measure where the singular, measure valued contribution to the solution is nontrivial (different from point mass). We remark that the situation is similar in fracture mechanics where a problem of closure at the continuum level can be addressed through the analysis of a discrete lattice (e.g. Truskinovsky, 1996). 

The simplest of these models which permits a detailed discussion of the decay of correlations is a random walk model in which a set of random walkers whose positions are initially correlated is allowed to diffuse the motion of any single random walker being independent of any other member of the set. Let us assume that there are r particles in the set and motion occurs on a discrete lattice. The state of the system is, therefore, completely specified by the probabilities Pr(nlf n2,..., nr /), (tij = — 1, 0, 1, 2,. ..) in which Pr(n t) is the joint probability that particle 1 is at n1( particle 2 is at n2, etc., at time l. We will also use the notation Nj(t) for the random variable that is the position of random walker j at time t. Reduced probability distributions can be defined in terms of the Pr(n t) by summation. We will use the notation P nh, rth,..., ntj I) to denote the distribution of random walkers iu i2,..., i at time t. We define... 

Twenty-two atoms were placed randomly on a discrete lattice at positions Xi, 0 < Xi < 100. The concentration curves are continuous and have areas that are approximately equal to the number of atoms in the random sample. Each atom contributes a unit area to the coarsegrained c(x). Broader convolution functions (higher values of B) produce greater degrees of coarse graining. 

Lastly, non-elementary several-stage reactions are considered in Chapters 8 and 9. We start with the Lotka and Lotka-Volterra reactions as simple model systems. An existence of the undamped density oscillations is established here. The complementary reactions treated in Chapter 9 are catalytic surface oxidation of CO and NH3 formation. These reactions also reveal undamped concentration oscillations and kinetic phase transitions. Their adequate treatment need a generalization of the fluctuation-controlled theory for the discrete (lattice) systems in order to take correctly into account the geometry of both lattice and absorbed molecules. As another illustration of the formalism developed by the authors, the kinetics of reactions upon disorded surfaces is considered. 

In computer simulations a discrete lattice of sites is considered, each site is occupied by not more than one particle. Particles A are localized in their sites for r seconds and then possess hops. Thus, the mean diffusion coefficient Da = a2/(2dr) could be introduced. We assume that particles B are immobile, Dq = 0, since it permits to reduce greatly simulation time. Moreover, for Da = Dq and d = 1,2 the kinetics turn out to be quite similar. A hop of particle A into the site occupied by particle B results in their instant recombination. 

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]. 

It should be noted in conclusion of this Section that preliminary results obtained by means of the discrete lattice formalism are presented in [85], This study demonstrates clearly the cooperative nature of the aggregation of two kinds of the Frenkel defects, vacancies and interstitials. 

In studying processes of accumulation of the Frenkel defects, one uses three different types of simple models the box, continuum, and discrete (lattice) models. In the simplest, box model, which was proposed first in [22], one studies the accumulation of complementary particles in boxes having a certain capacity, with walls impenetrable for diffusion of particles among the boxes. The continuum model treats respectively a continuous medium the intrinsic volume of similar defects at any point of the space is not bounded. In the model of a discrete medium a single cell (e.g., crystalline lattice site) cannot contain more than one defect (v or i). 

Surface reactions are very important in both theoretical and applied research. Experimental information on the individual reaction steps is difficult to obtain and the interpretation of the data is not easy. However, investigation of individual steps of a surface reaction can be obtained by using theoretical models, where discrete lattices are used to represent the surface. Depending on the number of different particles and on the adsorption and reaction steps, the models are classified as A + A 0, A + B 0, A + 2 —> 0,... 

The simulation takes place on a discrete lattice with coordination number 2. Each lattice site is given a lattice vector l. The state of the site l is represented by the lattice variable 07, which may depend on the state of the catalyst site (e.g., promoted or not) and on the coverage with a particle. Let us assume we deal with the simple case in which all catalyst sites are equal... 

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