Entropies and Dimensions

The invariant density is given by p(x) = 7r-y x(l — x es is easily verified by substitution. 

While there is, at present, no known CA analogue of a Froebenius-Perron construction, a systematic n -order approximation to the invariant probability distributions for CA systems is readily obtainable from the local structure theory (LST), developed by Gutowitz, et.al. [guto87a] LST is discussed in some detail in section 5.3. 

---

Generalized Renyi Entropies and Dimensions A hierarchy of generalized entropies and dimensions, SQ B,t), s[ B,t), S B,t),. .. - analogous to the hierarchy of fractal dimensions, Dq, Di,. .., introduced earlier in equation 4.94 for continuous systems may also be defined ... 

The entropies and dimensions introduced above were defined for purely spatial sequences of site values at given times as such, they can be used to characterize sets of CA configurations. An alternative behavioral characterization is of the purely temporal sequence of site-values at a given lattice site. 

Hence, according to the transition state theory, adsorption becomes more likely if the molecule in the mobile physisorbed precursor state retains its freedom to rotate and vibrate as it did in the gas phase. Of course, this situation corresponds to minimal entropy loss in the adsorption process. In general, the transition from the gas phase into confinement in two dimensions will always be associated with a loss in entropy and the sticking coefficient is normally smaller than unity. 

More quantitative chemical evidence for random coil configuration comes from cyclization equilibria in chain molecules (49). According to the random coil model there must be a very definite relationship among the concentrations of x-mer rings in an equilibrated system, since the cyclization equilibrium constant Kx should depend on configurational entropy and therefore on equilibrium chain and ring dimensions. Values of /Af deduced from experimental values on Kx for polydimethylsiloxane, both in bulk and in concentrated solution, agree very well with unperturbed dimensions deduced from dilute solution measurements(49). 

These expressions demonstrate that the change of entropy and internal energy on deformation under these conditions is both intra- and intermolecular in origin. Intramolecular (conformational) changes, which are independent of deformation, are characterized by the temperature coefficient of the unperturbed dimensions of chains d In intermolecular changes are characterized by the thermal expansivity a and are strongly dependent on deformation. The difference between the thermodynamic values under P, T = const, and V, T = const, is vefy important at small deformations since at X - 1 2aT/(/,2 + X — 2) tends to infinity. 

In these equations we have broken down the constant A into a product ve y where v now has the dimensions of the rate constant k and in analogy to the nomenclature for A is referred to also as a frequency factor. S and H are referred to respectively as the entropy and enthalpy of activation, and as we shall see later they are capable of association with more conventional entropy and enthalpy changes. 

The question may arise whether the same changes in material properties can also be obtained by stretching the filaments after the polymerization has taken place. In our experience the anisotropy was never so great in such cases, whereas the isotropic state was almost completely recovered when the samples were heated above the melting temperature. At this temperature the post-polymerization drawn filaments retained the original dimensions they possessed before stretching. Apparently some strain-induced crystallization yielded a metastable anisotropy which was lost under the combined action of entropy and strained crosslinks when the crystalline areas were melted. 

Fig. 17. The transition to chaos (from a to d) observed in the work function during CO oxidation on Pt(l 10) while decreasing CO pressure. Chaos in the upper time series (d) was characterized by the Liapunov exponent, Kolmogorov entropy, and the embedding dimension (From Ref. 68.)...

Molecular descriptors defined in order to model solvation entropy and describe dispersion interactions in solution. Taking into account the characteristic dimension of the molecules by atomic parameters, they are defined as ... 

Some sample properties may be obvious to the analyst, such as colour, shape and dimensions or may be measured easily, such as mass, density and mechanical strength. There are also properties which depend on the bonding, molecular structure and nature of the material. These include the thermodynamic properties such as heat capacity, enthalpy and entropy and also the structural and molecular properties which determine the X-ray diffraction and spectrometric behaviour. 

Orientation in crosslinked elastomers primarily reflects the configurational entropy and intramolecular conformational energy of the chains. However, as first shown by deuterium NMR experiments on silicone rubber (Deloche and Samulski, 1981 Sotta et al., 1987), unattached probe molecules and chains become oriented by virtue of their presence in a deformed network. This nematic coupling effect is brought about intermolecular interactions (excluded volume interactions and anisotropic forces) which can cause nematic coupling (Zemel and Roland, 1992a Tassin et al., 1990). The orientation is only locally effective, so it makes a negligible conttibution to the stress (Doi and Watanabe, 1991), and the chains retain their isotropic dimensions (Sotta et al., 1987). 

When oriented polymers are heated, they will try to reach their original high entropy, and shrinkage will occur as soon as the molecules can move sufficiently to recoil to their undisturbed dimensions. For amorphous polymers this will be the case when the glass transition point 7 is reached. For crystalline polymers the behaviour is more complicated. 

Macromolecules in solution may assmne various shapes for example statistical coils, worm-like, rods or globules. The coiled stracture is most suitable for molecular characterization by hquid chromatography. As indicated in section 11.2.1, polymer coils in solution possess large conformational entropy and any forced change of the coil dimension is accompanied with the entropy alteration. 

Mathematically, the entropy and Gibbs energy potentials are two sides of the same coin - one implies the other, as shown in Figure 4.15. Looking at it in still another way, you see from Figure 4.5 that a single system can have dU y = 0 and dSuy = 0 simultaneously. If we were not limited to three dimensions, we could show that the same system also has dGj- p = 0. Each condition implies all the others. 

Real chains in good solvents have the same universal features as selfavoiding walks on a lattice. These features are described by two critical exponents, y and v. All other exponents of interest can be expressed in terms of these two. The exponent y is related to chain entropy, and the exponent v is related to chain size. A real chain has a size (flp — M ), which is much larger than an ideal chain (/ o A ). For three dimensions the exponent v is very close to 3/5. 

---