Diffusive systems

Lengyel I and Epstein I R 1992 A chemical approach to designing Turing patterns in reaction-diffusion systems Proc. Natl Acad. Sc/. 89 3977-9... 

Toth A, Lagzi I and Florvath D 1996 Pattern formation in reaction-diffusion systems cellular acidity fronts J. Rhys. Chem. 100 14 837-9... 

Lee K-J, MoCormiok W D, Pearson J E and Swinney H L 1994 Experimental observation of self-replioating spots in a reaotion-diffusion system Nature 369 215-8... 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

These two processes provide examples of the moving boundary problem in diffusing systems in which a solid solution precedes the formation of a compound. The diickness of the separate phase of the product, carbide or... 

Strictly the diffusion coefficient D measured for any type of binary system A/B is in fact the resultant effect of two partial diffusivities and D, representing respectively the diffusivity of A into B and of B intO/4. For most practical purposes, however, a single diffusion coefficient is sufficient to define a given diffusion system. 

The extension of generic CA systems to two dimensions is significant for two reasons first, the extension brings with it the appearance of many new phenomena involving behaviors of the boundaries of, and interfaces between, two-dimensional patterns that have no simple analogs in one-dimension. Secondly, two-dimensional dynamics permits easier (sometimes direct) comparison to real physical systems. As we shall see in later sections, models for dendritic crystal growth, chemical reaction-diffusion systems and a direct simulation of turbulent fluid flow patterns are in fact specific instances of 2D CA rules and lattices. 

Plate 4. A snapshot of the Hodgepodge CA. The Hodgepodge rule, introduced by Ger-hardt and Schuster [gerh89], is chareicterized by its self-org tnized spiral structures and is a crude model of a famous oscillating chemical reeiction-diffusion system called the Belousov-Zh botinskii reaction. See Chapter 8. 

In this section we introduce several CA models of prototypical reaction-diffusion systems. Such systems, the first formal studies of which date back to Turing , often exhibit a variety of interesting spatial patterns that evolve in a self-organized fashion. 

Chitosan scaffolds were reinforced with beta-tricalciiun phosphate and calcium phosphate invert glass [177]. Along the same line, composites of Loligo beta-chitin with octacalcium phosphate or hydroxyapatite were prepared by precipitation of the mineral into a chitin scaffold by means of a double diffusion system. The octacalciiun phosphate crystals with the usual form of 001 blades grew inside chitin layers preferentially oriented with the 100 faces parallel to the surface of the squid pen and were more stable to hy-... 

In order to discuss the various techniques we must distinguish between diffusive and non-diffusive systems (J8). Diffusive systems, such as liquids, are characterized by the eventual diffusion of particles over all of the available space non-diffusive systems such as solids, glasses and macromolecules with a definite average structure are characterized by time independent average positions around which the atoms fluctuate. 

The problem is that for diffusive systems the multidimensional configuration space is so vast that it can never be integrated by simulation techniques. This is immediately clear from the occurrence of N in Z. The number of integrand evaluations should vastly exceed N , which for N 1000 and one evaluation per picosecond on the futuristic ultrasupercomputer requires vastly longer then the age of the universe. 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

Exchange diffusion systems are present in the membrane for exchange of anions against OH ions and cations against H ions. Such systems are necessary for uptake and output of ionized metaboUtes while preserv-... 

Some specific solutes diffuse down electrochemical gradients across membranes more rapidly than might be expected from their size, charge, or partition coefficients. This facilitated diffusion exhibits properties distinct from those of simple diffusion. The rate of facilitated diffusion, a uniport system, can be saturated ie, the number of sites involved in diffusion of the specific solutes appears finite. Many facihtated diffusion systems are stereospecific but, fike simple diffusion, require no metabolic energy. 

Diffusion Systems The liquid whose vapor is to be the contaminant of the gas phase is contained in a reservoir maintained at a constant temperature. The liquid is allowed to evaporate and the vapor diffuses slowly through the capillary tube into a flowing gas stream. If the rate of diffusion of the vapor and the flow rate of the diluent gas are known, the vapor concentration in the resultant gas mixture can be calculated. 

The diffusion system. Figure 8.31(B), is a useful and simple apparatus for preparing mixtures of volatile and moderately volatile vapors in a gas stream [388]. The method is based on the constant diffusion of a vapor from a tube of accurately known dimensions, producing a gas phase concentration described by equation (8.12). 

Diffusion systems are characterized by the release rate of a drug being dependent on its diffusion through an inert membrane barrier. Usually this barrier is an insoluble polymer. In general, two types or subclasses of diffusional systems are recognized reservoir devices and matrix devices. These will be considered separately. 

Reaction-diffusion systems can readily be modeled in thin layers using CA. Since the transition rules are simple, increases in computational power allow one to add another dimension and run simulations at a speed that should permit the simulation of meaningful behavior in three dimensions. The Zaikin-Zhabotinsky reaction is normally followed in the laboratory by studying thin films. It is difficult to determine experimentally the processes occurring in all regions of a three-dimensional segment of excitable media, but three-dimensional simulations will offer an interesting window into the behavior of such systems in the bulk. 

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