Chaotic diffusion

In the classical case, when the field intensity exceeds a threshold value, the electron s motion becomes chaotic and strong excitation and ionization takes place. In the quantum case instead, interference effects may lead to the suppression of the classical chaotic diffusion, the so-called quantum dynamical localization, and in order to ionize the atom, a larger field intensity is required (see Fig. 1). 

Considering the sensitivity of classical chaotic systems to external perturbations, and the ubiquitous nature of chaotic dynamics in larger systems, it is important to 1 establish that quantum mechanics allows for control in chaotic systems as well. [ One simple molecular system that displays quantirm chaos is the rotational exci- tation of a diatomic molecule using pulsed microwave radiation [227], Under the conditions adopted below, this system is a molecular analog of the delta-lacked ij rotor, that is, a rotor that is periodically lacked by a delta fiinction potential, which 4 is a paradigm for chaotic dynamics [228, 229], The observed energy absorption of such systems is called quantum chaotic diffusion. 

Evidently, the first two terms are incoherent since they do not depend on the value of the phase p in Eq. (6.69). They represent quantum dynamics associated with each of. the states j, 0) and j2, 0) independently. The last two terms represent interference effects due to initial-state coherence between ly), 0) and j2, 0). Hence, the absorp-, 2 tion of rotational energy in this system, that is, quantum chaotic diffusion, can be controlled by manipulating the quantum phase p in the initial state [Eq. (6.69)], which corresponds to manipulating the interference tenn in Eq. (6.73). V... 

In the Anderson picture the suppression of classical chaotic diffusion is understood as a destructive phase interference phenomenon that limits the spread of the rotor wave function over the available angular momentum space. The localization effect has no classical analogue. It is purely quantum mechanical in origin. The localization of the quantum rotor wave function in the angular momentum space can be demonstrated readily by plotting the absolute squares of the time averaged expansion amplitudes... 

Bayfield, J.E., Casati, G., Guameri, I. and Sokol, D.W. (1989). Localization of classically chaotic diffusion for hydrogen atoms in microwave fields, Phys. Rev. Lett. 63, 364-367. 

For systems of more than two degrees of freedom, however, their role is not so obvious because their dimension is not large enough to work as barriers. This problem is closely related to the phenomenon of Arnold diffusion. In other words, dynamical processes along nonlinear resonances may create a way to go around these tori. In particular, intersections of resonances would play a dominant role in IVR since chaotic diffusion... 

Figure 3.24(h) shows the opposite to Figure 3.24(a) in the sense that a web has dense intersections. Moreover, those intersections overlap in phase space creating global chaos. Here, chaotic diffusion takes place in all the directions on the equi-energy surface. Then, the IVR takes place uniformly involving all the degrees of freedom, which is the case where the statistical reaction theory can be applied. 

Figure 12. (a) Propagation of the preferential flow zones (fingers) along the fracture surface with time for different angles of the fracture model inclination (from vertical) (27), and (b) results of calculations of coefficient a for eq 3, which describes the process of chaotic diffusion, using the results of the Su et al. (27) and Tokunaga etal. (26) experiments. 

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. 

In tills chapter we shall examine how such temporal and spatial stmctures arise in far-from-equilibrium chemical systems. We first examine spatially unifonn systems and develop tlie tlieoretical tools needed to analyse tlie behaviour of systems driven far from chemical equilibrium. We focus especially on tlie nature of chemical chaos, its characterization and the mechanisms for its onset. We tlien turn to spatially distributed systems and describe how regular and chaotic chemical patterns can fonn as a result of tlie interjilay between reaction and diffusion. 

The local dynamics of tire systems considered tluis far has been eitlier steady or oscillatory. However, we may consider reaction-diffusion media where tire local reaction rates give rise to chaotic temporal behaviour of tire sort discussed earlier. Diffusional coupling of such local chaotic elements can lead to new types of spatio-temporal periodic and chaotic states. It is possible to find phase-synchronized states in such systems where tire amplitude varies chaotically from site to site in tire medium whilst a suitably defined phase is synclironized tliroughout tire medium 51. Such phase synclironization may play a role in layered neural networks and perceptive processes in mammals. Somewhat suriDrisingly, even when tire local dynamics is chaotic, tire system may support spiral waves... 

If tlie diffusion coefficients of tlie chemical species are sufficiently different, new types of chemical instability arise which can lead to tlie fonnation of chemical patterns and ultimately to spatio-temporal chaotic behaviour. 

If the diffusion coefficient of species A is less tlian tliat of B (D < D ) tlie propagating front will be planar. However, if is sufficiently greater than tire planar front will become unstable to transverse perturbations and chaotic front motion will ensue. To understand tire origin of tire mechanism of tire planar front destabilization consider tire following suppose tire interface is slightly non-planar. We would like to know if tire dynamics will tend to eliminate this non-planarity or accentuate it. LetZ)g The situation is depicted schematically in figure... 

This is strictly true only for two-dimensional area-preserving maps in dimensions iV > 4, chaotic orbits may leak through KAM surfaces by a process called Arnold diffusion (see [licht83j). 

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... 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

The rate of agitation, stirring, or flow of solvent, if the dissolution is transport-controlled, but not when the dissolution is reaction-con-trolled. Increasing the agitation rate corresponds to an increased hydrodynamic flow rate and to an increased Reynolds number [104, 117] and results in a reduction in the thickness of the diffusion layer in Eqs. (43), (45), (46), (49), and (50) for transport control. Therefore, an increased agitation rate will increase the dissolution rate, if the dissolution is transport-controlled (Eqs. (41 16,49,51,52), but will have no effect if the dissolution is reaction-controlled. Turbulent flow (which occurs at Reynolds numbers exceeding 1000 to 2000 and which is a chaotic phenomenon) may cause irreproducible and/or unpredictable dissolution rates [104,117] and should therefore be avoided. 

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