Sum Of Entropies Research Articles

Modification of adipic acid crystals. II. Influence of growth in the presence of oleic acid on crystal properties

Adipic acid (AA) was crystallized under defined conditions from water containing 0–110 μmol · dm −3 oleic acid (OA). With increasing concentration of OA, the water content of the crystals was constant (0.047 ± 0.005 mole fraction), while the uptake of OA increased linearly, and the dissolution rate (DR) and specific surface area (SSA) of a defined sieve fraction decreased. The decrease in DR exceeded the decrease in SSA. Washing the crystals with chloroform removed an appreciable proportion of the poorly water-soluble OA from the surface of the crystals and produced a marked increase in DR. OA at 15 μmol · dm −3 caused a doubling of DR, whereas higher concentrations reduced DR of the washed crystals to the original value. Increasing incorporation of OA into the AA crystals reduced their density, indicating a decrease in crystallinity. Increasing incorporation of OA also reduced the melting point, T m, the enthalpy of fusion, ΔH f, and the enthalpy of solution, ΔH s, of the crystals, indicating an increase in energy of the crystalline bulk, corresponding to an increase of lattice strain. Higher concentrations of OA reversed the energetic effects. The reduction in the entropy of fusion by small amounts of OA was greater than the increase in the sum of the ideal partial molar entropies of the components of the crystals by a factor approaching 1000. This indicates that OA greatly disrupts the order of the crystal lattice of AA.

Analysis of atomic electron momentum densities: Use of information entropies in coordinate and momentum space

The entropy maximization procedure has been extended to treat simultaneously the densities in coordinate and momentum space. The key quantity to be maximized is the sum of information entropies in complementary spaces rather than the entropy in one space alone. This modified procedure has been used to assess the quality of refined electron momentum densities for He, Be and H 2. The momentum density which maximizes the entropy sum yields good estimates of Compton lineshape and related momentum space expectation values.

Information processing in linear vector space

Strings of information units are written as naive matrices A and linear algebra is used to find quadratical forms A T A and AA T . These are interpreted as position vectors in n-and m -dimensional Euclidean space. Binary information relationships are treated as graphs and bipartite graphs with matrices (A + B) or (A|B). Their quadratical forms G T G and GG T are found and interpreted. Information matrices A and G have twofold symmetry connected with permutations of their rows and columns. The logarithmic measure of this symmetry is entropy. Its limit is the sum of Boltzmann and Shannon entropies.

Solid–solid phase transitions in azoxy compounds

Solid–solid phase transitions in diazabicyclo-octene N-oxide, diazatricyclononene N-oxide, 1,1-dimethylazoxyethane, 1,1-dimethylazoethane and di-t-butylvinylene have been investigated by differential scanning calorimetry. The entropies of transition were 34.2, 42.4, 31.1, 20.2 and 22.9 J K–1 mol–1, respectively. Diazabicyclo-octene N-oxide and di-t-butylvinylene have two solid-state transitions. The entropies given are the sum of the two entropy increments. The two cyclic compounds have small fusion entropies, indicating that they form plastic crystals below the melting point. The entropies for the cyclic compounds are compared with those for bicyclic hydrocarbons of similar structure. The entropies of fusion for the three di-t-butyl compounds are large, with the sum of entropies of fusion and transition in the range 60–70 J K–1 mol–1.

Interaction information in multivariate probability distributions

Abstract : It is shown in this note that the entropy of a multivariate distribution can be expressed in terms of the sum of onedimensional marginal entropies, the sum of transmitted information between each pair of component variables, the sum of interaction information in trivariate component distributions, and so on. Using this result, the author gives a class of multivariate distributions having specified component densities and some preassigned association measure between some component variables. Proofs of equations and statements which are not so evident are also given.

Entropy and the uncertainty principle

A minimum principle is obtained for the sum of entropies of two distributions related as the absolute squares of a Fourier transform pair. The minimum is shown to be attained for a Gaussian pair. The joint entropy is calculated for two other Fourier pairs of interest. Applications to the uncertainty principle are made by defining a joint entropy for position and momentum. A generalized uncertainty principle, for any set of observables not simultaneously measurable, is conjectured.