Chaos and statistics

Seligman, T. H., and Nishioka, H. (Eds.) (1986), Quantum Chaos and Statistical Nuclear Physics, Springer-Verlag, Berlin. 

Volume 22 Synergetics - From Microscopic to Macroscopic Order Editor E. Frehland Volume 23 Synergetics of the Brain Editors E. Ba ar, H. Flohr, H. Haken, and A. J. Mandell Volume 24 Chaos and Statistical Methods Editor Y Kuramoto Volume 25 Dynamics of Hierarchical Systems By J. S. Nicolis... 

This branch of polymer physics is closely connected to another of I.M. Lif-shitz favorite directions in physics, namely, the theory of disordered systems [73]. The situation when different samples have only statistical similarity is typical for the physics of chaos, and many concepts I.M. Lifshitz developed are also quite naturally applied in the physics of disordered polymers. The idea of self-averaging in general and self-averaging of free energy [74], in particular, are examples of such concepts. 

P. Gaspard, Chaos, Scattering, and Statistical Mechanics, Cambridge University Press, Cambridge, UK, 1998. 

Below we show how the energy-optimal control of chaos can be solved via a statistical analysis of fluctuational trajectories of a chaotic system in the presence of small random perturbations. This approach is based on an analogy between the variational formulations of both problems [165] the problem of the energy-optimal control of chaos and the problem of stability of a weakly randomly perturbed chaotic attractor. One of the key points of the approach is the identification of the optimal control function as an optimal fluctuational force [165],... 

G.F. Gribakin, A.A. Gribakina, V.V. Flambaum, Quantum chaos in multicharged ions and statistical approach to the calculation of electron-ion resonant radiative recombination, Aust. J. Phys. 52 (1999) 443. 

Recent years have also witnessed exciting developments in the active control of unimolecular reactions [30,31]. Reactants can be prepared and their evolution interfered with on very short time scales, and coherent hght sources can be used to imprint information on molecular systems so as to produce more or less of specified products. Because a well-controlled unimolecular reaction is highly nonstatistical and presents an excellent example in which any statistical theory of the reaction dynamics would terribly fail, it is instmctive to comment on how to view the vast control possibihties, on the one hand, and various statistical theories of reaction rate, on the other hand. Note first that a controlled unimolecular reaction, most often subject to one or more external fields and manipulated within a very short time scale, undergoes nonequilibrium processes and is therefore not expected to be describable by any unimolecular reaction rate theory that assumes the existence of an equilibrium distribution of the internal energy of the molecule. Second, strong deviations Ifom statistical behavior in an uncontrolled unimolecular reaction can imply the existence of order in chaos and thus more possibilities for inexpensive active control of product formation. Third, most control scenarios rely on quantum interference effects that are neglected in classical reaction rate theory. Clearly, then, studies of controlled reaction dynamics and studies of statistical reaction rate theory complement each other. 

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. 

We still lack a proper understanding of how classical chaos induces the universal fluctuations in the energy levels that are so accurately described by the statistics of random matrix ensembles. Berry (1985) was the first to prove semiclassically some important results on the connection between chaos and energy level correlation functions of classically chaotic systems. The fundamental question, however, of how classical chaos generates quantum spectra amenable to a random matrix description is still not completely solved in all its details. 

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. 

In order to make connections between chaotic properties and statistical behavior in reaction dynamics, one must first define chaotic properties of open systems, since all chemical reactions involve unbounded motion in at least one coordinate. A way of linking chaotic behavior in bound systems to that in open systems was discussed previously for classical unimolecular decay. However, in the quantum case, we do not attempt a similar link but rather establish the circumstances under which chaos in closed systems implies statistical behavior in open systems. 

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. 

The problems discussed above all relate to spectral properties and statistics. In fact, the classical definition of chaos is more concerned with dynamical properties one can define classical chaos in terms of the instar bility of trajectories under infinitesimal displacements, which leads to the exponential divergence of neighbouring trajectories in phase space. The Liapounov exponent is the argument of the exponential which determines the rate of this divergence, and is often taken as a measure. 

S. A. Rice, Overview of the dynamics of intramolecular transfer of vibrational energy, Adv. Chem. Phys. 47 117 (1981) P. Brumer, Intramolecular energy transfer theory for the onset of statistical behavior, Adv. Chem. Phys. 47 202 (1981) P. Brumer and M. Shapiro, Chaos and reaction dynamics, Adv. Chem. Phys. 70 365 (1988). 

We return to the conception of molecular chaos, of statistical equilibrium, and of a partition law defining the thermal balance between the various parts of a material system and making intelligible the idea of a temperature. The obvious task seems now to be the formulation of some quantitative rules about equilibria in systems of many atoms and molecules. 

The essential character of thermal phenomena becomes clear, the conditions of coexistence of solids, liquids, and gases in systems of any number of chemical components are explained, the dependence of equilibria upon concentrations, upon pressure, and upon temperature is defined. The conceptions of entropy and free energy, of statistical equilibrium and energy distribution, provide quantitative laws which describe the perpetual conflict of order and chaos, and which prescribe in a large measure not only the shapes assumed by the material world but also the pattern of its possible changes. 

The DSMC method is a molecule-based statistical simulation method for rarefied-gas flows introduced by Bird [3]. The method solves the dynamical equations for the gas flow numerically, using thousands of simulated molecules. Each simulated molecule represents a large number of real molecules. Assuming molecular chaos and a rarefied gas, only binary collisions need be considered, and so the molecular motion and the collisions are uncoupled if the computational time step is smaller than the physical collision time. Interactions with botmdaries and with other... 

Computational studies have indicated that chaotic behavior is expected in classical mechanical descriptions of the motion of highly excited molecules. As a consequence, intramolecular dynamics relates directly to the fundamental issues of quantum vs classical chaos and semiclassical quantization. Practical implications are also clear if classical mechanics is a useful description of intramolecular dynamics, it suggests that isolated-molecule dynamics is sufficiently complex to allow a statistical-type description in the chaotic regime, with associated relaxation to equilibrium, and a concomitant loss of controlled reaction selectivity. 

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