Distribution functions derivation

In the distribution function derived here, the dimensionless size p (=L/E) is introduced as y=jpP and L is unmeasurable and is difficult to estimate as mentioned... 

Both the osmotic pressure and the high-frequency shear modulus can be calculated analytically from the radial distribution function derived above. The osmotic pressure n is related to the radial distribution function and the energy potential... 

The molecules and infrared spectra selected for training have a profound influence on the radial distribution function derived from the CPG network and on the quality of 3D structure derivation. Training data are typically selected dynamically that is, each query spectrum selects its own set of training data by searching the most similar infrared spectra, or most similar input vector. Two similarity measures for infrared spectra are useful ... 

Stein (67) has used the basic theory of Nishijima for developing equations for predicting the values of various ratios of the fluorescence intensities of differing polarization expected from a stretched ideal rubber. Stein s theoretical efforts made use of Roe and Krigbaum s (55) expression for the complete orientation distribution function derived from the Kuhn and Griin (58) statistical segment model. Stein s equations have yet to be verified by experiment. 

Fig. 2.3.5. Orientational distribution functiony(cos ) as defined by (2.3.22) in the nematic phase of MBBA. Circles represent values from Raman measurements and the line gives the distribution function derived from a two-term Maier-Saupe potential in which the parameters are adjusted to give a good fit with the observed

Local order characterized by pair distribution functions (derived from electron scattering traces),... 

Another consequence of the treatment of dilute solute molecules as statistical particles is the conclusion that the equilibrium concentration of solute particles in a gravitational field should vary with the height in the solution. The distribution function derived by Einstein is ... 

In a subsequent analysis, Srolovitz et al. (22) showed that when the grain size is normalized to the average grain size, then the size distribution function becomes time invariant, as also predicted by the mean field theories. A comparison of the distribution function obtained from the computer simulations with the lognormal distribution, and the distribution functions derived by Hillert and that by Louat are shown in Fig. 9.13. As shown in Fig. 9.18, the distribution function can also provide an excellent fit to some experimental data. 

Figure 1.2 gives the radial distribution function derived from Fig. 1.1. This figure also shows correspondence between the principal peaks and the radial density for the solid crystal. Because of repulsive interactions between atoms, p(r) has zero magnitude out to a radius nearly equal to the atomic radius. At large radii, p r) is equal to ihe bulk density of the liquid. The mean co-ordination number for ihe liquid can be determined from this curve by integration. Table 1.4 gives some comparisons of data obtained from diffraction studies on liquids with the corresponding data for the crystalline solids. In general it is found that...

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