One-body distribution function

In the same approximation the one-body distribution function reads as... 

Equation 68 is referred to as the high-temperature approximation because the potential appears in the integral as the product 0 V. Thus, the approximation is valid for small perturbations or high temperatures. Higher-order terms can be included, in principle, to obtain more accuracy. However, the derivatives of the distribution functions with respect to X involve higher-order distribution functions 170 e.g., the first-order correction in X to the distribution function involves three- and four-body distribution functions which are usually difficult to obtain. In some cases, the superposition approximation or other approximate expressions for the higher-order distributions have been introduced.175 However, the first-order result is the one that has been employed in most applications.176... 

If we look at the behavior of any one particular molecule in a system of molecules, its dynamics is only influenced by nearby molecules, or its nearest neighbors, provided that the interaction forces decay rapidly with intermolec-ular separation distance (the usual case). Thus, it is not always necessary to know the full N -body distribution function, and a (much) lower-order distribution function involving only the nearest neighbors will often suffice. Although we lose information by such a contraction, it is information that, for all intents and purposes, has no bearing on the problem. The analogy with our manufacturing example would be that the sprocket mass does not affect its performance and is, therefore, information that is not needed. Ultimately, it s the physics of the problem that dictates what information is important or not important. 

Many distribution functions can be apphed to strength data of ceramics but the function that has been most widely apphed is the WeibuU function, which is based on the concept of failure at the weakest link in a body under simple tension. A normal distribution is inappropriate for ceramic strengths because extreme values of the flaw distribution, not the central tendency of the flaw distribution, determine the strength. One implication of WeibuU statistics is that large bodies are weaker than small bodies because the number of flaws a body contains is proportional to its volume. 

After this computer experiment, a great number of papers followed. Some of them attempted to simulate with the ab-initio data the properties of the ion in solution at room temperature [76,77], others [78] attempted to determine, via Monte Carlo simulations, the free energy, enthalpy and entropy for the reaction (24). The discrepancy between experimental and simulated data was rationalized in terms of the inadequacy of a two-body potential to represent correctly the n-body system. In addition, the radial distribution function for the Li+(H20)6 cluster showed [78] only one maximum, pointing out that the six water molecules are in the first hydration shell of the ion. The Monte Carlo simulation [77] for the system Li+(H20)2oo predicted five water molecules in the first hydration shell. A subsequent MD simulation [79] of a system composed of one Li+ ion and 343 water molecules at T=298 K, with periodic boundary conditions, yielded... 

Although Vj does not appear explicitly in Eqs. (3.32) and (3.34), the interaction influences the driving potential and the vector potential via in Eq. (3.29). In general, there is no one-body potential and vector potential that satisfies Eqs. (3.32) and (3.34), hence the driving potential in Eq. (3.32) is unrealistic. However, it turns out that the driving potential and the vector potential for rotation of the orientation and translation of the wave function distribution of one-component gas of charged particles without spin are the same as those for the one-particle system in Section 3.5.3. 

The change of the basic energy functional arises from the nonlinear nature of the effective Hamiltonian. This Hamiltonian has in fact an explicit dependence on the charge distribution of the solute, expressed in terms of (fM, which is the one-body contraction of I hf) ( hf > and lllus it is nonlinear. It must be added that this nonlinearity is of the first order, in the sense that the interaction operator depends only on the first power of ffM. 

Once a drug enters the body, from whatever route of administration, it has the potential to distribute into any one of three functionally distinct compartments of body water, or to become sequestered in some cellular site. 

With the above, a formal set of equations Is given, the elaboration of which requiring a solution for the problem that the recurrent relationships p p - p p, . .. diverge. Relatively simple densities, or distribution functions, are converted into more complex ones. A "closure" is needed to "stop this explosion". A number of such closures have been proposed, all involving an assumption of which the rigour has to be tested. Most of these write three-body interactions in terms of three two-body Interactions, weighted in some way. A well known example is Kirkwood s superposition closure, which reads ... 

Structures of liquids in general are dominated by influences of intermolecular repulsions. Intermolecular attractions have a comparatively minor effect on the radial distribution function and, in the case of asymmetric molecules, on intermolecular correlations as well. At the high density and close packing prevailing in the liquid state, the spatial arrangement of the molecules of a liquid can be satisfactorily described, therefore, by representing the molecules as hard bodies of appropriate size and shape whose only interactions are the excessive repulsions that would be incurred if one of them should overlap another. Once the liquid structure has been characterized satisfactorily on this basis, one may take account of the intermolecular attractions by averaging them over the molecular distribution thus determined. Mean-field theories are useful in this connection. 

In investigations of the structure and properties of disperse materials, particularly plastic foams, it is necessary to find out the distribution pattern of one phase in the material. It is simpler to consider the distribution of the gas phase in a solid body, i.e. the gas-filled cell distribution within a foam bulk] In their turn, the cells, as shown above, may be characterized by several parameters (size, shape, volume and surface area). The cell size distribution pattern is the most comprehensive characteristic of the dispersity of the gas structural elements of plastics. Furthermore, the cell size can, be determined by one of the methods which will be discussed below, and the foam dispersity is expressed in terms of the nominal cell diameter distribution function. 

Chatzis and Dullien (1977) were the first to use tubular bonds and spherical sites to simulate pore throats and bodies respectively, in network models of porous media. Previously, intersections (sites) were assumed not to have any volume. In their model, individual elements are represented as cylindrical tubes with spherical indentations in the middle (Fig. 3-17C). Bond lengths and bond and site radii can be drawn from independent distribution functions, or as is more common, correlated with each other so that only one distribution function is required (Ioanni-dis Chatzis, 1993). 

It is possible to show [45-47] that there is only one external potential V(r) that gives rise to a given equilibrium one-body density profile p(r) and to the N-body distribution firN), Since f(rN) is a functional of p(r), we can take Q[f] and A[f] to be functionals of p(r), the singlet density distribution. Thus, using p instead of f as the argument in the functionals in eq. (35), and since Q is a minimum at equilibrium,... 

An additional issue in the development of the density functional theory is the parameterization of the trial function for the one-body density. Early applications followed the Kirkwood-Monroe [17,18] idea of using a Fourier expansion [115-117,133]. More recent work has used a Gaussian distribution centered about each lattice site [122]. It is believed that the latter approach removes questions about the influence of truncating the Fourier expansion upon the DFT results, although departures from Gaussian shape in the one-body density can also be important as has been demonstrated in computer simulations [134,135]. 

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