Simultaneous distribution functions

Equations AlO and All allow us to regard Ufp,q) of Eq. A9 as the probability with which phonons can be found at coordinate vector q with momentum vector p (although it is not normalized yet). Since the coordinate and its conjugate momentum cannot be determined simultaneously at definite values in quantum mechanics [6], this probability must be approximate. In fact, it happens to have (nonphysical) negative values in some systems [22]. In phonon systems, fortunately, it is always positive, and it can be used as a semiclassical simultaneous distribution function for the coordinate q and the momentum p. After normalization, the Wigner distribution function thus obtained is given by... 

Second-Order Integral Equations for Associating Fluids As mentioned above in Sec. II A, the second-order theory consists of simultaneous evaluation of the one-particle (density profile) and two-particle distribution functions. Consequently, the theory yields a much more detailed description of the interfacial phenomena. In the case of confined simple fluids, the PY2 and HNC2 approaches are able to describe surface phase transitions, such as wetting and layering transitions, in particular see, e.g.. Ref. 84. 

Now let us add the possibility of collisions. Before we proceed, we make the following two assumptions (1) only binary collisions occur, i.e. we rule out situations in which three or more hard-spheres simultaneously come together (which is a physically reasonable assumption provided that the gas is sufficiently dilute), and (2) Boltzman s Stosszahlansatz, or his molecular chaos assumption that the motion of the hard-spheres is effectively pairwise uncorrelated i.e. that the pair-distribution function is the product of individual distribution functions ... 

To see the type of differences that arises between an iterative solution and a simultaneous solution of the coefficient equations, we may proceed as follows. Bor the thirteen moment approximation, we shall allow the distribution function to have only thirteen nonzero moments, namely n, v, T, p, q [p has only five independent moments, since it is symmetric, and obeys Eq. (1-56)]. For the coefficients, we therefore keep o, a, a 1, k2), o 11 the first five of these... 

There are many ways we could assign probability distribution functions to the increments N(t + sk) — N(t + tk) and simultaneously satisfy the independent increment requirement expressed by Eq. (3-237) however, if we require a few additional properties, it is possible to show that the only possible probability density assignment is the Poisson process assignment defined by Eq. (3-231). One example of such additional requirements is the following50... 

These two types of deviations occur simultaneously in actual reactors, but the mathematical models we will develop assume that the residence time distribution function may be attributed to one or the other of these flow situations. The first class of nonideal flow conditions leads to the segregated flow model of reactor performance. This model may be used... 

The formula for specialized distribution functions makes no such assumptions and hence involves g<3). It also involves g(4), g(5),.. . since correction is made for the possibility of a defect having two, three,. . . other partners simultaneously. Using Eq. (171) and the superposition approximation one finds that for the sodium chloride type lattice... 

The classical description is quite different from the quantum. In classical dynamics we describe the coordinates and momenta simultaneously as a function of time and can follow the path of the system as it goes from reactants to products during the collision. These paths, called trajectories, provide a fnotion picture of collision process. The results of any real collision can be represented by computing a large number of trajectories to obtain distribution of post-collisions properties of interest (e.g. energy or angular distribution). In fact, the trajectory calculation means the transformation of one distribution function (reagent distribution, pre-collision) into another (product distribution, post-collision), which is determined by PE function. 

Another approach is known as the Distributed Activation Energy Model (DAEM). This model recognizes that devolatilization occurs through many simultaneous reactions. To express this process in a mathematically tractable manner, these reactions are all presumed to be first order and to be describable by a continuous distribution of kinetic rates with a common pre-exponential and a defined distribution function of activation energy [43],... 

First let us recall the more important electron distribution functions and their origin in terms of corresponding density matrices. The electron probability density ( number density in electrons/unit volume) is the best known distribution function others refer to a pair of electrons, or a cluster of n electrons, simultaneously at given points in space. 

