Information theory, dynamical systems

The decay of the spatial block entropy, which gives the amount of information contained in a block of N contiguous site values ai,...,aN needed to predict the value (Jn+i is considerably slower than [block-length), and is therefore indicative of very long and complex correlations we will come back to this point later in chapter 4, following our discussion of dynamical system theory. 

As defined above, the Lyapunov exponents effectively determine the degree of chaos that exists in a dynamical system by measuring the rate of the exponential divergence of initially closely neighboring trajectories. An alternative, and, from the point of view of CA theory, perhaps more fundamental interpretation of their numeric content, is an information-theoretic one. It is, in fact, not hard to see that Lyapunov exponents are very closely related to the rate of information loss in a dynamical system (this point will be made more precise during our discussion of entropy in the next section). 

The above theory is usually called the generalized linear response theory because the linear optical absorption initiates from the nonstationary states prepared by the pumping process [85-87]. This method is valid when pumping pulse and probing pulse do not overlap. When they overlap, third-order or X 3 (co) should be used. In other words, Eq. (6.4) should be solved perturbatively to the third-order approximation. From Eqs. (6.19)-(6.22) we can see that in the time-resolved spectra described by x"( ), the dynamics information of the system is contained in p(Af), which can be obtained by solving the reduced Liouville equations. Application of Eq. (6.19) to stimulated emission monitoring vibrational relaxation is given in Appendix III. 

To illustrate an application of nonlinear quantum dynamics, we now consider real-time control of quantum dynamical systems. Feedback control is essential for the operation of complex engineered systems, such as aircraft and industrial plants. As active manipulation and engineering of quantum systems becomes routine, quantum feedback control is expected to play a key role in applications such as precision measurement and quantum information processing. The primary difference between the quantum and classical situations, aside from dynamical differences, is the active nature of quantum measurements. As an example, in classical theory the more information one extracts from a system, the better one is potentially able to control it, but, due to backaction, this no longer holds true quantum mechanically. 

We turn now to an analysis of English chemists who provided the first systematic interpretations of chemical reaction mechanisms in which the molecule was modeled as a dynamic system of positive nuclei and negative electrons. While their approach was informed by physical ideas and theories, it was unarguably a chemical approach, consistent with classical nineteenth-century chemistry, from which it developed, and with quantum chemistry, which it helped to construct. 

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. 

Recent years have also witnessed exciting developments in the active control of unimolecular reactions [30,31]. Reactants can be prepared and their evolution interfered with on very short time scales, and coherent hght sources can be used to imprint information on molecular systems so as to produce more or less of specified products. Because a well-controlled unimolecular reaction is highly nonstatistical and presents an excellent example in which any statistical theory of the reaction dynamics would terribly fail, it is instmctive to comment on how to view the vast control possibihties, on the one hand, and various statistical theories of reaction rate, on the other hand. Note first that a controlled unimolecular reaction, most often subject to one or more external fields and manipulated within a very short time scale, undergoes nonequilibrium processes and is therefore not expected to be describable by any unimolecular reaction rate theory that assumes the existence of an equilibrium distribution of the internal energy of the molecule. Second, strong deviations Ifom statistical behavior in an uncontrolled unimolecular reaction can imply the existence of order in chaos and thus more possibilities for inexpensive active control of product formation. Third, most control scenarios rely on quantum interference effects that are neglected in classical reaction rate theory. Clearly, then, studies of controlled reaction dynamics and studies of statistical reaction rate theory complement each other. 

Hamiltonian dynamical system theory is the mathematical framework on which TST rests many textbooks, of various mathematical sophistication, describe this branch of pure/applied mathematics. Some of the various flavors are [20-24]. Very little of this vast information will be needed here, and we shall try to be as self-consistent as possible. 

Not all systems can be modeled with discrete event simulation. Some events are continuous, such as the rate of evaporation. These occurrences can be modeled, but they require a different approach. For more information see Theory of Modeling and Sinnulation Integrating Discrete Event and Continuous Complex Dynamic Systems, second edition, by B. Zeigler, H. Praehofer, and T. C. Kim, New York Academic Press, 2000. 

The unique features of our system enable us to use three different theoretical tools — a molecular dynamics simulation, models which focus on the repulsion between atoms and a statistical approach, based on an information theory analysis. What enables us to use a thermodynamic-like language under the seemingly extreme nonequilibrium conditions are the high density, very high energy density and the hard sphere character of the atom-atom collisions, that contribute to an unusually rapid thermalization. These conditions lead to short-range repulsive interactions and therefore enable us to use the kinematic point of view in a useful way. 

The success of the maximum entropy procedure to predict the shattering of clusters encourages us to use it in more complicated systems, where very little is known about the potential energy surface. In the next section the results from both molecular dynamics simulations and information theory analysis for clusters made up of N2 and O2 molecules are presented. 

In the preceding sections, the possible rest points for the gradostat equations were determined and their stability analyzed. The problem that remains is to determine the global behavior of trajectories. In this regard, the theory of dynamical systems plays an important role. First of all, some information can be obtained from the general theorem on inequalities discussed in Appendix B. We illustrate this with an application to the gradostat equations. 

One of the most widely used indexes of community structure has been species diversity. Many measures for diversity are used, from such elementary forms as species number to measures based on information theory. A decrease in species diversity is usually taken as an indication of stress or impact upon a particular ecosystem. Diversity indexes, however, hide the dynamic nature of the system and the effects of island biogeography and seasonal state. As demonstrated in microcosm experiments, diversity is often insensitive to toxicant impacts. 

Smolensky, P. (1986b). Information processing in dynamical systems Foundations of harmony theory. In D. Rumelhart, J. McClelland,... 

Oscillations being qualitative characteristics of a dynamic system, global analysis in particular via bifurcation theory becomes a primary tool of detecting such characteristics of a dynamic system. On the other hand, quantitative aspects of oscillations such as their periods, and amplitudes would supply further detailed information about a particular oscillation. 

The dynamical methods presented in this chapter may be used in conjunction with electronic structure methods such as Density Functional Theory that can be used on various RDX fragments as well as on RDX itself. A key feature of the reaction path based methods is that the calculation does not scale unfavorably with the number of atoms, making the prospect of performing these calculations to get valuable dynamical information for larger systems tractable. 

There is still a gap between our models of liquid-state reactions and the often bewildering complexity of real chemical systems. Progress in shortening the gap will probably come only from the application of a variety of methods to this problem. The full promise of picosecond spectroscopy techniques for studying the details of the dynamics of reactive events in liquids has yet to be realized. How deeply can these methods probe the dynamics Computer simulations, another source of experimental information in reacting systems, are only beginning to be exploited. "" The description by direct computer simulation of both primary and secondary recombination dynamics, for example, would yield a wealth of information that could be used to test theories. 

The engineering sciences (such as statics, dynamics, strength of materials, transport processes, and electricity) can be described as refinements or adaptations of physical fundamentals. Some others, however (such as information theory and control systems), are conceptual in nature and lend themselves well to mathematical manipulation. Hence relevant mathematical attributes have been included here along with engineering sciences not directly related to physics. 

In control theory, open systems are viewed as interrelated components that are kept in a state of dynamic equilibrium by feedback loops of information and control. The plant s overall performance has to be controlled in order to produce the desired product while satisfying cost, safety, and general quality constraints. 

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