Entropy vectors

Figure 11.2 Thermodynamic vectors for a simple fluid, represented in a two-dimensional diagram in which lengths and angles are expressed in terms of experimental properties for example, cos 0St — (Cy/Cp)1 2 and cos 6Sy = aP(TW/CPpT) 2. The thermodynamically conjugate temperature and volume vectors T), V) are perpendicular, as are the pressure and entropy vectors P), S). A number of thermodynamic relationships among the experimental quantities can be read off directly from the diagram.

Figure 11.5 compares the fluid entropy vectors, whose lengths range from about 0.25 (ideal gas) to about 0.75 (ether). As expected, the entropy vectors exhibit an approximate inverted or complementary (conjugate) relationship to the corresponding T vectors of Fig. 11.3. The length of each S vector reflects resistance to attempted temperature change (under isobaric conditions), i.e., the capacity to absorb heat with little temperature response. The lack of strict inversion order with respect to the T lengths of Table 11.3 reflects subtle heat-capacity variations between isochoric and isobaric conditions, as quantified in the heat-capacity or compressibility ratio...

Figure 11.5 Entropy vectors (SI units) of common laboratory liquids at 20°C, 1 atm.

In contrast to the unit dependence of the thermodynamic vector lengths and metric eigenvalues, the thermodynamic angles are pure dimensionless numbers. Figure 11.9 exhibits the angle 6Sy that each entropy vector S) makes with respect to the volume (abscissa) and temperature (ordinate) axes. 

Figure 11.9 Entropy vectors S) of common laboratory fluids at 20°C, 1 atm, showing each vector in its proper orientation with respect to orthogonal V) and T) directions.

We mentioned above that a typical problem for a Boltzman Machine is to obtain a set of weights such that the states of the visible neurons take on some desired probability distribution. For example, the task may he to teach the net to learn that the first component of an Ai-component input vector has value +1 40% of the time. To accompli.sh this, a Boltzman Machine uses the familiar gradient-descent technique, but not on the energy of the net instead, it maximizes the relative entropy of the system. 

We now apply the same principle to calculate the entropy of a spectrum (or any other signal). The entropy, 5 of a spectrum given by the vector y is defined as... 

The name of the distribution is due to the fact that the saddle point X can also be obtained as the vector ofLagrange multipliers needed to find the distribution q = qME for which the relative entropy,... 

In 1983, Sasaki et al. obtained rough first approximations of the mid-infrared spectra of o-xylene, p-xylene and m-xylene from multi-component mixtures using entropy minimization [83-85] However, in order to do so, an a priori estimate of the number S of observable species present was again needed. The basic idea behind the approach was (i) the determination of the basis functions/eigenvectors V,xv associated with the data (three solutions were prepared) and (ii) the transformation of basis vectors into pure component spectral estimates by determining the elements of a transformation matrix TsXs- The simplex optimization method was used to optimize the nine elements of Tixi to achieve entropy minimization, and the normalized second derivative of the spectra was used as a measure of the probability distribution. 

When a wormlike spherocylinder is in the liquid crystal phase, its tangent vector a at each contour point should align more or less to the preference direction of the phase specified by the director n. This alignment induces the orientational entropy decrease — Sor from the entropy in the isotropic state. Since the orientation of the tangent vector stretches the wormlike spherocylinder, — Sor includes a conformational entropy loss of the spherocylinder. 

According to (12.82), 771 must vanish whenever the determinantal row or column vectors become linearly dependent, i.e., when there exists some linear combination Y of the variables Xf+ 2, Xf+ 3,..., Xc+2 such that F(A) = -A) for every phase A. For example, when p = 2, 771 must vanish when values of and c+2 are proportional in the two phases. The vanishing of 77 signals a type of redundancy of the extensive variable Xb as though the system could be prepared from one fewer chemical component than had been supposed. However, the vanishing of an 77 could arise from other special linear relationships connecting entropies or volumes, as well as composition variables. 

Such structuring is necessarily an intermolecular effect. The simplest type of an intermolecular effect, which should be treated first, is due to the crosslinks between the chains themselves. Dobson and Gordon (50) have remarked that most crosslinks are actually short chains of one or several links, which upon straining the network, become oriented but cannot be stretched. As a result an additional entropy force should arise, which has not yet been accounted for in the Gaussian theory. This force can be calculated on the basis of the Kuhn and Grun (114) chain vector orientation argument, which yields in extension... 

The backbone dynamics of 4-oxalocrotonate tautomerase, a 41-kDa homo-hexamer with 62 residues per subunit, and its complex with a substrate analogue have been analyzed by the model-free formalism.60 Binding of the analogue freezes the motion of some of the backbone NH vectors in the active site, leading to a loss of entropy (Chapter 2). 

Schirmer et al. (7.) indicate that the constants and E j may be derived from physical or statistical thermodynamic considerations but do not advise this procedure since theoretical calculations of molecules occluded in zeolites are, at present, at least only approximate, and it is in practice generally more convenient to determine the constants by matching the theoretical equations to experimental isotherms. We have determined the constants in the model by a method of parameter determination using the measured equilibrium data. Defining the entropy constants and energy constants as vectors... 

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