Ordering distribution

The higher order distributions are defined analogously. 1 2 contains through the identity... 

Differentiation of Eq. (8) with respect to the position of a molecule gives a hierarchy of integro-differential equations, each of which relates a distribution function to the next higher order distribution function. Specifically,... 

Integral equations provide a satisfactory formalism for the study of homogeneous and inhomogeneous fluids. If the usual OZ equation is used, the best results are obtained from semiempirical closures such as the MV and DHH closures. However, this empirical element can be avoided by using integral equations that involve higher-order distribution functions, but at the cost of some computational complexity. 

Probability. Some slates are much more probable than others. If you shake red and black marbles with each other, ihe random distribution at the left is much more probable than the highly ordered distribution at the right. 

J.E. Bjorkholm, C.V. Shank, Higher-order distributed feedback oscillators, Appl. Phys. Leu. 1972, 20. 306. 

These simple values for the first few coefficients are the result of our choice of number density and temperature as multiplicative factors in the zero-order distribution function, and of defining the expansion variable in terms of the mean velocity and temperature. 

We begin our discussion of random processes with a study of the simplest kind of distribution function. The first-order distribution function Fx of the time function X(t) is the real-valued function of a real-variable defined by6... 

One way of avoiding the difficulty of having to measure, or in some way specify, a complete distribution function is to restrict attention to those theorems or relationships that do not depend on the detailed shape of the distribution function, but rather depend only on certain more easily measured parameters of the distribution function. A convenient and widely used set of such parameters is the momenta of the distribution function. In the case of first order distribution... 

All averages of the form (3-96) can be calculated in terms of a canonical set of averages called joint distribution functions by means of an extension of the theorem of averages proved in Section 3.3. To this end, we shall define the a order distribution function of X for time spacings rx < r2 < < by the equation,... 

We conclude this section by introducing some notation and terminology that are quite useful in discussions involving joint distribution functions. The distribution function F of a random variable associated with time increments fnf m is defined to be the first-order distribution function of the derived time function Z(t) = + fn),... 

A few minutes thought should convince the reader that all our previous results can be couched in the language of families of random variables and their joint distribution functions. Thus, the second-order distribution function FXtx is the same as the joint distribution function of the random variables and 2 defined by... 

As we shall see shortly, all possible finite order distributions of Y(t) are uniquely determined in terms of the known finite order distributions of the increments of N(t). Before proceeding, however, it is perhaps wise to discuss the meaning of the integral in Eq. (3-239) in more detail. 

A comparison of Eq. (3-268) with Eq. (3-208) shows that the finite order distribution pYmtTm is an m-dimensional gaussian distribution60 with the covariance matrix... 

The Gaussian Process.—A gaussian process was defined in the last section to be a process all of whose finite-order distributions are multi-dimensional gaussian distributions. This means that the multi-dimensional characteristic function of Px.fK must be of the form... 

It is now a simple matter to conclude our argument by showing that all finite-order distribution functions of Y(t) are gaussian. To accomplish this, we write... 

Now, our previous result shows that F0(f), being the result of passing X(t) through the linear, time-invariant filter h0(t )> must have a gaussian first-order distribution therefore,... 

Equation (3-325), along with the fact that Y(t) has zero mean and is gaussian, completely specifies Y(t) as a random process. Detailed expressions for the characteristic function of the finite order distributions of Y(t) can be calculated by means of Eq. (3-271). A straightforward, although somewhat tedious, calculation of the characteristic function of the finite-order distributions of the gaussian Markov process defined by Eq. (3-218) now shows that these two processes are in fact identical, thus proving our assertion. 

An ordered distribution of spheres of different sizes always allows a better filling of space the atoms are closer together, and the attractive bonding forces become more effective. As for the structures of other types of compound, we observe the validity of the principle of the most efficient filling of space. A definite order of atoms requires a definite chemical composition. Therefore, metal atoms having different radii preferentially will combine in the solid state with a definite stoichiometric ratio they will form an inter-metallic compound. 

For the structures of M2C and M2N the question arises is there an ordered distribution of occupied and unoccupied octahedral holes There are several possibilities for an ordered distribution, some of which actually occur. For example, in W2C occupied and unoccupied octahedral holes alternate in layers this is the Cdl2 type. In /3-V2N there are alternating layers in which the octahedral holes are one-third and two-thirds occupied. The question of ordered distributions of occupied interstices is the subject of the following sections. 

