Distribution Functions and Moments

The moments of the molecular weight distribution are defined as either 

An experimental determination of the molecular weight distribution conceptually sorts the polymer molecules into bins, with one bin for each degree of 

The ratio of weight-to-number average chain lengths is the polydispersity, 

This dimensionless number measures the breadth of the molecular weight distribution. It is 1 for a monodisperse population (e.g., for monomers before reaction) and is 2 for several common polymerization mechanisms. 

An experimental determination of the MWD conceptually sorts the polymer molecules 

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Mutual information is thus a random variable since it is a real valued function defined on the points of an ensemble. Consequently, it has an average, variance, distribution function, and moment generating function. It is important to note that mutual information has been defined only on product ensembles, and only as a function of two events, x and y, which are sample points in the two ensembles of which the product ensemble is formed. Mutual information is sometimes defined as a function of any two events in an ensemble, but in this case it is not a random variable. It should also be noted that the mutual... 

Langenbucher F. Handling of computational in vitro/in vivo correlation problems by Microsoft Excel, part II. Distribution functions and moments. Eur J Pharm Biopharm 2002 53 1—7. 

Experimental investigations of the kinetics of stationary nucleation presuppose the determination of the rate J as a function of temperature and pressure. Information on the nucleation rate may be obtained from data on distribution functions and moments of appearance of the first critical nucleus. It requires repeated experiments with one sample or measurements with a system of equivalent samples. 

The other class of fluctuation phenomena, well-known since the work of Gibbs (done in 1902, see Gibbs (1948)) and Einstein (1910), is equilibrium fluctuation. The theory of equilibrium (thermostatic) fluctuations considers the equilibrium state as a stationary stochastic process (see, for example, Tisza Quay (1963) and Tisza (1966)). By thermostatic fluctuation theory the statistical character (e.g. the distribution functions and moments derived from it) can be computed. 

The Characteristic Function.—The calculation of moments is often quite tedious because of difficulties that may be encountered in evaluating the pertinent integrals or sums. This problem can be simplified quite often by calculation of the so-called characteristic function of the distribution from which, as we shall see, all moments can be derived by means of differentiation. This relationship between the characteristic function and moments is sufficient reason for studying it at this time however, the real significance of the characteristic function will not become apparent until we discuss the central limit theorem in a later section. 

Note 1 An infinite number of molar-mass averages can in principle be defined, but only a few types of averages are directly accessible experimentally. The most important averages are defined by simple moments of the distribution functions and are obtained by methods applied to systems in thermodynamic equilibrium, such as osmometry, light scattering and sedimentation equilibrium. Hydrodynamic methods, as a rule, yield more complex molar-mass averages. 

For any quantity that is a function of time we can describe its properties in terms of its distribution function and the moments of this function. We first define the probability distribution function p(t) as the probability that a molecule entering the reactor wih reside there for a time t. This function must be normalized... 

Early numerical estimates of ternary moments [402] were based on the empirical exp-4 induced dipole model typical of collision-induced absorption in the fundamental band, which we will consider in Chapter 6, and hard-sphere interaction potentials. While the main conclusions are at least qualitatively supported by more detailed calculations, significant quantitative differences are observed that are related to three improvements that have been possible in recent work [296] improved interaction potentials the quantum corrections of the distribution functions and new, accurate induced dipole functions. The force effect is by no means always positive, nor is it always stronger than the cancellation effect. 

Fluid Model of Discharges. An important question is whether it makes sense to attempt to solve for distribution functions or moments in die absence of a commensurate accuracy in the treatment of neutral-species chemistry. As already stated, modeling of the chemically reacting plasma requires solutions to the bulk gas momentum and energy balance equations and continuity equations for each reacting neutral species. Surface chemistry is... 

Next, we define a parallel set of NPD function in continuous flow recirculating systems. We restrict our discussion to steady flow systems. Here, as in the case of RTD, we distinguish between external and internal NPD functions. We define fk and 4 as the fraction of exiting volumetric flow rate and the fraction of material volume, respectively, that have experienced exactly k passages in the specified region of the system. The respective cumulative distribution functions, and /, the means of the distributions, the variances, and the moments of distributions, parallel the definitions given for the batch system. 

According to Eq. (4.251), the quadratic term is determined by the odd-rank moments of the equilibrium distribution function, and is absent if the latter is even in x. Thus, for the existence of x(2) the presence of a bias field in Eq. (4.245) is mandatory. Otherwise, the next-to-linear response term would be cubic in the probing field amplitude see Section III. 

When controlling the PSD during crystallization, some characteristics of the PSD of the product are often the focus of processing rather than exactly matching the full size distribution. These characteristics include moments of the PSD, number-mean crystal size, weight-mean crystal size, variance of the distribution function, and the coefficient of variation. Weight-mean crystal size is the most common PSD characteristic used in practice. For the case where the particles have... 

The moments have physical meaning. The zeroth order moment (iq is the total number of particles per unit volume (or mass, depending on the basis). The first-order moment fii is the total length of the particles per unit volume, with the particles lined up along the characteristic length. The second-order moment is proportional to the total surface area, and the third-order moment is proportional to the total volume. Many physical characteristics of the particles such as the number-mean crystal size, weight-mean crystal size, the variance of the distribution function, and the coefficient of variation also can be represented in terms of the lower order moments of the distribution. 

The weight-average molar mass is the ratio of the second to the first moment of the number fraction distribution function and is equal to the first moment of the weight fraction distribution function ... 

From these distribution functions, their moments related to sol and gel fractions and to various averages of the degree of polymerization [Eqs (6.45) and (6.46)] may be calculated. Results for functionality/= 3 are presented here. The sol fraction is defined as the fraction of all sites belonging to finite molecules... 

Table 7.1. Common daughter distribution functions and their moment transforms.

The mean square end-to-end distance, < r >, is the second moment of the radial distribution function and so is defined by the integral (Young and Lovell, 1990)... 

Poland, D. Time evolution of polymer distribution functions from moment equations and maximum-entropy methods. J. Chem. Phys. 111(17), 8214-8224 (1999)... 

The distribution function, the moments, and the averages can be given in terms of the molar masses rather than in terms of the degrees of polymerization. Theoretically, a description in terms of the degrees of polymerization is more desirable, but molar masses are obtained experimentally. 

How can this formal treatment of the distribution function (and resulting order parameters) be generalized to include the smectic-A structure We find the clue in Kirkwood s treatment of the melting of crystalline solids. In a crystal the density distribution function (the translational molecular distribution function) is periodic in three dimensions and can be expanded in a three-dimensional Fourier series. Kirkwood does this and then identifies the order parameters of the crystalline phase as the coefficients in the Fourier series. For simplicity let us consider a one-dimensionally periodic structure (such as the smectic-A but with the orientational order suppressed for the moment). The distribution function, which describes the tendency of the centers of mass of molecules to lie in layers perpendicular to the z-direction, can be expanded in a Fourier series ... 

The most probable value of r, occurring at the maximum in W(r), is not zero and is found to be equal to 1/j . Figure 8 shows a plot of the radial distribution function W(r) for a six-link random chain with / = 1 in arbitrary units. The mean-square end-to-end distance is the second moment of the radial distribution function, and can thus be calculated from equation (18), which yields the results given in equations (8) and (19). 

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