The Concept of Ergodicity

The formal definition of ergodicity takes one of several forms. For our current purposes, we opt for a definition that is close to our intended goal (the computation of averages). 

Definition 5.1 A Hamiltonian system is said to be ergodic on an energy surface X e (microcanonically ergodic) if, for almost all trajectories F emanating from initial conditions on He, and for any observable g, the following holds 

Alternatively, we may say that a dynamical system as ergodic if the probability measure of any invariant set A of the flow is either 0 or 1. An invariant set A of the flow is one such that 

On the other hand, the average of x on Se is just the probability measure of Se A. If we assume the system is ergodic, we must have Pt(I7e A) = 0, thus Pr(.A) = 1. Since, according to the definition, equality of time and space averages is allowed to fail only on a set of probability measure zero, we have that either Pr(A) = OorPr(A) = 1. 

the strict definition of ergodicity must be relaxed to accommodate the realistic modelling setting for example only some portion of the eno gy surface may be accessible by trajectories in the time-interval available, due to the cost of computing steps with a numerical method. For a more detailed discussion of these and other ergodicity issues in the context of deterministic molecular dynamics, see the articles of Tupper [379, 381]. 

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Central to many developments in this book is the concept of ergodicity. Let us consider a physical system consisting of N particles. Its time evolution can be described as a path, or trajectory, in phase space. If the system was initially in the state... 

The nature of the intramolecular motion of a molecule is intimately related to the concept of ergodicity. To quote Maxwell, the ergodic hypothesis .. . is that the... 

The concept of ergodicity, and the method of analyzing it, can be understood using an analogy to a simpler finite model, a discrete Markov Chain. 

Mathematically, the concept of mixing is rather unwieldy, and is built upon the concepts of ergodicity and dynamical systems theory. The reader is referred to [1] for an introduction to the mathematical treatment of mixing, and... 

To characterize these invariant structures and the changes of reaction coordinates, the concept of finite-time Lyapunov exponents can be useful [44]. The original definition of the Lyapunov exponents needs ergodicity (see, e.g.. Ref. 45) to make sure that the time average of the exponents converges. However, for chaotic itinerancy, the exponents would not converge. Moreover, the finite-time Lyapunov exponents can be more useful to detect whether... 

Here we also include the contribution of Okushima, in which the concept of the Lyapunov exponents is extended to orbits of finite duration. The mathematical definition of the Lyapunov exponents requires ergodicity to ensure convergence of the definition. On the other hand, various attempts have been made to extend this concept to finite time and space, to make it applicable to nonergodic systems. Okushima s idea is one of them, and it will find applications in nonstationary reaction processes. 

Does the concept of chemical equilibrium make sense in this context It is not clear that the concept of equilibrium makes sense, even for a closed thermodynamic system. Matter might flow persistently across the reaction graph for the lifetime of the universe non-ergodically, without ever reaching an equilibrium distribution. 

The dynamical origin of statistical physics has been one of the main topics in the study of chaos. The very existence of statistical averages, the concept of equilibrium and the mechanism of approaching equilibrium are among the most important. Here, the idea of ergodicity plays a crucial role. However, the current study of chaos is not limited to these topics related to equilibrium statistical physics. It also extends to entail dynamical behavior in systems far from equilibrium and the properties of those systems that do not necessarily show ergodicity. 

An even stronger property than the ergodic property is the concept of a mixing system. For a mixing system, the finite time density p(z, t) converges, in the weak sense, to the invariant distribution Poo(z), as f 00. That is, we have, for all test functions (p in some chosen space... 

There is a key difference between the concept of microcanonical ergodicity and canonical ergodicity. For deterministic dynamics, the distribution solves the Liouville equation... 

Nos developed a deterministic approach to constant temperature simulations based on an extended Lagrangian which does not disturb the dynamic properties of the system. Since then, modifications of this approach have been developed to generate constant pressure and temperature simulations. These extended system methods have been known to suffer from stability problems, as well as occasional failures in ergodicity. Klein and co-workers introduced the concept of Nos6-Hoover chains to overcome these problems, and used a Liouville operator formalism to obtain reversible integrators generating these chains. 

Concepts of Coherence (1.18-21) The correlation term in the expression for the intensity of superimposed helds, and in general the ergodic mean of the product of held amplitudes sampled at different points in space and time, is called mutual intensity. If the conditions are temporally stationary - which they would be if the observed sources are stable with time - it is the time difference r which matters. 

Thus, the well-known concept of stationary reaction rates limitation by "narrow places" or "limiting steps" (slowest reaction) should be complemented by the ergodicity boundary limitation of relaxation time. It should be stressed that the relaxation process is limited not by the classical limiting steps (narrow places), but by reactions that may be absolutely different. The simplest example of this kind is an irreversible catalytic cycle the stationary rate is limited by the slowest reaction (the smallest constant), but the relaxation time is limited by the reaction constant with the second lowest value (in order to break the weak ergodicity of a cycle two reactions must be eliminated). 

One should be careful to distinguish between the following two concepts (a) an ergodic system, (b) ergodic density distribution in the T-space. For the relationship of (a) to (b), see note 101. 

In cases like D2CO or NO2 comparison with experimental data on a state-specific level are ruled out entirely and one has to retreat to more averaged quantities like the average dissociation rate, (fc), and the distribution of rates, Q(k). If the dynamics is ergodic — the basic assumption of all statistical theories — one can derive a simple expression for Q k), which had been established in nuclear physics in order to describe the neutron emission rates of heavy nuclei [280]. These concepts have since developed into the field of random matrix theory (RMT) and statistical spectroscopy [281-283] and have also found applications in the dissociation of energized molecules [121,284-286]. 

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. 

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. 

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