Chaos theory approach

Thus far, quantum chaos theory has mainly concentrated on the statistical description of real energy levels and their spacings (see Section 4.1.1). The complex helium resonances with their two-dimensional character require a different approach. It may be based on Ginibre s ensemble of complex random matrices (Ginibre (1965)). The investigation of the statistical properties of the helium resonances in the complex plane is certainly a good subject for future research. 

Figure 39. Classical and nonclassical approaches to chaos theories [5-7, 72e, f]-...

Abstract. Quantum chaos at finite-temperature is studied using a simple paradigm, two-dimensional coupled nonlinear oscillator. As an approach for the treatment of the finite-temperature a real-time finite-temperature field theory, thermofield dynamics, is used. It is found that increasing the temperature leads to a smooth transition from Poissonian to Gaussian distribution in nearest neighbor level spacing distribution. 

In this chapter we have reviewed the development of unimolecular reaction rate theory for systems that exhibit deterministic chaos. Our attention is focused on a number of classical statistical theories developed in our group. These theories, applicable to two- or three-dimensional systems, have predicted reaction rate constants that are in good agreement with experimental data. We have also introduced some quantum and semiclassical approaches to unimolecular reaction rate theory and presented some interesting results on the quantum-classical difference in energy transport in classically chaotic systems. There exist numerous other studies that are not considered in this chapter but are of general interest to unimolecular reaction rate theory. 

In the first part, our aim is to discuss how we can apply concepts drawn from dynamical systems theory to reaction processes, especially unimolecular reactions of few-body systems. In conventional reaction rate theory, dynamical aspects are replaced by equilibrium statistical concepts. However, from the standpoint of chaos, the applicability of statistical concepts itself is problematic. The contribution of Rice s group gives us detailed analyses of this problem from the standpoint of chaos, and it presents a new approach toward unimolecular reaction rate theory. 

Y. L. Klimontovich, Turbulent Motion and the Structure of Chaos A New Approach to the Statistical Theory of Open Systems, Kluwer, Dordrecht, NL, 1992. 

Traditional statistical theories are based on strong collision regions that are well-defined regions in phase space, an approach which does not utilize insights afforded by studies of chaos. Consider, as an alternative, defining... 

Thus, one is interested in a rather special kind of statistics, viz. the statistics of a dense population of interacting levels. This is the fundamental distinction between chaos and complexity there may arise situations in which levels do not necessarily all interact (they might have different quantum numbers) but are simply present in large numbers, so that their analysis is not possible in practice but could be performed in principle. These are called unresolved transition arrays (UTAs). One can develop [526] a theory of UTAs which yields general theorems about them as a whole. Such theories are a statistical approach to the interpretation of spectra, but are not related to the problem of quantum chaology. 

Winfree, A.T. 1991b. Varieties of spiral wave behavior An experimentalist s approach to the theory of excitable media. Chaos 1 303-34 and (Erratum) 2 273 (1992). 

Klimontovich YuL (1990) Turbulent motion and chaos structure a new approach to the theory of open systems. Nauka, Moscow (English translation Klimontovich YL (1991) Turbulence motion and Chaos structure. Kluwer, Dordrecht)... 

Winfree, A. T. 1991. Varieties of Spiral Wave Behavior An Experimentalist s Approach to the Theory of Excitable Media, Chaos 1, 303-334. 

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