Types of Distribution Functions

Distribution functions are mostly named after the person who discovered them. Only the mathematical consequences of the distribution functions will be discussed in this chapter. The application to mechanisms or molar mass averages will be discussed in Chapters 16-23. 

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The larger the value of n, the more uniform is the size distribution. Other types of distribution functions can be found in Reference 1. Distribution functions based on two parameters sometimes do not accurately match the actual distributions. In these cases a high order polynomial fit, using multiple parameters, must be considered to obtain a better representation of the raw data. 

In order to obtain properly averaged results, either collision parameters for each trajectory must be selected by Monte Carlo methods or, when starting values are systematically chosen, the final results must be integrated over complete statistical distributions. The purpose of a Monte Carlo selection technique is to ensure that the distributions of each parameter within a sample of trajectories approach the true statistical distributions as the size of the sample grows. Some examples of how this can be done for different types of distribution function will be described below. Before starting the integration, it is generally necessary to transform the selected values of the collisional... 

It must be kept in mind, that S represents just a first-order approximation of the distribution function, and this under the additional premise of complete cylindrical symmetry only. It might be an acceptable measure when comparing cases for which a mean-field model applies. However, comparing the order parameters of liquid crystals with those of other partially ordered phases, such as stretched polymers or tribological samples can be misleading due to possibly different types of distribution functions. 

For low polydispersities, the parameter cra corresponds to the relative standard deviation. The choice of the type of distribution function is of minor importance [78,79]. The intraparticular scattering function of spheres with the radius a is given as [42]... 

Two types of distribution functions have been discussed so far the spatial distribution of molecules in a gravitational field, the Boltzmann distribution, and the speed distribution in a gas (the Maxwell distribution). These can be written in a combined form, the Maxwell-Boltzmann distribution law. 

In the present author s view, the Gaussian distribution function based on the solvent fluctuation model, which is developed for a simple redox couple, is used too often even when the basic assumption is not valid. For example, this type of distribution function is often drawn for the hydrogen evolution reaction where the oxidized state is H+ and reduced state is H2.105 Certainly the nature of the solvation is completely different between H+ and H2. Moreover, when one considers the kinetics of the hydrogen evolution reaction, one should consider not the energy level of H+/H2 but that of H+/H(a) as Gurney did. 

The accuracy of the first-order perturbation theory expressions depends on the transport theory formulation and on the type of distribution functions used. We shall now elaborate on the above statements. 

In synthesizing polymers in vivo and in vitro, molecular homogenous ( monodisperse ) polymers (i.e., those in which every macromolecule has the same molar mass or molecular weight ) occur only under quite specific conditions. The overwhelming majority of polymer syntheses proceed more or less randomly, and the resulting macromolecular substances have more or less broad molar mass distributions. The kind of molar mass distribution obtained depends on the nature of the polymerization, which may be either thermodynamically or kinetically controlled. Each kind of distribution is characterized by a definite relationship between the mole fraction x and the degree of polymerization X. Consequently, it is possible in many cases to deduce the kind of polymerization involved from the type of distribution function obtained. 

Consequently, when stating a type of distribution function, the property on which it is based, i.e., amount of substance, mass, etc., should also be given. 

For molecular fluids, it is convenient to define different types of distribution functions, correlation functions and related quantities. In particular, in the pair-wise additive theory of homogeneous fluids (see eq. [8.74]), a central role is played by the angular pan-correlation function g(ri2C0i(02) proportional to the probability density of finding two molecules with position rj and rj and orientations (Oi and (Dj (a schematic representation of such function is reported in Figure 8.5). 

Equation (81) suggests that the surface heterogeneity is induced solely by the MPSD. This approach is more fundamentally sound than the approach of energy distribution in Eq. (55), since the distribution of energy sites, F(e), may not follow any particular type of distribution function. On the other hand, the microporous structure of the activated carbon results from the random processes of carbonization and activation and therefore may possibly be characterized by some distribution functions. Hence, assuming a function f r) is more reasonable than arbitrarily imposing a function F e). In the MPSD approach, F e) can be derived from /(r) according to the relationship... 

Alexander et al. [1981] have obtained solutions to this type of equation for various types of distribution functions of W for an initial delta function input in P. In particular, they considered distribution functions for which a mean transition rate (IT) could be defined, and functions which were of the form... 

There are many proposals in the literature for curve fitting and digital differentiation, and the most important ones are listed in Table 3-9. Morrey [119], for example, fitted the digital data of a segment of a spectrum to a quartic equation and differentiated this convolute. A least-squares analysis of this equation gives rise to a set of linear equations. Peak positions were obtained without consideration of the type of distribution function representing the peak. 

The distribution function has no extrema. Although this type of distribution function describes a vast majority of association phenomena such as hydrogen bond chain association and base stacking in several systems it does not apply to micellization or other association phenomena that are characterized by extrema in the distribution function. 

Polymer molecules are chains of covalently bonded monomer units. Some polymers are linear, like beads on a necklace others are branched (see Figure 31.1). Statistical mechanics has a prominent role in predicting the properties of polymers because two types of distribution function are intrinsic to them. First, polymer chains have distributions of conformations. Second, synthetic polymers have distributions of chain lengths. Many of the unique properties of polymeric solutions and solids, such as the elasticity of rubber and the viscoelasticity of slimy liquids, are consequences of the entropies that arise from the conformational freedom of chain moiecules. We begin with the properties of polymer mixtures. 

Analytically speaking, molecular weight distribution of oligomers is usually described by the three types of distribution functions Floiy, Schultz, and Poisson. 

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