Compounds which have the NiAs structure often exhibit a certain phase width in that metal atom positions can be vacant. The composition then is M X. The vacancies can have a random or an ordered distribution. In the latter case we have to deal with superstructures of the NiAs type they are known, for example, among iron sulfides such as Fe9S10 and Fe10Sn. If metal atoms are removed from every other layer, we have a continuous series from Mj 0X with the NiAs structure down to M0 5X (= MX2) with the Cdl2 structure phases of this kind are known for Co Te (CoTe NiAs type CoTe2 Cdl2 type). 

The pKa of a molecule, a charge-state-related parameter, is a descriptor of an acid-base equilibrium reaction [34,35]. Lipophilicity, often represented by the octanol-water partition coefficient Kp is a descriptor of a two-phase distribution equilibrium reaction [36]. So is solubility [37-39]. These three parameters are thermodynamic constants. On the other hand, permeability Pe is a rate coefficient, a kinetics parameter, most often posed in a first-order distribution reaction [40-42]. 

The order parameter is directly available from the calculations and the SCF results are given in Figure 17. The absolute values of the order parameter are a strong function of head-group area. Unlike in most SCF models, we do not use this as an input value it comes out as a result of the calculations. As such, it is somewhat of a function of the parameter choice. The qualitative trends of how the order distributes along the contour of the tails are rather more generic, i.e. independent of the exact values of the interaction parameters. The result in Figure 17 is consistent with the simulation results, as well as with the available experimental data. The order drops off to a low value at the very end of the tails. There is a semi-plateau in the order parameter for positions t = 6 — 14,... 

Together with Eq. (66), this equation describes exactly the linear response of the system to an external field, with arbitrary initial conditions. Its physical meaning is very simple and may be explained precisely as for Eq. (66) 32 the evolution of the velocity distribution results in two effects (1) the dissipative collisions between the particles which are described by the same non-Markoffian collision operator G0o(T) 35 1 the field-free case and (2) the acceleration of the particles due to the external field. As we are interested in a linear theory, this acceleration only affects the zeroth-order distribution function It is... 

By the mid-nineteenth century, approximately sixty-five elements were known When Graham planned a chemistry course in the mid-nineteenth century, he divided the elements into "groups or natural families," based on their properties he divided the metals into nine "orders" distributed among three "classes" (alkalis and alkali-earths metals of earths and metals proper, divided according to affinity for oxygen).49 The classes of elements are not abruptly separated, he stated, but shade "into each other in their characters, like the classes created by the naturalists for the objects of the organic world. "50... 

Figure 2.3. Schemes of ordered substitutions, (a) Model of cluster formation, (b) Ordered distribution of the two types of atoms.

As a footnote to these observations, we also have to mention that frequently structural distortions (axial ratio and/or inter-axial angle variations) accompany the formation of derivative structures, especially because of the ordered distribution of atoms of different sizes or of vacant sites. 

In the preceding paragraphs examples of a number of so-called superstructures have been considered. Generally, it has been observed that a derivative structure has fewer symmetry operations than the reference structure it has either a larger cell or a lower symmetry (or both) than the reference structure. Typically the passage from the reference structure to the derivative structure (superstructure) may be related to the fact that a set of equipoints of a certain structure (the reference one) has to be subdivided into two (or more) subsets in order to obtain the description of the other structure. The structure of the Cu type (cF4 type), for instance, corresponds to 4 Cu atoms in the unit cell, placed in 0, 0, 0 14, 14, 0 14, 0, 14 0, 14, 14, whereas in the cP4-AuCu3 type structure the same atomic sites are subdivided, in another space group, into two sets with an ordered distribution of the two atomic species (1 Au atom in 0, 0, 0 and 3 Cu atoms in 14, 14, 0 14, 0,14 0,14,14). 

Notice that not in all the compilations and not by all the researchers are such structural variants considered as separate structure types. It seems however useful to speak of, say, the cI58-Ti5Re24 type structure when the elements are orderly distributed in selected sites of the Mn type, whereas it could really be more appropriate to speak of the cI58 a -Mn type even in the case of a binary combination (a solid solution) in which all the sites of this structure are occupied, at random in the same proportion, by the different components. 

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