Pattern entropy

The discretized symbol-sequence defined in equation 8.39 suggests that we might use two other familiar measures of complexity to characterize the various dynamical regimes of behavior namely, the pattern entropy and dynamical entropy. 

Pattern Entropy The pattern entropy, S-p, is a purely static measure, and is defined by... 

Regime Spatial Power Spectra S(k) Pattern Distribution Q(.D) Pattern Entropy Sp Dynamical Entropy Sd... 

Cycloheptatrienes are in many cases in rapid equilibrium with an isomeric bicy-clo[4.1.0]heptadiene. The thermodynamics of the valence isomerism has been studied in a number of instances, and some of the data are given below. Calculate the equilibrium constant for each case at 25°C. Calculate the temperature at which K= for each system. Are the signs of the enthalpy and entropy as you would expect them to be Can you discern any pattern of substituent effects from the data ... 

In order to quantitatively trace behavior as a function of A, it is clear that we need to look at statistical measures that distinguish between ordered and random behavior. To this end, consider the spreading rates of differenc e patterns and entropy. 

Dynamical Entropy In order to capture the dynamics of a CML pattern, Kaneko has constructed what amounts to it mutual information between two successive patterns at a given time interval [kaneko93]. It is defined by first obtaining an estimate, through spatio-temporal samplings, of the probability transition matrix Td,d = transition horn domain of size D to a domain of size D. The dynamical entropy, Sd, is then given by... 

These patterns illustrate the incongruity that exists between mathematically precise notions of entropy (see below), or the amount of disorder in a system, and intuitive notions of complexity. 

Fig. 12.1 Three patterns illustrating the incongruity between mathematically precise notions of entropy and intuitive notions of complexity. Whereas pattern (b) is intuitively the most complex, it has neither the highest (c) or lowest (a) entropy. See text.

Whereas pattern (b) is intuitively the most complex of the three patterns, it has neither the highest entropy (which belongs to pattern (c)) or the lowest (which belongs to pattern (a)). Indeed, were we to plot our intuitive sense of complexity as a function of the amount of order or disorder in a system, it would probably look something like that shown in figure 12.2. The problem is to find an objective measure of the complexity of a system that matches this intuition. 

The difference in these patterns probably reflects that the hydrate entropies are related simply to the net positive charge on the cationic species (i.e., +2 for Pu022) while the hydrolysis reaction is the result of interaction of a water molecule with the metal atom itself — i.e., Pu in Pu022. If this is a valid explanation, the hydrolysis order indicates that the charge on Pu in Pu022 is actually between +3 and +4 and probably about +3.3. 

S, a measure of disorder. Low entropy means little disorder high entropy means great disorder. It follows that we can express the pattern that we have identified as follows ... 

The pattern we have identified is one version of the second law of thermodynamics. The natural progression of a system and its surroundings (which together make up the universe") is from order to disorder, from lower to higher entropy. For practical measurements, a small isolated region, such as a thermally insulated, sealed flask or a calorimeter, is considered to represent the universe. 

Table 7.1 lists the standard entropies of vaporization of a number of liquids. These and other data show a striking pattern many values are close to 85 J-K 1-mol h This observation is called Trouton s rule. The explanation of Trouton s rule is that approximately the same increase in positional disorder occurs when any liquid is converted into vapor, and so we can expect the... 

This pardaxin model is not unique. We have developed several similar models that are equally good energetically and equally consistent with present experimental results. It is difficult to select among these models because the helices can be packed a number of ways and the C-terminus appears very flexible. Our energy calculations are far from definitive because they do not include lipid, water, ions, membrane voltage, or entropy and because every conformational possibility has not been explored. The model presented here is intended to illustrate the general folding pattern of a family of pardaxin models in which the monomers are antiparallel and to demonstrate that these models are feasible. 

These two examples show that regular patterns can evolve but, by definition, dissipative structures disappear once the thermodynamic equilibrium has been reached. When one wants to use dissipative structures for patterning of materials, the dissipative structure has to be fixed. Then, even though the thermodynamic instability that led to and supported the pattern has ceased, the structure would remain. Here, polymers play an important role. Since many polymers are amorphous, there is the possibility to freeze temporal patterns. Furthermore, polymer solutions are nonlinear with respect to viscosity and thus strong effects are expected to be seen in evaporating polymer solutions. Since a macromolecule is a nanoscale object, conformational entropy will also play a role in nanoscale ordered structures of polymers. 

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