Chaos theory, chemical reaction dynamics

M. Toda, T. Komatsuzaki, T. Konishi, R. S. Berry and S. A. Rice, Geometrical Structures of Phase Space in Multidimensional Chaos Applications to Chemical Reaction Dynamics in Complex Systems, Advances in Chemical Physics, John-Wiley Sons, Hoboken, NJ, 2005, vol. 130, parts A and B. T. Komatsuzaki, R. S. Berry and D. M. Leitner, Advancing Theory for Kinetics and Dynamics of Complex, Many-Dimensional Systems Clusters and Proteins, Advances in Chemical Physics, John-Wiley Sons, Hoboken, NJ, 2011, vol. 145. 

Intramolecular dynamics and chemical reactions have been studied for a long time in terms of classical models. However, many of the early studies were restricted by the complexities resulting from classical chaos, Tlie application of the new dynamical systems theory to classical models of reactions has very recently revealed the existence of general bifurcation scenarios at the origin of chaos. Moreover, it can be shown that the infinite number of classical periodic orbits characteristic of chaos are topological combinations of a finite number of fundamental periodic orbits as determined by a symbolic dynamics. These properties appear to be very general and characteristic of typical classical reaction dynamics. 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

Oscillating reactions, a common feature of biological systems, are best understood within the context of nonlinear chemical dynamics and chaos theory based models that are used to predict the overall behavior of complex systems. A chaotic system is unpredictable, but not random. A key feature is that such systems are so sensitive to their initial conditions that future behavior is inherently unpredictable beyond some relatively short period of time. Accordingly, one of the goals of scientists studying oscillating reactions is to determine mathematical patterns or repeatable features that establish relationships to observable phenomena related to the oscillating reaction. 

Modern theories and concepts describing the initial dynamic evolution of fluid motions leading to the onset of turbulence, explaining experimental observations of instability phenomena, are t3rpically rather theoretical and usually not considered by chemical reaction engineers. The interested reader is referred to texts on chaos theory and non-linear d3mamics (e.g., [155], chap 5). 

The existence of chaotic oscillations has been documented in a variety of chemical systems. Some of the earliest observations of chemical chaos have been on biochemical systems like the peroxidase-oxidase reaction [12] and on the well known Belousov-Zhabotinskii (BZ) [13] reaction. The BZ reaction is the Ce-ion-catalyzed oxidation of citric or malonic acid by bromate ion. Early investigations of the BZ reaction used the techniques of dynamical systems theory outlined above to document the existence of chaos in this reaction. Apparent chaos in the BZ reaction was found by Hudson et al [14] and the data were analysed by Tomita and Tsuda [15] using a return-map method. Chaos was confirmed in the BZ reaction carried out in a CSTR by Roux et al [16,17] and by Hudson and... 

Subsection A contains a summary of the formal definitions of chaotic behavior, derived from ergodic theory detailed discussions of this topic may be found elsewhere.11 We comment, in this section, on the gap that must be bridged in order to apply these concepts to chemical dynamics. Subsection B discusses some recent developments in computational signatures of chaos. In Subsection C we review a number of studies that have provided some of these links and that, in some instances, have resulted in new useful computational methods for treating the dynamics of reactions displaying chaotic dynamics. In addition, it includes a subsection on connection between formal ergodic theory and statistical behavior in unimolecular decay. 

Oscillations and chaos are observed frequently in chemical systems. Most of the experimental investigations of these phenomena have been carried out for well-stirred systems where spatial degrees of freedom are assumed to play no role. If this is the case the system may be described in terms of chemical rate equations for a small number of macroscopic chemical concentrations. The periodic or chaotic attractors typically have low dimensions and can be characterized using the tools of dynamical systems theory [20]. Chemical systems may also display spatiotemporal oscillations and chaos. If spatial degrees of freedom are important the appropriate macroscopic model is the reaction-diffusion equation. The attractors may have high dimension and the theoretical description of such spatiotemporal states is less well developed. 

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