Statistical theories dynamical aspects

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. 

FLUID DYNAMICAL ASPECTS AND MACROSCOPIC THEORY. The following section shows that one can join statistical mechanics with fluid dynamics in the spirit of the global simulations this link is essential. The conceptual, intellectual and practical importance of this link is equally important and we are confident to have opened an important path to further understand physical phenomena. 

Classical trajectory calculations, performed on the PES1 and PESl(Br) potential energy surfaces described above, have provided a detailed picture of the microscopic dynamics of the Cl- + CH3Clb and Cl" + CH3Br SN2 nucleophilic substitution reactions.6,8,35-38 In the sections below, different aspects of these trajectory studies and their relation to experimental results and statistical theories are reviewed. 

It is most remarkable that the entropy production in a nonequilibrium steady state is directly related to the time asymmetry in the dynamical randomness of nonequilibrium fluctuations. The entropy production turns out to be the difference in the amounts of temporal disorder between the backward and forward paths or histories. In nonequilibrium steady states, the temporal disorder of the time reversals is larger than the temporal disorder h of the paths themselves. This is expressed by the principle of temporal ordering, according to which the typical paths are more ordered than their corresponding time reversals in nonequilibrium steady states. This principle is proved with nonequilibrium statistical mechanics and is a corollary of the second law of thermodynamics. Temporal ordering is possible out of equilibrium because of the increase of spatial disorder. There is thus no contradiction with Boltzmann s interpretation of the second law. Contrary to Boltzmann s interpretation, which deals with disorder in space at a fixed time, the principle of temporal ordering is concerned by order or disorder along the time axis, in the sequence of pictures of the nonequilibrium process filmed as a movie. The emphasis of the dynamical aspects is a recent trend that finds its roots in Shannon s information theory and modem dynamical systems theory. This can explain why we had to wait the last decade before these dynamical aspects of the second law were discovered. 

Elementary processes in chemical dynamics are universally important, besides their own virtues, in that they can link statistical mechanics to deterministic dynamics based on quantum and classical mechanics. The linear surprisal is one of the most outstanding discoveries in this aspect (we only refer to review articles [2-7]), the theoretical foundation of which is not yet well established. In view of our findings in the previous section, it is worth studying a possible origin of the linear surprisal theory in terms of variational statistical theory for microcanonical ensemble. 

In the preceding sections several statistical approaches for calculating reaction rates and distributions of quantum states were described and illustrated using recent studies. We conclude with some remarks on the dynamical aspects leading to and creating deviations from statistical theories. 

Several aspects of this formulation require emphasis. First, defining the SCR yields a particular statistical theory hence, a wide variety of theories are possible. Second, one should anticipate that a typical scattering event will have both direct and statistical components. Third, an analysis of the degree of statistical behavior in an exact dynamical calculation requires three sequential steps (1) the SCR must first be defined (2) the component of the scattering that does not pass through the SCR is then eliminated from consideration and (3) the product distribution associated with the component passing through the SCR is compared with the results predicted from Eq. (2.17) the latter often requires additional numerical computations. 

On a modest level of detail, kinetic studies aim at determining overall phenomenological rate laws. These may serve to discriminate between different mechanistic models. However, to it prove a compound reaction mechanism, it is necessary to determine the rate constant of each elementary step individually. Many kinetic experiments are devoted to the investigations of the temperature dependence of reaction rates. In addition to the obvious practical aspects, the temperature dependence of rate constants is also of great theoretical importance. Many statistical theories of chemical reactions are based on thermal equilibrium assumptions. Non-equilibrium effects are not only important for theories going beyond the classical transition-state picture. Eventually they might even be exploited to control chemical reactions [24]. This has led to the increased importance of energy or even quantum-state-resolved kinetic studies, which can be directly compared with detailed quantum-mechanical models of chemical reaction dynamics [25,26]. 

It is not the aim of this book to give a full account of the present stage of the collision and statistical theory by considering all various approaches to the solution of the dynamic problems involved. Instead, it attempts to present a detailed discussion of the relations between both theories from a unified point of view. Therefore, attention is paid not so much to computational techniques as to the fundamental aspects of the problem. Their complete elucidation is possible only by means of exact definitions of the concepts and by accurate formulations of the theories. Computational approaches are certainly of great importance for the practical application of any physical theory. In particular, the physical chemist is much interested in how to calculate the reaction velocities, which requires an estimation of various parameters entering the rate equations. Very often, however, we ask about the procedure of evaluating some quantities which are not well defined, for instance, the quantum correction to a classical (or semiclassical), collision or statistical theory. As a consequence, large discrepancies between the results of different approaches arise mainly... 

