Ergodic theory

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

P. Walters. An Introduction to Ergodic Theory. Springer Verlag, Berlin, Heidelberg, New York, Tokyo (1981)... 

In its extreme form the ergodic hypothesis is clearly untenable. Only probability statements can be made in statistical mechanics and these have nothing to do with sequences in time [117]. Not surprisingly, a totally convincing proof of the ergodic theorem in its many guises has not been formulated. The current concensus still is that an axiomatic basis, completely independent of ergodic theory should be chosen [115] for the construction of statistical mechanics. 

The implimentation of quantum statistical ensemble theory applied to physically real systems presents the same problems as in the classical case. The fundamental questions of how to define macroscopic equilibrium and how to construct the density matrix remain. The ergodic theory and the hypothesis of equal a priori probabilities again serve to forge some link between the theory and working models. 

Cornfeld I. P, Fomin S. V. and Sinai Ya. G. Ergodic Theory. Springer (Berlin, Heidelberg) (1982). 

Vol. 1514 U. Krengel, K. Richter, V. Warstat (Eds.), Ergodic Theory and Related Topics III, Proceedings, 1990. VIII, 236 pages. 1992. 

Lebowitz, J.L. and Penrose, O. (1973). Modem ergodic theory, Phys. Today 26(2), 23-29. 

Here, the t,-independency of the exponents is shown by the multiplicative ergodic theory of Oseledec [8],... 

C. Schiitte, W. Huisinga, and P. Deuflhard (2001) Transfer operator approach to conformational dynamics in biomolecular systems. In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (B. Fiedler, ed.). Springer Berlin Heidelberg, pp. 191-223... 

I. G. Sinai and V. Scheffer, Introduction to Ergodic Theory, Princeton University Press, Princeton, NJ, 1976. 

K. Petersen, Ergodic Theory, Cambridge University Press, Cambridge, UK, 1983. 

A. Input from Classical Ergodic Theory What is Chaos ... 

Conditions on system properties, for example, potential surfaces, state densities, masses, and so forth, which are necessary for relaxation and which emerge from quantum ergodic theory, would be used to identify properties of molecular systems necessary and sufficient to ensure statistical reaction dynamics. 

The number of difficulties associated with carrying out the preceding program is formidable. First and foremost, there is no satisfactory theory of chaotic motion in quantum ergodic theory, either for isolated bound or unbound, systems. Indeed, the qualitative concept of relaxation that we traditionally associate with statistical theories is only consistent with results for ideal systems in the classical ergodic theory of bound systems. Second,... 

The clearest results have been obtained for classical relaxation in bound systems where the full machinery of classical ergodic theory may be utilized. These concepts have been carried over empirically to molecular scattering and decay, where the phase space is not compact and hence the ergodic theory is not directly applicable. This classical approach is the subject of Section II. Less complete information is available on the classical-quantum correspondence, which underlies step 4. This is discussed in Section III where we introduce the Liouville approach to correspondence, which, we believe, provides a unified basis for future studies in this area. Finally, the quantum picture is beginning to emerge, and Section IV summarizes a number of recent approaches relevant for a quantum-mechanical understanding of relaxation phenomena and statistical behavior in bound systems and scattering. 

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. 

Classical ergodic theory defines a number of formal ideal model bound systems that display a heirarchy of properties of increasing statistical nature. Specifically, consider a dynamical system defined on a compact phase space (e.g., a bound molecule) with coordinates and momenta q, p. Of interest is the dynamics of a state, defined as a distribution /(p,q,t = 0) at time t = 0 that evolves under the influence of the Liouville operator (see Section III) Lc to... 

We first reformulate unimolecular decay in terms of symbolic dynamics so as to permit utilization of modern concepts in ergodic theory. In doing so we, at least initially, replace the continuous time dynamics by a discrete time mapping. Specifically, we consider dynamics at multiples of a fixed time increment S, defining T"x as the propagation of a phase space point x for n time increments [i.e., x(t = nS) = T"x]. In what follows, time parameters associated with the discrete dynamics are measured in units of S. These include t, t, t, and t<, which are also used in connection with the flow. In the later context the conversion to -independent units is implicit. Note that within the assumed discrete dynamics, S and S+ are broadened from surfaces to volumes S-s and S+s comprising all points that enter or have left R during a time interval b. 

Further characterization of P(t) and Pg(t) requires details of the dynamics, here formulated in terms of ergodic-theory-based assumptions. However, since ergodic theory11 considers dynamics on a bounded manifold, it is not directly applicable to the unbounded phase space associated with molecular decay. To resolve this problem we first introduce a related auxiliary bounded system upon which conditions of chaos are imposed, and then determine their effect on the molecular decay details of this construction are provided elsewhere.46 What we show is that adopting this condition leads to a new model for decay, the delayed lifetime gap model (DLGM) for P(t) and Pg(t). The simple statistical theory assumption that Pt(t) and P(t) are exponential with rate ks(E) is shown to arise only as a limiting case. 

These results constitute the first major steps in formalizing statistical theories of reaction dynamics and relating statistical molecular behavior to ergodic theory. Specifically, they demonstrate that by invoking a mixing condition on a well-chosen R we obtain an analytically soluble model for P(t) which is asymptotically well approximated by exponential decay with rate K. The rate of decay is directly affected by the relaxation time t and equals ks(R) in the limit t - 0. A similar approach can be used46 to provide an ergodic theory basis for product distributions. 

P. R. Halmos, Lectures on Ergodic Theory. Mathematical Society of Japan, Tokyo, 1953. 

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