Order distribution function

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. 

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... 

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... 

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... 

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... 

As it was already written above, we would like to study structural changes in the charge distribution between macroscopic objects, that is caused by the image forces, and depends on the wall-to-wall distance. To obtain direct structural information about the system, we will introduce a configurational analogue of the phase-space distribution function. At equilibrium, the definition of an fth order distribution function given by Eq. (12) can be applied to the equilibrium probability density [Eq. (13)], and the integration with respect to impulses can easily be carried out. We write for the rth order local density... 

It was indicated that statistical methods are the most promising. The notion of bulk liquid structure can be made quantitative in terms of radial distribution functions g(r) (for simple liquid), direction-dependent distribution functions g r,0] or higher order distribution functions r, ...) (for more complicated... 

Higher order correlation functions 5 (r,.r2...r ) are defined from higher-order distribution functions p" . For the pair correlation function PopCrj.rj) we can write... 

Figure 1 The first-order distribution function of the Landau free energy hypersurface, a cluster cluster of size 14.5 A, showing the transformation...

In the same way, one may define higher-order distribution functions. For instance, / (rj, r2, r3) is the probability of finding a particle at r3 given that there are already particles at ri and r2. A defining relationship for this function is... 

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... 

Thus / (l, t) = n (X, 0- An exact equation of motion for the distribution function can be written in terms of a hierarchy that relates the lower order distribution functions to higher order ones. The first member of the hierarchy is well known and for hard-sphere interactions may be written in the following form" (cf. also Appendix B). 

Fig. 5,11. Order distribution-functions of various parts of a polyacrylate with side-chain mesogen (phenylene-4-hydroxybenzoate) separated by a flexible spacer (hexamethylene). Determined from the angular dependence of the deuterium NMR line-shape. Presented by H. W. Spiess, Dechema Tagung 1987, see also Refs. The angle p is defined between the ma etic field and the...

It should be mentioned that actually represents the distribution of the mathematical expected value of the particles, and (as in kinetic theory) higher-order distribution functions are required to rigorously determine the system evolution-see Ramkrishna, Borwanker, and Shah [115, 116, 117]. However, Eq. 12.6-1 or 2 seem to be adequate for a variety of problems. 

In the foregoing section the distribution function in the complete phase space, /(x, t), was discussed and the general equation of change presented. In the sections to follow we often present results in terms of averages with respect to lower-order distribution functions, specifically averages involving the phase space or the configuration space of one molecule or a pair of molecules. This section is devoted to the definitions of these contracted distributions functions. 

Average Values in Terms of Lower-Order Distribution Functions... 

In this section we present some formulas and notation that will be very useful in subsequent sections, where averages with respect to the full phase space distribution are reduced to averages with respect to lower-order distribution functions. By formalizing the reduction procedure some economy of presentation can be achieved. For those not particularly interested in the mathematical details, the principal results are given in Eqs. (5.9), (5.10), (5.19) and (5.20). 

All the it s on the right side have dimensions of energy per unit volume. The individual contributions can then be written in terms of lower-order distribution functions by usmg Eqs. (5.9) and (5.19) ... 

This example seems to be representative of a general result When a given quantity is expressed in terms of GMDF s, it requires in general a lower-order distribution function compared with the corresponding expression in terms of ordinary MDF s. We shall encounter other examples later. It should be realized that although the GMDF s may seem to be more complex than ordinary MDF s, this does not necessarily imply that their computation, either analytically or numerically, should be more difficult. 

Here e. is the internal energy of a molecule at the fth vibrational and /th rotational levels, k is the Boltzmann constant, = u — v is the peculiar velocity. The zero-order distribution function for atomic species reads... 

First-order distribution functions are obtained in Nagnibeda Kustova (2009) in the form... 

The right hand sides of Eqs. (66) are specified by the zero-order distribution function. The... 

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. 

To preview the results somewhat, it will be shown that the general form of the transport equations contains expressions for the property flux variables (momentum flux P, energy flux q, and entropy flux s) involving integrals over lower-order density functions. In this form, the transport equations are referred to as general equations of change since virtually no assumptions are made in their derivation. In order to finally resolve the transport equations, expressions for specific lower-ordered distribution functions must be determined. These are, in turn, obtained from solutions to reduced forms of the Liouville equation, and this is where critical approximations are usually made. For example, the Euler and Navier-Stokes equations of motion derived in the next chapter have flux expressions based on certain approximate solutions to reduced forms of the Liouville equation. Let s first look, however, at the most general forms of the transport equations. 

Now, let s look at each flux term utilizing the distribution function, Eq. (5.92). We require both the first-order and second-order distribution functions, which under local equilibrium conditions follow Eq. (5.92) as [see Eqs. (4.47), (4.66), and (4.72)]... 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

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