Distribution function, theoretical calculation

In the next two figures we discuss the pair-correlation functions as obtained from the two-density theory and computer simulations. First, in Fig. 3 we compare the counterion-counterion pair-distribution function as obtained theoretically (lines) and from simulations (symbols). The numerical calculations apply to cp = 0.0001 and ce = 0.005 mol dm-3 the results show that the theory underestimates the counterion-counterion correlation. Next, in Fig. 4 the macroion-counterion pair-distribution is shown for the same set of parameters. Finally, in Fig. 5 the macroion-macroion pair-distribution functions are calculated by both theoretical approaches at cp = 0.0001 mol dm-3 solution and for zp = —10 and —30. 

The general structure of the theoretical tool of ST should now be quite clear. For each set of independent variables we define a partition function. This partition function is related to a thermodynamic quantity through one of the fundamental relationships. On the other hand, each of the summands in the PF is proportional to the probability of realizing the specific value of the variable on which the summation is carried out. Having the probability distribution, for each set of independent variables, one can write down various averages over that distribution function. The calculation of such averages consists of the main outcome of ST. 

As described before nine existing identical machines have been analysed. The results consist of the safety-related empirical and theoretical failure probability of the safety-related function. Further, the failure density function of the theoretical distribution function is calculated and illustrated to clarify how the maximum of the failure density function is obtained. The received maximum values can finally be compared with the quantitative requirements of the EN ISO 13849-1 and the Performance Level can be determined. 

Theoretical results of similar quality have been obtained for thermodynamics and the structure of adsorbed fluid in matrices with m = M = 8, see Figs. 8 and 9, respectively. However, at a high matrix density = 0.273) we observe that the fluid structure, in spite of qualitatively similar behavior to simulations, is described inaccurately (Fig. 10(a)). On the other hand, the fluid-matrix correlations from the theory agree better with simulations in the case m = M = S (Fig. 10(b)). Very similar conclusions have been obtained in the case of matrices made of 16 hard sphere beads. As an example, we present the distribution functions from the theory and simulation in Fig. 11. It is worth mentioning that the fluid density obtained via GCMC simulations has been used as an input for all theoretical calculations. 

While thin polymer films may be very smooth and homogeneous, the chain conformation may be largely distorted due to the influence of the interfaces. Since the size of the polymer molecules is comparable to the film thickness those effects may play a significant role with ultra-thin polymer films. Several recent theoretical treatments are available [136-144,127,128] based on Monte Carlo [137-141,127, 128], molecular dynamics [142], variable density [143], cooperative motion [144], and bond fluctuation [136] model calculations. The distortion of the chain conformation near the interface, the segment orientation distribution, end distribution etc. are calculated as a function of film thickness and distance from the surface. In the limit of two-dimensional systems chains segregate and specific power laws are predicted [136, 137]. In 2D-blends of polymers a particular microdomain morphology may be expected [139]. Experiments on polymers in this area are presently, however, not available on a molecular level. Indications of order on an... 

The probability distribution functions of pairs and singletons were calculated from Z according to theoretical expressions and taking into account Eqs. 5 and 6 ... 

A straightforward Fourier transform of the EXAFS signal does not yield the true radial distribution function. First, the phase shift causes each coordination shell to peak at the incorrect distance second, due to the element-specific backscattering amplitude, the intensity may not be correct. The appropriate corrections can be made, however, when phase shift and amplitude functions are derived from reference samples or from theoretical calculations. The phase- and amplitude-corrected Fourier transform becomes ... 

FIG. 10. Theoretical calculations reveal that in the case of adsorption of Xe on Ni the resonance associated with Xe(6s) state is broadened significantly with a long tail that extends to the Ni Fermi level. STM images are determined by the LDOS at the Fermi level. Although the contribution of Xe to the LDOS is small, it significantly extends the spatial distribution of the electronic wave function further away from the surface thereby acting as the central channel for quantum transmission to the probe tip. (From Ref. 71.)... 

Table 4.2 Two-dimensional displacement distribution function W on W 110 heating periods were 60 s each experimental values were averaged over equivalent directions. Theoretical values calculated from the experimental mean square displacments are listed in... 

The present theoretical approach to rubberlike elasticity is novel in that it utilizes the wealth of information which RiS theory provides on the spatial configurations of chain molecules. Specifically, Monte Carlo calculations based on the RIS approximation are used to simulate spatial configurations, and thus distribution functions for end-to-end separation r of the chains. Results are presented for polyethylene and polydimethylsiloxane chains most of which are quite short, in order to elucidate non-Gaussian effects due to limited chain extensibility. 

The data, except for the 0-0.1 /x fraction, fall on a straight line. However, the value of f calculated from the intercept is larger by about a factor of 100 than the value of r calculated from the slope. On the basis of this limited trial, the fit of the data to this form of distribution function appears to be quite unsatisfactory. A correct form of distribution function should apply to the entire class of airburst populations, and additional work now underway is devoted largely to resolving the problem of determining an appropriate form of distribution function to apply to airburst particle size distributions. However, there may be no simple function which reflects adequately the over-all behavior of the particle population. Indeed, Johnson (5) was able to demonstrate that his experimental results on isotope distribution with particle size were compatible with theoretical distributions obtained by following a modified version... 

Fractionation Data and Distribution Analysis of HEC After One Day of Cellulase Attack. In Table III, the fractionation data of HEC are given after one day of enzymic hydrolysis. As after one hour of enzymic hydrolysis, no theoretical distribution function accorded well with the fractionation data, but we evaluated the parameters by numerical analysis, using the Gauss-Laguerre method (43,44). This method has one advantage over other numerical methods, e.g., (45)—all the calculations involved can be done manually without the need of high-speed computers. 

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