JEST-M, Vol. 2, Issue 1, 2013 - MVJ College of Engineering

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35 JEST-M, Vol. 2, Issue 1, 2013 As conceptual vehicles for studying pattern formation and complexity As original models of fundamental physics A. CA as Powerful Computation Engines СA allow very efficient parallel computational implementations to be made of lattice models in physics and thus for a detailed analysis of many concurrent dynamical processes in nature. Indeed, dedicated hardware represents one of the most promising practical applications of CA modeling. With the help of such hardware, many heretofore intractable but technologically important problems such as fluid flow near and around airplane wings, are becoming computationally accessible for the first time. B. CA as Discrete Dynamical System Simulators CA allows systematic investigation of complex phenomena by embodying any number of desirable physical properties. Reversible CA, for example, can be used as laboratories for studying the relationship between microscopic rules and macroscopic behavior- exact computability ensuring that the memory of the initial state is retained exactly for arbitrarily long periods of time. Discrete models of turbulence are particularly impressive in that they clearly show that very simple finite dynamical implementations of local conservation laws (defined so that the discrete system is computationally universal) are capable of exactly reproducing continuum system behavior on the micro scale. C. CA as Conceptual Vehicles for Exploring Pattern Formation CA can be treated as abstract discrete dynamical systems embodying intrinsically interesting, and potentially novel, behavioral features. The central motivation is to abstract the general principles governing self-organizing structure formation. Related interests include formal classification of the dynamical behavior of complex systems; a better understanding of the relationship between the dynamics of continuous and discrete systems; and the quantification of complexity-as a system property. All of these questions are particularly accessible through studying the generic behavior of СA systems. D. CA as Original Models of Fundamental Physics CA allows studies of radically new discrete dynamical approaches to microscopic physics, exploring the possibility that nature locally and digitally processes its own future states. This paper is devoted to a prolonged discussion of such potentially ground breaking models of physics. Using the fact that computationally universal systems are capable of arbitrarily complicated behavior (in the sense that they can mimic any computation performed by a conventional computer), the idea is to construct fundamentally discrete field theories to compete with existing continuous models. The emphasis in this class of models is emphatically not to construct a lattice-gauge-like theory; rather, in the same way as lattice gas CA successfully reproduce continuous fluid flow despite never having heard of the Navier-Stokes equations, so the hope is to abstract a set of microphysical laws that reproduce known behavior on the macro scale. IV. REQUIREMENTS OF CELLULAR AUTOMATA [2] A Cellular Automation generally requires A regular lattice of cells covering a portion of a d- dimensional space. A set ф (r', t) = { ф 1(r', t), ф 2(r', t), ф 3(r', t)................ ф m(r', t)} of Boolean variables attached to each site r of the lattice and giving the local state r’ of each cell at the time t = 0, 1, 2,.... A rule R = {R 1 , R 2 ..... R m } which specifies the time evolution of the state's ф (r',t) in the following way as shown in Eq. (4.1). (r ', t 1) R ( (r ', t), (r ' 1, t), (r ' 2, t),.... (r ' q, t)) j j Where in Eq. (1) r' k designate the cells belonging to a given neighborhood of cell r'. In the above definition, the rule R is identical for all sites and is applied simultaneously to each of them, leading to a synchronous dynamics. It is important to notice that the rule is homogeneous, that is it cannot depend explicitly on the cell position r. However, spatial (or even temporal) in homogeneities can be introduced by having some (r ') systematically 1 in some given locations of the lattice j to mark particular cells for which a different rule applies. Boundary cells are a typical example of spatial in homogeneity. Similarly, it is easy to alternate between two rules by having a bit which is 1 at even time steps and 0 at odd time steps. In our definition, the new state at time t + 1 is only a function of the previous state at time t. It is sometimes necessary to have a longer memory and introduce a dependence on the states at time t - l, t - 2,..., t -k. Such a situation is already included in the definition if one keeps a copy of the previous states in the current state. V. NEIGHBOURHOODS (NH) A cellular automata rule is local, by definition. The updating of a given cell requires one to know only the state of the cells in its vicinity. The spatial region in which a cell needs to search is called the neighborhood. In principle, there (1) Sireesha Kraleti,Ravi Pullepudi
36 JEST-M, Vol. 2, Issue 1, 2013 is no restriction on the size of the neighborhood, except that it is the same for all cells. However, in practice, it is often made up of adjacent cells only. If the neighborhood is too large, the complexity of the rule may be unacceptable as the complexity usually grows exponentially fast with the number of cells in the neighborhood (NH). For two-dimensional cellular automata, two neighborhoods are often considered, the von Neumann neighborhood, which consists of a central cell (the one which is to be updated) and its four geographical neighbors north, west, south and east as shown in Fig. 1. The Moore neighborhood contains, in addition, second nearest neighbors north-east, north-west, south-east and south-west, that is a total of nine cells as shown in Fig. 2 Fig. 1 Von Neumann NH Where in Figs.1 and 2, the shaded region indicates the central cell which is updated according to the state of the cells located within the domain marked with the bold line. Another useful neighborhood is the so-called Margolus neighborhood which allows a partitioning of space and a reduction of rule complexity. The space is divided into adjacent blocks of two-by-two cells. The rule is sensitive to the location within this so-called Margolus block, namely upper-left, upper-right, lower-left and lower-right. The way the lattice is partitioned changes as the rule is iterated. It alternates between an odd and an even partition, as shown in Fig. 3. As a result, information can propagate outside the boundaries of the blocks, as evolution takes place. The key idea of the Margolus neighborhood is that when updating occurs, the cells within a block evolve only according to the state of that block and do not immediately depend on what is in the adjacent blocks. To understand this property, consider the difference with the situation of the Moore neighborhood shown in Fig. 4.3 After iteration, the cell located on the west of the shaded cell will evolve relatively to its own Moore neighborhood. Therefore, the cells within a given Moore neighborhood evolve according to the cells located in a wider region. The purpose of space partitioning is to prevent these "long distance" effects from occurring. In Fig.3, the lattice is partitioned in 2 x 2 blocks. Two partitioning are possible - odd and even partitions, as shown by the blocks delimited by the solid and broken lines, respectively. The neighborhood alternates between these two situations, at even and odd time steps. Within a block, the cells are distinguishable and are labeled ul, ur, 11 and lr for upperleft, upper-right, lower-left and lower-right. The cell labeled lr in the figure will be become ul, at the next iteration, with the alternative partitioning. VI. BOUNDARY CONDITIONS Fig. 2 Moore NH In practice, when simulating a given cellular automata rule, one cannot deal with an infinite lattice. The system must be finite and have boundaries. Clearly, a site belonging to the lattice boundary does not have the same neighborhood as other internal sites. In order to define the behavior of these sites, a different evolution rule can be considered, which sees the appropriate neighborhood. This means that the information of being, or not, at a boundary is coded at the site and, depending on this information, a different rule is selected. Following this approach, it is also possible to define several types of boundaries, all with different behavior. Instead of having a different rule at the limits of the system, another possibility is to extend the neighborhood for the sites at the boundary. Fig. 3 Margolus NH Sireesha Kraleti,Ravi Pullepudi
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JEST-M, Vol. 2, Issue 1, 2013 - MVJ College of Engineering

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