The TST, as Eyring s theory is known, is a stadstical-mechanical theory to calculate the rate constants of chemical reactions. As a statistical theory it avoids the dynamics of colUsions. However, ultimately, TST addresses a dynamical problem the proper defmition of a transition state is essentially dynamic, because this state defines a condition of dynamical instability, with the movement on one side of the transition state having a different character from the movement on the other side. The statistical mechanics aspect of the theory comes from the assumption that thermal equilibrium is maintained all along the reaction coordinate. We will see how this assumption can be employed to simplify the dynamics problem. 

Although the collision and transition state theories represent two important methods of attacking the theoretical calculation of reaction rates, they are not the only approaches available. Alternative methods include theories based on nonequilibrium statistical mechanics, stochastic theories, and Monte Carlo simulations of chemical dynamics. Consult the texts by Johnson (62), Laidler (60), and Benson (59) and the review by Wayne (63) for a further introduction to the theoretical aspects of reaction kinetics. 

Determination of the statistical characteristics of natural catastrophes in their historical aspect, selecting categories and determining spatiotemporal scales of catastrophic changes in habitats. Analysis of the history of disasters is important for understanding the present dependences of crises both in nature and in society. The statistical characteristics of the dynamics of natural disasters enable formulation of the basis for the mathematical theory of catastrophes and to determine top-priority directions of studies. 

TYoe, J. (1992). Statistical aspects of ion-molecule reactions, in State-Selected and State-to-State Ion-Molecule Reaction Dynamics, Part 2 Theory, ed. M. Baer and C.Y. Ng (Wiley, New York). 

We have added several appendices that give a short introduction to important disciplines such as statistical mechanics and stochastic dynamics, as well as developing more technical aspects like various coordinate transformations. Furthermore, examples and end-of-chapter problems illustrate the theory and its connection to chemical problems. 

With these theories in mind, we now turn to a number of examples of organic reactions studied using direct dynamics. In all of these cases, some aspect of the application of the statistical approximation is found to be in error. At minimum, the collected weight of these trajectory studies demonstrates the caution that need be used when applying TST or RRKM. More compelling though is that these studies question the very nature of the meaning of reaction mechanism. 

Computer simulations, Monte Carlo or molecular dynamics, in fact appear to be the actual most effective way of introducing statistical averages (if one decides not to pass to continuous distributions), in spite of their computational cost. Some concepts, such as the quasi-structure model introduced by Yomosa (1978), have not evolved into algorithms of practical use. The numerous versions of methods based on virial expansion, on integral equation description of correlation functions, on the application of perturbation theory to simple reference systems (the basic aspects of these... 

It is natural to conceive that this short-time behavior should be due to some time interval for a trajectory to spend to look for exit ways to the next basins in the complicated structure of phase space. In the next section, we will propose a geometrical view that shows what this complexity is. Hence we consider that the hole of Na- b(t) in the short-time region should be a reflection of chaos, which is just opposite to the behavior arising from nonchaotic direct paths as observed in Hj" dynamics. The present effect is therefore expected to be more significant as the molecular size increases or the potential surface and corresponding phase-space structure become more complicated. Another important aspect of the hole in Na-,b t) is an induction time for a transport of the flow of trajectories in phase space. It is of no doubt that the RRKM theory does not take account of a finite speed for the transport of nonequilibrium phase flow from the mid-area of a basin to the transition states. Berblinger and Schlier [28] removed the contribution from the direct paths and equate the statistical part only to the RRKM rate. One should be able to do the same procedure to factor out the effect of the induction time due to transport. We believe that the transport in phase space is essentially important in a nonequilibrium rate theory and have reported a diffusion model to treat them [29]. 

Troe, J. Statistical aspects of ion-molecule reactions. In State-selected and state-to-state ion-molecule reaction dynamics theory Baer, M., Ng, C.-Y., Eds. John Wiley New York, 1992, 485. 

Theoretical chemistry is the discipline that uses quantum mechanics, classical mechanics, and statistical mechanics to explain the structures and dynamics of chemical systems and to correlate, understand, and predict their thermodynamic and kinetic properties. Modern theoretical chemistry may be roughly divided into the study of chemical structure and the study of chemical dynamics. The former includes studies of (1) electronic structure, potential energy surfaces, and force fields (2) vibrational-rotational motion and (3) equilibrium properties of condensed-phase systems and macromolecules. Chemical dynamics includes (1) bimolecular kinetics and the collision theory of reactions and energy transfer (2) unimolecular rate theory and metastable states and (3) condensed-phase and macromolecular aspects of dynamics. 

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