Most probable distribution methods

By application of least-squares methods, most probable values and standard deviations for the parameters Em and a of this distribution have been calculated, and from the former a value for the temperature-independent rate constant 0 has been derived ... 

Compared with chromatographic methods, this countercurrent distribution method is probably less efficient, the apparatus required is very much more complicated, and the labor involved is considerably greater nevertheless, it possesses several definite advantages over chromatographic methods. The chief of these is the absence of any solid phase, which may act as an adsorbent for the solutes. In partition chromatographic methods this adsorption may often lead to distortion of the bands and render fractionation very inefficient. The behavior of a solute on a countercurrent distribution depends only on its partition coefficient, which in most cases is constant, so that it is possible to calculate the exact theoretical distribution curve and this may be used as a very sensitive test for purity. 

While the average state of a system should be determined by an average over all distributions, the most probable distribution is very much more probable than any other. Therefore, it is a good approximation to just consider this distribution. This may be demonstrated using the more rigorous method of steepest descents [1]. To obtain the maximum probability, equation 2A.2 is recast into logarithmic form. Thus... 

One chain-end is typically unsaturated due to chain transfer and termination mechanisms. Mol wts can range from several hundred to several million. There is no long-chain branching unless special synthesis methods ate employed. The mol wt distribution is commonly the most probable,... 

Whereas multi-wall carbon nanotubes require no catalyst for their growth, either by the laser vaporization or carbon arc methods, catalyst species are necessary for the growth of the single-wall nanotubes [156], while two different catalyst species seem to be needed to efficiently synthesize arrays of single wall carbon nanotubes by either the laser vaporization or arc methods. The detailed mechanisms responsible for the growth of carbon nanotubes are not yet well understood. Variations in the most probable diameter and the width of the diameter distribution is sensitively controlled by the composition of the catalyst, the growth temperature and other growth conditions. 

For the usual accurate analytical method, the mean f is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating at random. The standard deviation, s, depends upon these small causes and may assume any value mean and standard deviation are wholly independent, so that an infinite number of distribution curves is conceivable. As we have seen, x-ray emission spectrography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve, not an error curve in the strictest sense there is no true value of N such as that presumably corresponding to a of Section 10.1—there is only a most probable value N. 

Given that the assumption of normally distributed data (see Section 1.2.1) is valid, several useful and uncomplicated methods are available for finding the most probable value and its confidence interval, and for comparing such results. 

If the most-probable orientation of the structural entities is no longer parallel to the fiber axis, we may observe a clearly inclined streak. Such orientations are frequently observed in herbal and animal natural fibers [45,260,261], If the split nature of the orientation distribution is clearly detected, the streak method can be applied or adapted. 

Distributions of structures obtained by fitting the intensity data can be compared to a most probable distribution of the sixteen structures assumming equal a priori probabilities subject to the constraint that the correct Si/Al ratio must be given. A method for calculating the most probable distribution of these structures has been previously reported (7.). 

The results of the various methods used to determine the molecular weight and degree of functionality of the diamine are shown in Table II. All of this evidence is consistent with the conclusion that the purified, isolated diamine possesses high difunctionality. When the diamine was chain extended with terephthaloyl chloride, a polymer product with a symmetrical molecular weight distribution was obtained (D.P. a 10). Inexact stoichiometry is most probably responsible for the limited extent of chain extension. 

The relationship between elimination half-life (h/2) body weight across species results in poor correlation, most probably because of the hybrid nature of this parameter [27]. A better approach may be to estimate volume of distribution and clearance by the most appropriate method and then estimate half-life indirectly from the relationship ... 

The non-equivalence of the statistical and kinetic methods Is given by the fact that the statistical generation Is always a Markovian process yielding a Markovian distribution, e.g. In case of a blfunc-tlonal monomer the most probable or pseudo-most probable distributions. The kinetic generation Is described by deterministic differential equations. Although the Individual addition steps can be Markovian, the resulting distribution can be non-Markovian. An Initiated step polyaddltlon can be taken as an example the distribution Is determined by the memory characterized by the relative rate of the Initiation step ( ). ... 

The maximum entropy method (MEM) is an information-theory-based technique that was first developed in the field of radioastronomy to enhance the information obtained from noisy data (Gull and Daniell 1978). The theory is based on the same equations that are the foundation of statistical thermodynamics. Both the statistical entropy and the information entropy deal with the most probable distribution. In the case of statistical thermodynamics, this is the distribution of the particles over position and momentum space ( phase space ), while in the case of information theory, the distribution of numerical quantities over the ensemble of pixels is considered. 

The method of the most probable distribution that underlies both Boltzmann-type and Gibbs-type formulations is outlined with admirable clarity in Chapter II of Schrodinger s book. 

The important aspect of (13.70b) is that each pa=Pa(U, V, N) has maximal ( most probable ) character with respect to the natural control variables of S. The constrained maximization procedure to find this optimal distribution by the method of Lagrange undetermined multipliers [see Schrodinger (1949), Sidebar 13.4, for further details] is very similar to that described in Section 5.2. In particular, the pa must be maximal with respect to variations in each control variable, leading to the usual second-derivative curvature conditions such as... 

Lin et al. (1971) used inelastic scattering of plane-polarized light of 632.8-nm wavelength from a He-Ne laser to determine the diffusion coefficient and thereby the hydrodynamic radii of monodisperse caseinate micelle fractions from milk. The cumulative distribution curve of the weight fraction of micelles revealed that about 80% of the casein occurs in micelles with radii of 50 to 100 nm and 95% between 40 and 220 nm, with the most probable radius at about 80 nm. This method has the advantage that the micelles are examined in their natural medium. 

On the other hand, the Boltzmann method of calculating the most probable distribution, used in the theoretical model, precludes an explicit consideration of actual values of transition probabilities (rate constants). This would only be possible if plasma reactions are considered as a multi-channel transport problem. However, the knowledge of a large number of various transition probabilities is necessary for such... 

In our paper [133] we have performed calculations of the heats of formation using all three parametrizations (MNDO, AMI, PM3) and both types of the variation wave function (SLG and SCF). Empirical functions of distribution of errors in the heats of formation [141] for the SLG-MNDO and SCF-MNDO methods are remarkably close to the normal one. That means that the errors of these two methods, at least in the considered data set, are random. In the case of the SLG-MNDO method, the systematic error practically disappears for the most probable value of the error... 

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