Statistical distribution function

A statistical distribution function is required in order to determine the effective properties of a mixture through the volume fractions and properties of its individual components. Different effective material properties may correspond to different forms of statistical distribution functions. Even one certain effective property may be not only dominated by these two parts (volume fractions and properties of its individual components), but also depend on the geometric characteristics and morphology associated with the material mixture. This is understandable, because the possible ways in which two materials may be mixed together in specified volume proportions is infinite, the resulting properties of the mixture would not be always identical. Because of the difficulties in the theoretical and experimental determination of the geometric and morphology effects on the effective properties 

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Weibull, W. 1951 A Statistical Distribution Function of Wide Applicability. Journal of Applied Mechanics, 73, 293-297. 

Weibull, W. A Statistical Distribution Function of Wide Application. 7. Appl. Mech., Vol. 18, 1951, pp. 293. 

The results of the three-dimensional random walk, based on the freely-jointed chain, has permitted the derivation of the equilibrium statistical distribution function of the end-to-end vector of the chain (the underscript eq denotes the equilibrium configuration) [24] ... 

Additional software has been developed to merge data from various data collection steps and to model the data using suitable statistical distribution functions. We are working on software to perform corrections for absorption, specimen shape, and misalignment. Library routines for 2-diraensional data smoothing and integration are being adapted to the calculation of orientation functions and other moments of the probability distributions. 

The uncertainties in the model inputs were elaborated using the statistic distribution functions for the initial parameters and also the Monte Carlo simulation. 

TFL.5. R. L. Anderson, R. Herman, and 1. Prigogine, On the statistical distribution function theory... 

Weibull. W. (1951). A statistical distribution function of wide applicability. J. Appl. Mech. 18, 293-297. Wells, J.K. and Beaumont, P.W.R. (1985). Debonding and pull-out processes in fibrous composites. J. Mater. Sci. 20, 1275-1284. 

Weibull, W., A statistical distribution function of wide applicability , J. Appl. Mech., 18, 293-297 (1951). 

Clearly, the physical reason for the indeterminacy relations of Eqs. (12) and (8) is due to the statistical character of the drift velocity vd as defined by Eq. (2) through the statistical distribution function n. Thereby, vd becomes complementary to q. 

The Fermi-Dirac and Maxwell-Boltzmann statistical distribution functions are widely used in semiconductor physics, with the latter commonly used as an approximation to the former. The point of this problem is to make you familiar with these distribution functions their forms, their temperature dependencies, and under what conditions they become interchangeable. Throughout this problem, use the energy of silicon s valence band (Evb) as the zero of your energy scale. 

In the Poisson and binomial distributions, the mean and variance are not independent quantities, and in the Poisson distribution they are equal. This is not an appropriate description of most measurements or observations, where the variance depends on the type of experiment. For example, a series of repeated weighings of an object will give an average value, but the spread of the observed values will depend on the quality and precision of the balance used. In other words, the mean and variance are independent quantities, and different two parameter statistical distribution functions are needed to describe these situations. The most celebrated such function is the Gaussian, or normal, distribution ... 

Statistical Molecular Distribution Functions in a Static and in an Alter-natii Electric Field.— We now shall apply the statistical distribution function expansion (100) to a molecular gas (or dilute solution of polar molecules in a non-polar solvent) immersed in an external electric field E. Not taking into consideration the mutual correlations of molecules but solely their interaction with the field E, we are justified by equation (81) in writing the potential energy to within the square of the field as follows ... 

Electric Saturation of the Kerr Effect. When calculating the dectric reorientation function of molecules (172) for molecular gases, it was justifiable to use the approximation (169a) for the perturbed statistical distribution function. In the general case, calcidations have to be carried out with a distribution function of the form (169) where, for axially... 

In some cases the external potential energy Up is small compared with the internal energy U and thermal energy kT, and the (perturbed) statistical distribution function... 

By insertion of the statistical distribution function expansion (100) into the perturbed statistical average (98) and with regard to the definition (129) of a fluctuation, we obtain to within the third approximation incluavely ... 

On inserting the energy (175) into the e Q)ansion of the perturbed statistical distribution function (100), we obtain to within the square of the electric field = zE, in place of (169a),... 

At moderate non-linearities of the medium, the statistical distribution function can be expanded in a series [cf. formulae (122) and (169a)] ... 

In the case now under consideration, the statistical distribution function is of the form ... 

Therefore molecular dynamics is a deterministic technique given an initial set of positions and velocities, the subsequent time evolution is in principle completely determined. The computer calculates a trajectory in a 6A-dimensional phase space (3A positions and 3A momenta). However, such trajectory is usually not particularly relevant by itself. Molecular dynamics is a statistical mechanics method. Like Monte Carlo, it is a way to obtain a set of configurations distributed according to some statistical distribution function, or statistical ensemble. 

E.E.Nikitin and L.Yu.Rusin, Statistical distribution functions of products of exoergic reactions, Khimiya Vysokhikh Energii 2, 124 (1975)... 

The classical Danckwerts surface-renewal model is analogous to the penetration theory. The improvement is in the view of the eddy replacement process. Instead of Higbies assumption that all elements have the same recidence time at the interface, Danckwerts [29] proposed to use an averaged exposure time determined from a postulated time distribution. The recidence time distribution of the surface elements is described by a statistical distribution function E(t), defined so that E(t)d,t is the fraction of the interface elements with age between t and t + dt. The rest of the formulation procedure is similar to that of the penetration model. 

Our starting point is a restatement of the energy as a function of atomic positions, now stated in terms of the type of statistical distribution functions introduced in chap. 3. The purpose of the first part of our discussion is to establish a means for evaluating the quantity (H — Hq)). For simplicity, imagine that the potential energy of interaction between the various atoms is characterized by a pair potential of the form y( r — r ). Our claim is that the average value of the energy of... 

The. second procedure makes use of a. statistical distribution function, the Wcibull distribution. Breakdown voltages are often statistically distributed according to the function. 

Where (s) = is infinitesimal strength statistical distribution function, taking the form of Weibull distribution, namely ... 

Reduced wave packets are statistical distribution functions taken to be real. => Observations transform systems in the instance that wave packets are reduced. Therefore, this argument is meaningless. 

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