Diffusion-reaction systems

In contrast to these basic approaches at the macroscopic and mesoscopic levels, one can consider a class of models that does not rely on a knowledge of the detailed rate law or reaction mechanism but instead abstract certain generic features of the behavior. These simplified models often provide insight into the system s dynamics and isolate the minimal features needed to rationalize complex phenomena. Cellular automata (CA) and coupled map lattices (CML) are two examples of such abstract models that we shall discuss. In the following sections, we discuss each of these models, give some of the background that led to their formulation, and provide an introduction to how they are constructed. The presentation will focus on a few examples instead of providing an exhaustive overview. 

For simplicity, suppose we consider a single chemical species X with local concentration cx(r, t). (We may suppose that other chemical species participating in the reaction are pool species whose concentrations are either in large excess or are held fixed by flows of reagents into and out of the system.) If we approximate the time and space derivatives in Eq. [1] by finite differences we obtain, 

DAt/ Axf Ijld, where d is the dimension of the system, this scheme will yield a faithful solution to the reaction-diffusion equation. The grid points need not lie on a cubic lattice and the size of the neighborhood used to 

We suppose that the A and B species are pool species, that is, their concentrations are very large. If the concentrations of these species are taken to be constant, only the concentration of species X can change. Following mass action kinetics, the reaction rates corresponding to the two steps in the mechanism (Eq. [3]) are 

If the system is homogeneous in space, there are no concentration gradients and the reaction-diffusion equation [1] reduces to the chemical rate law. 

For classical evolutions, we merely substitute crj for p. Looking at plots of N fi, p vs. v/N, it is clear that although the quantum dynamics generally appears to preserve the characteristic structure of the classical spectrum, particular structural details tend to be washed-away [ilachSSbj. If high or low frequency components are heavily favored in the classical evolution, for example, they will similarly be favored in the quantum model discrete peaks, however, will usually disappear. White-noise spectra, of course, will remain so in the quantum model. 

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. 

A large class of reaction-diffusion reactions may be expressed by the (deceptively simple) equation 

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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... 

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 ... 

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. 

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. 

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. 

MORPHOLOGY OF SURFACES IN MESOSCOPIC POLYMERS, SURFACTANTS, ELECTRONS, OR REACTION-DIFFUSION SYSTEMS METHODS, SIMULATIONS, AND MEASUREMENTS... 

In what follows we will discuss systems with internal surfaces, ordered surfaces, topological transformations, and dynamical scaling. In Section II we shall show specific examples of mesoscopic systems with special attention devoted to the surfaces in the system—that is, periodic surfaces in surfactant systems, periodic surfaces in diblock copolymers, bicontinuous disordered interfaces in spinodally decomposing blends, ordered charge density wave patterns in electron liquids, and dissipative structures in reaction-diffusion systems. In Section III we will present the detailed theory of morphological measures the Euler characteristic, the Gaussian and mean curvatures, and so on. In fact, Sections II and III can be read independently because Section II shows specific models while Section III is devoted to the numerical and analytical computations of the surface characteristics. In a sense, Section III is robust that is, the methods presented in Section III apply to a variety of systems, not only the systems shown as examples in Section II. Brief conclusions are presented in Section IV. 

At the mesoscopic level, the reaction-diffusion system is described by a set of partial differential equations,... 

Morphology of Surfaces in Mesoscopic Polymers, Surfactants, Electrons, or Reaction-Diffusion Systems Methods,... 

INVERSE PROBLEMS FOR REACTION-DIFFUSION SYSTEMS WITH APPLICATION TO GEOGRAPHICAL POPULATION GENETICS... 

Reaction-diffusion systems have been studied for about 100 years, mostly in solutions of reactants, intermediates, and products of chemical reactions [1-3]. Such systems, if initially spatially homogeneous, may develop spatial structures, called Turing structures [4-7]. Chemical waves of various types, which are traveling concentrations profiles, may also exist in such systems [2, 3, 8]. There are biological examples of chemical waves, such as in parts of glycolysis, heart... 

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