Distribution function Terms Links

But any complete description of the evolution of perturbations in the universe will link all of these terms initial velocity and density perturbations to the various components (baryons, dark matter, photons) evolve prior to last scattering as discussed above, and so photon overdensities occur in potential wells, and velocity perturbations occur in response to gravitational and pressure forces. Indeed, to solve this problem in its most general form, we must resort to the Boltzmann equation. The Boltzmann equation gives the evolution of the distribution function, fi(xp,Pp) for a particle of species i with position Xp, and momentum p/(. In its most general form, the Boltzmann equation is formally... 

Sharaf, M. A. Mark, J. E., Elastomers Having High Functionality Cross Links Viewed as Bimodal Networks. An Alternative Interpretation in Terms of Bi-modal Chain-Length Distributions. J. Polym. Set, Polym. Phys. Ed. 1995, 33, 1151-1165. 

The multimedia model present in the 2 FUN tool was developed based on an extensive comparison and evaluation of some of the previously discussed multimedia models, such as CalTOX, Simplebox, XtraFOOD, etc. The multimedia model comprises several environmental modules, i.e. air, fresh water, soil/ground water, several crops and animal (cow and milk). It is used to simulate chemical distribution in the environmental modules, taking into account the manifold links between them. The PBPK models were developed to simulate the body burden of toxic chemicals throughout the entire human lifespan, integrating the evolution of the physiology and anatomy from childhood to advanced age. That model is based on a detailed description of the body anatomy and includes a substantial number of tissue compartments to enable detailed analysis of toxicokinetics for diverse chemicals that induce multiple effects in different target tissues. The key input parameters used in both models were given in the form of probability density function (PDF) to allow for the exhaustive probabilistic analysis and sensitivity analysis in terms of simulation outcomes [71]. 

Another aspect of matching output to user needs involves presentation of results in a statistical framework—namely, as frequency distributions of concentrations. The output of deterministic models is not directly suited to this task, because it provides a single sample point for each run. Analytic linkages can be made between observed frequency distributions and computed model results. The model output for a particular set of meteorologic conditions can be on the frequency distribution of each station for which observations are available in sufficient sample size. If the model is validated for several different points on the frequency distribution based on today s estimated emission, it can be used to fit a distribution for cases of forecast emission. The fit can be made by relating characteristics of the distribution with a specific set of model predictions. For example, the distribution could be assumed to be log-normal, with a mean and standard deviation each determined by its own function of output concentrations computed for a standardized set of meteorologic conditions. This, in turn, can be linked to some effect on people or property that is defined in terms of the predicted concentration statistics. The diagram below illustrates this process ... 

The electron density is a continuous function that is experimentally observable, hence uniquely defined, at all points in space. Its topology can be described in terms of the distribution of its critical points, i.e. the points at which the electron density has a zero gradient in all directions. There are four kinds of critical point which include maxima (A) usually found near the centres of atoms, and minima (D) found in the cavities or cages that lie between the atoms. In addition there are two types of saddle point. The first (B) represents a saddle point that is a maximum in two directions and a minimum in the third, the second (C) represents a saddle point that is a minimum in two direction and a maximum in the third. One can draw lines of steepest descent connecting the maxima (A) to the minima (D), lines whose direction indicates the direction in which the electron density falls off most rapidly. Of the infinite number of lines of steepest descent that can be drawn there exists a unique set that has the property that, in passing from the maximum to the minimum, each line passes successively through a B and a C critical point. This set forms a network whose nodes are the critical points and whose links are the lines of steepest descent connecting them. 

The dual site-bond description (DD) of disordered stmctures [3] allows a proper modeling of the porous structure. In the context of this treatment, two kinds of alternately intercormected void entities are thought to conform the porous network, i.e. the sites (cavities) and the bonds (capillaries, necks). C bonds meet into a site and each bond is the link between two sites. Thus a twofold distribution of sites and bonds is required to construct a porous network. For simplicity, the size of each entity can be measured in terms of a quantity, R, defined as follows for sites, considered as hollow spheres, R is the radius of the sphere while for bonds, idealized as hollow cylinders open at both ends, R is the radius of the cylinder. Under the DD scheme, FgfR) andFg(R)are the size distribution density functions, for sites and bonds respectively, on a number of elements basis and normalized so that the probabilities to find a site or a bond having a size R or smaller are ... 

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