Complexity theory, self-organizing

At the mesoscopic scale, interactions between molecular components in membranes and catalyst layers control the self-organization into nanophase-segregated media, structural correlations, and adhesion properties of phase domains. Such complex processes can be studied by various theoretical tools and simulation techniques (e.g., by coarse-grained molecular dynamics simulations). Complex morphologies of the emerging media can be related to effective physicochemical properties that characterize transport and reaction at the macroscopic scale, using concepts from the theory of random heterogeneous media and percolation theory. 

In Table 1 are summarized representative examples of self-exchange rate constant data for a variety of different types of redox couples based on metal complexes, organometallic compounds, organics and clusters. Where available the results of temperature dependence studies are also cited. For convenience, data obtained from temperature dependence studies are presented as enthalpies and entropies of activation as calculated from the reaction rate theory expression in equation (14). 

For example, the standard synergetic approach [52-54] denies the possibility of any self-organization in a system with with two intermediate products if only the mono- and bimolecular reaction stages occur [49] it is known as the Hanusse, Tyson and Light theorem. We will question this conclusion, which in fact comes from the qualitative theory of non-linear differential equations where coefficients (reaction rates) are considered as constant values and show that these simplest reactions turn out to be complex enough to serve as a basic models for future studies of non-equilibrium processes, similar to the famous Ising model in statistical physics. Different kinds of auto-wave processes in the Lotka and Lotka-Volterra models which serve as the two simplest examples of chemical reactions will be analyzed in detail. We demonstrate the universal character of cooperative phenomena in the bimolecular reactions under study and show that it is reaction itself which produces all these effects. 

E. Brandas, Complex Symmetry, Jordan Blocks and Microscopic Selforganization An Examination of the Limits of Quantum Theory Based on Nonself-adjoint Extensions with Illustrations from Chemistry and Physics, in N. Russo, V. Ya. Antonchenko, E. Kryachko (Eds.), Self-Organization of Molecular Systems From Molecules and Clusters to Nanotubes and Proteins, NATO Science for Peace and Security Series A Chemistry and Biology, Springer Science+Business Media B.V., Dordrecht, 2009, p. 49. 

Stuart Kauffman of the Santa Fe Institute is a leading proponent of complexity theory. Simply put, it proposes that many features of living systems are the result of self-organization—the tendency of complex systems to arrange themselves in patterns—and not natural selection ... 

Complex, self-organizing systems continuously adapt to and change with their environments but do so in ways that are impossible to predict. Because of this, people have begun to realize that industrial firms operate very much like ecosystems (Boston, 2001). The result of this has been complexity theory, explaining how a complex adaptive systems approach can help make policy, business, education, and research decisions. Studying how ants find the closest food source, for instance, has helped industries to find efficient solutions to marketing soap, schedule movement of casks of whiskey, and control crowds at amusement parks (Figure 8.2.8). 

Eor ferrocene sites at the end of long alkanethiols self-organized at gold electrodes and diluted with unsubstituted thiols with the redox moiety in contact with the electrolyte (Fig. 4a), Chidsey has reported [34] curved Tafel plots (Fig. 4b), which could be fitted by equations derived from Marcus theory with values of k = 0.85 eV and Z = 6.73 x 10 s"l eV" for a reaction rate of A = 2.5 s at in Fig. 4(b). Similar curvature in Tafel plots has been reported by Faulkner and coworkers [35] for adsorbed osmium complexes at ultramicro-electrodes (UME). The temperature dependence of the rate coefficient could also be fitted from Marcus equation and electron states in the metal and coupling factors given by quantum mechanics. 

First elementary reaction steps at an isolated reaction center have been considered and then the increasing complexity of the catalytic stem when several reaction centers operate in parallel and communicate. This situation is common in heterogeneous catalysis. On the isolated reaction center, the key step is the self repair of the weakened or disrupted bonds of the catalyst once the catalytic cycle has been concluded. Catalytic systems which are comprised of autocatalytic elementary reaction steps and communication paths between different reaction centers, mediated through either mass or heat transfer, may show self-organizing features that result in oscillatory kinetics and spatial organization. Theory as well as experiment show that such self-organizing phenomena depend sensitively on the size of the catalytic system. When the system is too small, collective behavior is shut down. 

Recent advances in polymerization techniques have made possible the synthesis of complex, including hierarchically branched, maaomolecules with perfectly controlled architecture. Analysis of interplay between complex maaomolecular topology and multiple competing interactions on conformations, routes of self-organization in solutions and at interfaces, and properties of resulting supramolecular stractures presents a major cuuent and future challenge for polymer theory. 

Coming back to limit cycle oscillations shown by systems of ordinary differential equations, this simple mode of motion still seems to deserve some more attention, especially in relation to its role as a basic functional unit from which various dynamical complexities arise. This seems to occur in at least two ways. As mentioned above, one may start with a simple oscillator, increase [x, and obtain complicated behaviors this forms, in fact, a modern topic. However, another implication of this dynamical unit should not be left unnoticed. We should know that a limit cycle oscillator is also an important component system in various self-organization phenomena and also in other forms of spatio-temporal complexity such as turbulence. In this book, particular emphasis will be placed on this second aspect of oscillator systems. This naturally leads to the notion of the many-body theory of limit cycle oscillators we let many oscillators contact each other to form a field , and ask what modes of self-organiza-tion are possible or under what conditions spatio-temporal chaos arises, etc. A representative class of such many-oscillator systems in theory and practical application is that of the fields of diffusion-coupled oscillators (possibly with suitable modifications), so that this type of system will primarily be considered in this book. 

A general mathematical treatment of the precellular self-organization of matter and the evolution of macromolecules was presented some time ago by Eigen. > A set of selectivity and evolutionary principles were derived for alternative populated states of macromolecules, leading towards states of higher complexity and information content. It would be highly desirable to analyze and correlate the three models presented thus far (molecular, co-acervate-microsphere and mathematical), select the realistic and cooperative portions of these models, and integrate them into a theory of precellular evolution which could be experimentally tested. 

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