Entropy mutual

Joint Entropy Conditional Entropy Mutual Entropy... 

In Eq. (15) the second term reflects the gain in entropy when a chain breaks so that the two new ends can explore a volume Entropy is increased because the excluded volume repulsion on scales less than is reduced by breaking the chain this effect is accounted for by the additional exponent 9 = y — )/v where 7 > 1 is a standard critical exponent, the value of 7 being larger in 2 dimensions than in 3 dimensions 72 = 43/32 1.34, 73j 1.17. In MFA 7 = 1, = 0, and Eq. (15) simplifies to Eq. (9), where correlations, brought about by mutual avoidance of chains, i.e., excluded volume, are ignored. 

Fig. 3.43 Schematic form of a plot of mutual information Xm vs. average single-site entropy H.

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... 

Table 9 shows that the value of AGn of the cooperative interaction between bonding centers is within the error in the determination of integral AG values. This fact can either indicate the slight mutual influence of the centers or be caused by the compensation between the enthalpy and entropy components of Gibbs free energy. 

Any entropy which the system may possess in virtue of the mutual actions of the masses is taken as included in Sni. 

The theorem of Nernst applies only to chemically homogeneous condensed phases the entropy of a condensed solution phase has at absolute zero a finite value, owing to the mutual presence of the different components. 

Gee and Orr have pointed out that the deviations from theory of the heat of dilution and of the entropy of dilution are to some extent mutually compensating. Hence the theoretical expression for the free energy affords a considerably better working approximation than either Eq. (29) for the heat of dilution or Eq. (28) for the configurational entropy of dilution. One must not overlook the fact that, in spite of its shortcomings, the theory as given here is a vast improvement over classical ideal solution theory in applications to polymer solutions. 

The net retention volume and the specific retention volume, defined in Table 1.1, are important parameters for determining physicochemical constants from gas chromatographic data [9,10,32]. The free energy, enthalpy, and. entropy of nixing or solution, and the infinite dilution solute activity coefficients can be determined from retention measurements. Measurements are usually made at infinite dilution (Henry s law region) in which the value of the activity coefficient (also the gas-liquid partition coefficient) can be assumed to have a constant value. At infinite dilution the solute molecules are not sufficiently close to exert any mutual attractions, and the environment of each may be considered to consist entirely of solvent molecules. The activity... 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

FVP treatment of both isolated samples 300 and 301 resulted in a mixture containing 300 + 301 this finding revealed that these two compounds can be mutually transformed into each other under forced conditions. Upon some kinetic measurements (determination of a negative activation entropy), the authors concluded that the ring transformation probably proceeds via a concerted sigmatropic shift to an antiaromatic diazirine intermediate. 

As in the previous experiments, at each epoch we would like to select a waveform (or really the error covariance matrix associated with a measurement using this waveform) so that the measurement will minimize the uncertainty of the dynamic model of the target. We study two possible measures entropy of the a posteriori pdf of the models and mutual information between the dynamic model pdf and measurement history. Both of these involve making modifications to the LMIPDA-IMM approach that are described in [5]. Since we want to minimize the entropy before taking the measurement, we need to consider the expected value of the cost. To do this we replace the measurement z in the IMM equations by its expected value. In the case of the second measure, for a model we have... 

V -dimethylformamide (DMF) and for the reaction of the same compound with anthracene anion radical in the same solvent.12 The results are shown in Figure 3.3. In the electrochemical case, the values predicted for the cyclic voltammetric peak potential (at 0.2 V/s) and the entropy of activation are plotted as functions of the ratio C/AS. Validation of the theory derives from the observation that the agreement between theoretical and experimental values is reached for the same value of AS C/AS for the peak potential and the entropy of activation. The same is true for the homogeneous reactions. That this common value of AS°F c/ASi]h- is smaller in the latter case than in the former falls in line with the fact that the presence of anthracene renders more difficult the mutual displacement of the R and X moieties within the solvent cage. 

Consider two liquid substances that are rather similar, such as benzene and toluene or water and ethylene glycol. When moles of the one are mixed with B moles of the other, the composition of the liquid mixture is given by specification of the mole fraction of one of them [e.g., Xa, according to Eq. (2.2)]. The energy or heat of the mutual interactions between the molecules of the components is similar to that of their self interactions, because of the similarity of the two liquids, and the molecules of A and B are distributed completely randomly in the mixture. In such mixtures, the entropy of mixing, which is a measure of the change in the molecular disorder of the system caused by the process of mixing the specified quantities of A and B, attains its maximal value ... 

CVM is a more powerful formalism which can include the mutual interaction of all the atoms in sets of clusters, so as to properly reflect a greater variety of atomic interactions (de Fontaine 1979, 1994). The smallest cluster that should be used to describe a three-dimensional lattice is the tetrahedron (Kikuchi 1951), but the complete calculation of entropy using CVM requires a consideration of all subsidiary clusters (van Baal 1973). This means pairs and planar triangles have to be included for the tetrahedron approximation, and all these in turn have to be included when larger clusters such as octahedra are considered. Various mathematical techniques... 

The principle of the liquid chromatography under critical conditions (LC CC) was elucidated in Section 16.3.3. The mutual compensation of the exclusion—entropy and the interaction—enthalpy-based retention of macromolecules can be attained when applying in the controlled way the interactions that lead to either adsorption or enthalpic partition. The resulting methods are called LC at the critical adsorption point (LC CAP) or LC at the critical partition point (LC CPP), respectively. The term LC at the point of exclusion-adsorption transition (LC PEAT) was also proposed for the procedures employing compensation of exclusion and adsorption [161]. It is anticipated that also other kinds of enthalpic interactions, for example the ion interactions between column packing and macromolecules can be utilized for the exclusion-interaction compensation. 

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