Euclidean lattices

The BBM gas consists of an arbitrary number of hard spheres (or balls) of finite diameter that collide elastically both among themselves and with any solid walls (or mirrors) that they may encounter during their motion. Starting out on some site of a two-dimensional Euclidean lattice, each ball is allowed to move only in one of four directions (see figure 6.10). The lattice spacing, d = l/ /2 (in arbitrary units), is chosen so that balls collide while occupying adjacent sites. Unit time is... 

The one dimensional rules given in equations 8.105 and 8.106 can be readily generalized to a d dimensional Euclidean lattice. Let T] r, t) be the d dimensional analogue of the one dimensional local slope at lattice point r at time Addition of sand is then generated by the rule... 

Fig. 8,20 First five iterations of an SDCA system starting from a 4-neighbor Euclidean lattice seeded with a single non-zero site at the center. The global transition rule F consists of totalistic-value and restricted totalistic topology rules C — (26,69648,32904)[3 3[ (see text for rule definitions). Solid sites have <7=1.

Vants represent the one of the simplest - and therefore, most persuasive - examples of emergence of high-level structures from low-level dynamics. Discovered by Langton [lang86], vants live on a two-dimensional Euclidean lattice and come in two flavors, red and bine. Each vant c an move in any of four directions (E,W,N,S). Each lattice site is either empty or contains one of two types of food, green food or yellow food. Vants arc fundamentally solitary creatures so that there is a strict conservation of the number of vants. 

To set up the problem and in order to appreciate more fully the difficulty in quantifying complexity, consider figure 12.1. The figure shows three patterns (a) an area of a regular two-dimensional Euclidean lattice, (b) a space-time view of the evolution of the nearest-neighbor one-dimensional cellular automata rule RllO, starting from a random initial state,f and (c) a completely random collection of dots. 

Another simple example is the traiditional two-dimensional random-walk on a four-neighbor Euclidean lattice [toff89]. Despite the fact that the underlying lattice is symmetric only with respect to rotations that are multiples of 90 deg, the probability distribution p(s, y) for a particle that begins its random walk at the origin becomes circularly symmetric in the limit as time t —> oo p x,y,t) —> (see figure 12.12). 

Fig. 12.12. A circularly symmetric Gaussian probability distribution p x,y) describing a two-dimensional random walk emerges for large times on the macroscopic level, despite the fact that the underlying Euclidean lattice is anisotropic.

Jourjine [jour85] generalizes Euclidean lattice field theory on a d-dimensional lattice to a cell complex. He uses homology theory to replace points by cells of various dimensions and fields by functions on cells, the cochains, in hopes of developing a formalism that treats space-time as a dynamical variable and describes the change in the dimension of space-time as a phase transition (see figure 12.19). 

In this section are displayed graphically the numerically exact results that have been obtained for unbiased, nearest-neighbor random walks on finite d = 2,3 dimensional regular. Euclidean lattices, each of uniform valency v, subject to periodic boundary conditions, and with a single deep trap. These data allow a quantitative assessment of the relative importance of changes in system size N, lattice dimensionality d, and/or valency v on the efficiency of diffusion-reaction processes on lattices of integral dimension, and provide a basis for understanding processes on lattices of fractal dimension or fractional valency. 

Figure 4.9. The A = 123 Sierpinski gasket, a two-dimensional uncountable set with zero measure and Hausdorff (fractal) dimension 3/ 2 = 1.584962... the companion Euclidean lattice referred to in the text is a space filling triangular lattice of (interior valency v = 6.

Figure 4.18. Influence of a short-range chemical/cage effect on the efficiency of an irreversible, encounter-controlled reaction at a centrally located reaction center, relative to a purely random process. Plotted is the ratio n) C)/ n) RW) versus system size for a d-dimensional Euclidean lattice of valency v = 2d. The abscissa in the figure is the edge length f overall, the reaction space has N — I sites where N =. The lower curve (A) corresponds...

Yet another important aspect is the change in the fractal dimension of polymers when they are simulated on fractal rather than Euclidean lattices. This fact is also important from the practical standpoint for multicomponent polymer systems. The introduction of a dispersed filler into a polymer matrix results in structure perturbation in terms of fractal analysis, this is expressed as an increase in the fractal dimension of this structure. As shown by Novikov and co-workers [25], the particles of a dispersed filler form in the polymer matrix a skeleton which possesses fractal (in the general case, multifractal) properties and has a fractal dimension. Thus, the formation of the structure of the polymer matrix in a filled polymer takes place in a fractal rather than Euclidean space this accounts for the structure modifications of the polymer matrix in composites. 

The classical problem in chemical kinetics is the diffusive processes influence on this kinetics. In diffusion-limited reactions their rate is defined by the diffusion duration, which is necessary for, the reagents to reach one another. Similar reactions simulation on Euclidean lattices gave the following results. The reactions were considered [7] ... 

The classical problem in chemical kinetics is a diffusive processes influence on this kinetics [8, 9, 32, 33]. In diffusion-controlled reactions, their rate is defined by diffusion time, which is necessary for reagents to reach one another. Simulation of similar reactions on Euclidean lattices has shown that reactions of the type (Eq. (2) of Chapter 2) and (Eq. (3) of Chapter 2) are described by the (Eq. (43) of Chapter 2) and (Eq. (53) of Chapter 2), respectively. As it is known [36], the change in space type from Euclidean to fractal strongly changes the chemical reaction course. In this case the reactions (Eq. (2) of Chapter 2) and (Eq. (3) of Chapter 2) are described by the (Eq. (6) of Chapter 2) and (Eq. (7) of Chapter 2), respectively. 

For all lattices, series analysis yielded a simple pole as the singularity of the SAW generating function. This is of course different to the Euclidean situation, where the corresponding exponent 7 =. Furthermore, the growth constant for SAW and SAPs for a given lattice is different. Surprisingly, the exponent for SAP appears to be the same as for Euclidean lattices, that is a =. For the 6,3 lattice it was possible to obtain an exact solution. This lattice has a tree-like dual structure, which accounts for the solvability. For SAP on this lattice the generating function was found to be... 

A good model for this phenomenon is a SAW on some lattice with an absorbing surface (boundary). Every site on surface visited by the polymer contributes an energy —E,. This model has widely been studied on various lattices and via a number of techniques that include exact enumeration [48,49], Monte Carlo [50], transfer matrix [51], renormalization group [52] etc. For a 2-d Euclidean lattice, exact value of found from conformal field theory is 1/2 [53]. 

As it is known [5], the fractal dimension of an object is a function of space dimension, in which it is formed. In the computer model experiment this situation is considered as fractals behavior on fractal (but not Euclidean) lattices [6], The space (or fractal lattice) dimension D can be determined with the aid of the following equation [ 1] ... 

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