On the other hand, the Monte Carlo method enables us to simultaneously obtain the time-dependent decay curve and the time-dependent distribution function. Therefore we adopted Monte Carlo simulation [18,21,84,85] for the analysis. The geminate ion recombination is also described as follows. 

Alternatively, we can work in momentum-space with the momentum distribution given by the square of the modulus of the momentum wavefunc-tion. However, because of Heisenberg s uncertainty relation it is impossible to specify uniquely the coordinates and the momenta simultaneously. Either the coordinates or the momenta can be defined without uncertainty. In classical mechanics, on the other hand, the coordinates as well as the momenta are simultaneously measurable at each instant. In particular, both the coordinates and the momenta must be specified at t — 0 in order to start the trajectory. Thus, we have the problem of defining a distribution function in the classical phase-space which simultaneously weights coordinates and momenta and which, at the same time, should mimic the quantum mechanical distributions as closely as possible. 

Figures 4.29 and 4.30 display the co-evolution of the formamide dipole moment and of the oxygen(formamide)-oxygen(water) radial distribution function during the polarization process [12]. As the dipole moment of the formamide increases, the position of the first peak of the RDF is shifted inward and its height increases. Once the dipole moment has reached its equilibrium value, it begins to fluctuate. Fluctuations are related to the statistical error associated with the finite length of the simulations. From Figures 4.29 and 4.30 it is clear that (1) ASEP/MD permits one to simultaneously equilibrate the...

The theory was tested with the aid of an ample data array on low-frequency magnetic spectra of solid Co-Cu nanoparticle systems. In doing so, we combined it with the two most popular volume distribution functions. When the linear and cubic dynamic susceptibilities are taken into account simultaneously, the fitting procedure yields a unique set of magnetic and statistical parameters and enables us to conclude the best appropriate form of the model distribution function (histogram). For the case under study it is the lognormal distribution. 

The ratio kao=/ka0 is a measure of the spread in energy associated with the distribution function/( ). Equations (19)—(21) may be readily extended to include systems having two or more simultaneous decomposition reactions. For two decomposition paths having specific rate constants k, and k,, ... 

Distribution Functions and Hydrogen-Deuterium Isotope Effects in Thermal Activation Systems. The isotope effects in thermal activation systems are determined to a large degree by the energy distribution function and kt and /(e) must be simultaneously considered. The equilibrium high-pressure effect is considerably different from the non-equilibrium low-pressure case, and they are discussed separately. 

In a gas g(r) is related to the pair distribution function, the probability of finding two molecules simultaneously at given dis-... 

The above described lack of smoothness at y = ya is essential. It refers to the characteristic power law distribution functions of cluster sizes in percolation, indicating that the most frequent number Lx of singly connected bonds is unity. This leads to a spontaneous fast decline of G when y exceeds the value yapp> since all L-N-B-chains with Lx=1 break simultaneously at this amplitude. Experimental results show that a smooth transition of G with varying strain amplitude appears that cannot be described by a power law distribution function or the assumed exponential type of/lfl (y). 

The application of the composite isotherms enables us to model the ion exchange on heterogeneous surfaces, such as rocks and soils. When the structure and composition of the sorbent is well known, we can choose the most probable site affinity distribution function. If not, it is desirable to fit the composite isotherm by different models. The just-described four isotherms provide an opportunity for this. In addition, when adsorption and ion exchange can take place simultaneously, adsorption and ion-exchange isotherms and site distribution functions can be combined (Cernik et al. 1996). 

AS, These two conditions cannot be satisfied simultaneously without choosing distribution functions for molecular density and velocity which are functions of Z. To avoid such a choice, which would lead to an almost impossibly complicated calculation, let us rather make a crude but simpler calculation which leads to results of the correct order of magnitude and is in qualitative agreement with experiment. 

Figure 15 gives a schematic of a simultaneous reaction-separation model. To include separation in a reactor targeting model, we postulate a separation function vector (7) analogous to a residence time distribution function for homogeneous reactors. Here, however, each species has its own residence time distribution function, which is dependent on its separation function 7c-... 

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