Chaotic relaxation

Consideration of bound-state dynamics affords one advantage not shared by systems undergoing reaction or decay. Specifically, since formal ergodic conditions require a compact phase space, ideal chaotic systems exist for bound systems but not for bimolecular collisions or unimolecular decay. Studies of these ideal bound systems therefore provide a route for analyzing statistical behavior in circumstances where the system is fully characterized. Furthermore, these ideal system results can be compared with the behavior of model molecular systems to assess the degree to which realistic systems display chaotic relaxation. 

Consider now bound-state dynamics from the viewpoint of quantum mechanics. The essential problem associated with chaotic dynamics has previously been alluded to, that is, since the quantum Liouville spectrum is discrete, the dynamics cannot display true chaotic relaxation. A number of proposals have been made as to what constitutes the quantum analogue of classical chaos.63 We now focus on an approach we have recently been developing and discuss its link to statistical behavior in reaction dynamics. 

It was further demonstrated that the dynamics of the modulation by the coupled local modes could be very complex, in which additional quasi-periodic and chaotic relaxation oscillations have been observed. In addition, there were also periodic short-time quasi-(2-switched spikes, which were attributed to the presence of foreign inclusions with SA properties likely at the grain boundaries [295]. Experimental results indicated that such local modes are present in coarse-grained ceramics and absent in fine-grained samples. 

Most chemically reacting systems tliat we encounter are not tliennodynamically controlled since reactions are often carried out under non-equilibrium conditions where flows of matter or energy prevent tire system from relaxing to equilibrium. Almost all biochemical reactions in living systems are of tliis type as are industrial processes carried out in open chemical reactors. In addition, tire transient dynamics of closed systems may occur on long time scales and resemble tire sustained behaviour of systems in non-equilibrium conditions. A reacting system may behave in unusual ways tliere may be more tlian one stable steady state, tire system may oscillate, sometimes witli a complicated pattern of oscillations, or even show chaotic variations of chemical concentrations. 

Analogous considerations apply to spatially distributed reacting media where diffusion is tire only mechanism for mixing chemical species. Under equilibrium conditions any inhomogeneity in tire system will be removed by diffusion and tire system will relax to a state where chemical concentrations are unifonn tliroughout tire medium. However, under non-equilibrium conditions chemical patterns can fonn. These patterns may be regular, stationary variations of high and low chemical concentrations in space or may take tire fonn of time-dependent stmctures where chemical concentrations vary in botli space and time witli complex or chaotic fonns. 

The perturbation theory presented in Chapter 2 implies that orientational relaxation is slower than rotational relaxation and considers the angular displacement during a free rotation to be a small parameter. Considering J(t) as a random time-dependent perturbation, it describes the orientational relaxation as a molecular response to it. Frequent and small chaotic turns constitute the rotational diffusion which is shown to be an equivalent representation of the process. The turns may proceed via free paths or via sudden jumps from one orientation to another. The phenomenological picture of rotational diffusion is compatible with both... 

Fig. 13.19. Relaxation-chaotic oscillations a(t), fe(t), and c(t) for the Hudson-Rossler model.

Summary. The semiclassical Boltzmann-Langevin method is extended to calculations of higher cumulants of current. Rs efficiency is demonstrated for mesoscopic diffusive contacts and chaotic cavities. We show that in addition to a dispersion at the inverse RC time characteristic of charge relaxation, higher cumulants of noise have a low-frequency dispersion at the inverse dwell time of electrons in the system. 

Thus, specific interactions directly determine the spectroscopic features due to hydrogen bonding of the water molecules, while unspecific interactions arise in all or many polar liquids and are not directly related to the H-bonds. Now it became clear that the basis of four different processes (terms) used in Ref. [17] and mentioned above could rationally be explained on a molecular basis. One may say that specific interactions are more or less cooperative in their nature. They reveal some features of a solid state, while unspecific interactions could be understood in terms of a liquid state of matter, if we consider chaotic gas-like motions of a single polar molecule, namely, rotational motions of a dipole in a dense surroundings of other molecules. The modem aspect of the spectroscopic studies leads us to a conclusion that both gas-like and solid-state-like effects are the characteristic features of water. In this section we will first distinguish between the following two mechanisms of dielectric relaxation ... 

When the temperature rises, the Debye relaxation time Td and the fitted mean reorientation time xor decrease, since intermolecular interactions weaken and become more chaotic. 

In an open system such as a CSTR chemical reactions can undergo self-sustained oscillations even though all external conditions such as feed rate and concentrations are held constant. The Belousov-Zhabotinskii reaction can undergo such oscillations under isothermal conditions. As has been demonstrated both by experiments [1] and by calculations 12,3] this reaction can produce a variety of oscillation types from simple relaxation oscillations to complicated multipeaked periodic oscillations. Evidence has also been given that chaotic behavior, as opposed to periodic or quasi-periodic behavior, can take place with this reaction [4-12]. In addition, it has been shown in recent theoretical studies that chaos can occur in open chemical reactors [11,13-17]. 

In contrast, conventional reaction rate theory replaces the dynamics within the potential well by fluctuations at equilibrium. This replacement is made possible by the assumption of local equilibrium, in which the characteristic time scale of vibrational relaxation is supposed to be much shorter than that of reaction. Furthermore, it is supposed that the phase space within the potential well is uniformly covered by chaotic motions. Thus, only information concerning the saddle regions of the potential is taken into account in considering the reaction dynamics. This approach is called the transition state theory. 

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