Differential correlation function

Figure 2.12 (a) Comparison of the differential correlation function for Finney s drp of monosized spheres and the experimental curve for a-Ni76P24- (b) the origin of second split peak. Dhs is the diameter of the hard sphere. (After Cargill, 1975)... 

Correlation functional due to Wilk, Vosko and Nusair Zero differential overlap... 

Differentiation of Eq. (1.71) shows that the initial value of R(t) is positive. Thus one may consider R(t) to have typical correlation function properties for t = 0 it is equal to the correspondingly determined dispersion Ro and it decreases to zero in a time of the order tc. 

Fig. 1.11. The normalized angular momentum correlation function Kj(t)/Kj(0) at k — 0.25 in differential (curve a), integral (curve b) and impact (curve c) theories.

The pair correlation function of the velocities and the pair correlation functions of some time derivatives of the velocity are sometimes taken into account.75 However, the validity of this description in the nonadiabaticity regions also has to be proved. The dynamic description or the description using the differentiable random process is more rigorous in this region.76... 

To complete the description and get the connection with the solute emission and absorption spectra, there is need of the correlation functions of the dipole operator pj= (a(t)+af(t))j and, consequently, the differential equation for the one solute mode has to be solved. The reader is referred to [133] for detailed analysis of this point as well as the equations controlling the relaxation to equilibrium population. The energy absorption and emission properties of the above model are determined by the two-time correlation functions ... 

Owing to the sensitivity of the chemical source term to the shape of the composition PDF, the application of the second approach to model molecular mixing models in Section 6.6, a successful model for desirable properties. In addition, the Lagrangian correlation functions for each pair of scalars (( (fO fe) ) should agree with available DNS data.130 Some of these requirements (e.g., desirable property (ii)) require models that control the shape of /, and for these reasons the development of stochastic differential equations for micromixing is particularly difficult. 

In order to perform calculations of the correlation function of particles 1 and 2 (which is the same quantity as the density of the 1st pseudoparticle, g ( )) using basis (49), we need the matrix elements of the delta function, 5(ri — ). They can be evaluated by replacing Ay Ay + uJn in expression (64) and then differentiating p times with respect to u. 

The computer-reconstructed catalyst is represented by a discrete volume phase function in the form of 3D matrix containing information about the phase in each volume element. Another 3D matrix defines the distribution of active catalytic sites. Macroporosity, sizes of supporting articles and the correlation function describing the macropore size distribution are evaluated from the SEM images of porous catalyst (Koci et al., 2006 Kosek et al., 2005). Spatially 3D reaction-diffusion system with low concentrations of reactants and products can be described by mass balances in the form of the following partial differential equations (Koci et al., 2006, 2007a). For gaseous components ... 

On this way we arrive at Bom-Green-Ivon, Percus-Yevick and hyperchain equations [5, 9], all having a general form (x,Vx,n,T) = 0. These non-linear integro-differential equations are close with respect to the joint correlation function, and Percus-Yevick equation gives the best approximation amongst known at present. An important point is that the accuracy of... 

However, a question arises - could similar approach be applied to chemical reactions At the first stage the general principles of the system s description in terms of the fundamental kinetic equation should be formulated, which incorporates not only macroscopic variables - particle densities, but also their fluctuational characteristics - the correlation functions. A simplified treatment of the fluctuation spectrum, done at the second stage and restricted to the joint correlation functions, leads to the closed set of non-linear integro-differential equations for the order parameter n and the set of joint functions x(r, t). To a full extent such an approach has been realized for the first time by the authors of this book starting from [28], Following an analogy with the gas-liquid systems, we would like to stress that treatment of chemical reactions do not copy that for the condensed state in statistics. The basic equations of these two theories differ considerably in their form and particular techniques used for simplified treatment of the fluctuation spectrum as a rule could not be transferred from one theory to another. 

It is convenient to divide a set of fluctuation-controlled kinetic equations into two basic components equations for time development of the order parameter n (concentration dynamics) and the complementary set of the partial differential equations for the joint correlation functions x(r, t) (correlation dynamics). Many-particle effects under study arise due to interplay of these two kinds of dynamics. It is important to note that equations for the concentration dynamics coincide formally with those known in the standard kinetics... 

Substitution of equation (2.3.62) into a set of equations (4.1.13) to (4.1.16) for noncharged (neutral) particles (Uvil r) = 0) does not affect equations (4.1.18) and (4.1.19) whereas the linear equation (4.1.23) describing the correlation dynamics splits now into three integro-differential equations. Main stages of the passage from general equations (4.1.14)—(4.1.16) for the joint densities to those for the joint correlation functions have been demonstrated earlier, see (4.1.20) and (4.1.21). Therefore let us consider only those terms which are affected by the use of superposition approximation. Hereafter we use the relative coordinates f=f — f(, f = r 2 — r[ and... 

The numerical methods for solving equations like (8.2.17), (8.2.22) and (8.2.23) are discussed in Section 5.1. In practice the conservative difference schemes are widely used for solving differential equations with the accuracy of the order 0(At + Ar2) [21, 26, 27] used as well 0(Af2 4- Ar2) [25], Unlike mathematically similar equations for the A + B —> 0 reaction (Section 5.1), where the correlation functions vary monotonously in time, the... 

The quantities in (4.40) are single time quantities. According to Eq. (4.38) we need the special correlation function (8pay(PP t)8 Uya( p pt)) this function is closely connected to the correlation function (SpSp). According to the discussions presented in Refs. 12 and 28 for the determination of such correlation functions, one has to start from a differential equation for the corresponding two-time correlation functions and use the relevant single-time correlation function as initial condition. From the equation for the fluctuations 8p, which reads... 

These equations are representation independent and are valid for an arbitrary time-dependence of the system Hamiltonian as, for example, in the case of strong laser driving. In a suitable representation, the coupled set of differential equations Eqs. (33) to (35) can be solved numerically, for example, using a Runge-Kutta scheme. As already mentioned in the Introduction, Yan and coworkers [33-35] independently developed a similar approach starting at a correlation function which is assumed to be of the form Eqs. (5) and (6). 

For time-dependent systems again the purely exponential time-dependence of the correlation function allows the derivation of a set of differential equations for the auxiliary operators... 

Becke exchange and LYP correlation functional Complete Active Space Self-Consistent Field Configuration Interaction Complete Neglect Differential Overlap Density Functional Theory... 

Equation (8) can also be derived using the techniques of functional differentiation. The first term in a functional expansion in correlation functions of order higher than pairs [56] reads... 

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