Average Entropy

In the volume elements describing individual subchains, the x, y, and z dimensions will be different, so Eq. (3.32) must be averaged over all possible values to obtain the average entropy change per subchain. This process is also easily accomplished by using a result from Chap. 1. Equation (1.62) gives the mean-square end-to-end distance of a subchain as n, 1q, and this quantity can also be written as x + y + z therefore... 

This expression gives the average entropy change per chain to get the average for the sample, we multiply by the number v of subchains in the sample. The total entropy change is... 

Fig. 3.26 Time evolution of the average entropy for elementary rule R32 see text.

For reven sible systems, evolution almost always leads to an increase in entropy. The evolution of irreversible systems, one the other hand, typically results in a decrease in entropy. Figures 3.26 and 3.27 show the time evolution of the average entropy for elementary rules R32 (class cl) and R122 (class c3) for an ensemble of size = 10 CA starting with an equiprobable ensemble. We see that the entropy decreases with time in both cases, reaching a steady-state value after a transient period. This dc crease is a direct reflection of the irreversibility of the given rules,... 

Although it is obviously impossible to enumerate all possible configurations for infinite lattices, so long as the values of far separated sites are statistically independent, the average entropy per site can nonetheless be estimated by a limiting procedure. To this end, we first generalize the definitions for the spatial set and spatial measure entropies given above to their respective block-entropy forms. 

While Sgf B,t) is determined directly by the number of distinct sequences of length X generated at time t , Smeas B,t) gives the average entropy per site by taking into account whatever correlations exist in blocks of length up to B. 

The average entropy production (5p) is deflned by averaging Sp along an infinite number of paths. Dividing Eq. (51) by (5p) we get... 

The average entropy decrease on adsorption of approximately 0.75 erg per sq. cm. when converted to a molar basis from the volume adsorbed at saturation vapor pressure is 1.8 0.4 e.u. less than liquid water at 25° C. or about that of liquid water at 0° C. The large uncertainty is due to difficulty in extrapolating to P/P° = 1. Since the entropy of adsorbed water molecules in the outermost layer must approximate that of liquid water at 25° C., then the inner layers, to maintain the average of 1.8 e.u., must be icelike with loss of translational modes. 

Scheme 1.2 The flexible-input generalization of the two-AO channel of Scheme 1.1a for the promolecular reference distribution p° = (1/2,1/2). The corresponding partial and average entropy/information descriptors of the chemical bond are also reported.

In the software badge, we account for model uncertainty by averaging the results of the posterior inference conditional on the Gamma and lognormal distributions for the gene expression data. As a parallel with the sample size determination when the inference process is based on model averaging, we therefore introduce the Average Entropy, denoted by Enta(-), and defined as... 

This quantity averages the Shannon entropies conditional on the Gamma and lognormal models, with weights given by their posterior probabilities. In Appendix B, we show that the average entropy is a concave function on the space of probability distributions which is monotone under contractive maps (Sebastiani and... 

Theorem 1 (Concavity). The average entropy Enta(9) is a concave function of the set of probability distributions for 9. 

Proof. The result follows from the fact that Shannon entropy is concave in the space of probability distribution (DeGroot, 1970), and the average entropy is a convex combination of Shannon entropies. 

Theorem 3 (Decomposability). The average entropy of the random vector 9 = 0i, 02 can be decomposed as... 

Proof. Let ML and Mu denote lognormal distributions for the expression values of two genes, and let wk and w2 be the posterior probability assigned to the models ML and ML2. When we decompose the average entropy of 0i and 02 we need to consider the space of model combinations... 

Only the orbits infinitesimally close to the steady state may be considered stable, according to Liyapunov s theory of stability. However, at a finite distance from the steady state, two neighboring points belonging to two distinct cycles tend to be far apart from each other because of differences in the period. Such motions are called stable in the extended sense of orbital stability. The average concentrations of X and Y over an arbitrary cycle are equal to their steady-state values (Xs = 1 and Y.. A 1). Under these conditions, the average entropy production over one period remains equal to the steady-state entropy production. 

Calculate the standard transfomed entropies of formation of the species and then calculate the mole fraction-weighted average entropy of the reactant. ) pHtermS = nH 8.31451 10 -3 Log [10 -pH] ... 

The experimental. 4-factors for the alcohol elimination reactions are all in the range, log. 4 = 11.8 0.3, which agrees well with transition state estimates log4js, = 11.5+0.3. Activation entropies are therefore, AS —8 + 1.4 cal. deg mole This gives an average entropy loss of —2.7 eu per internal rotation restricted in the transition state. The experimental data for these reactions all seem quite reliable. Data for these reactions and the four-center reactions are given in Table 37. 

The latter equation is of great importance as it allows a calculation of the average affinity - (AG)2r from a knowledge of the average heat (AH)t,2) average entropy of reaction ASYrr r ... 

Figure 5.120 presents, next, the entropy of melting of 20 alkali halides. All of these crystals have the F 3 space group of the NaCl stracture which is depicted in Fig. 5.2. The average entropy of fusion of these 20 salts is 24.43 1.7 J K moT. Since their formula mass refers to two ions, the average positional entropy of fusion is 12.2 J K" mol", in good accord with Richards s rale. A total of 76 other salts, with up to four ions per formula, was similarly analyzed [41]. The majority of these salts follow the same rule of constant entropy of fusion per mole of ions, irrespective of the...

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