Closure relations

Either of these Vineyard-like approximations, along with an additional closure relation, will allow the exact results for A (t), F(k, t), and F %k, t) to constitute a closed set of equations. The closure relation consists of an independent approximate determination of the self irreducible memory function O Kk, t). One inmitive notion behind the proposed closure relation is the expectation that the -dependent self-diffusion properties, such as F k, t) itself or its memory function O k, t), should 

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In the two-electronic-state Bom-Huang expansion, the fulTHilbert space of adiabatic electoonic states is approximated by the lowest two states and furnishes for the corresponding electronic wave functions the approximate closure relation... 

The general equation can be further reduced to the case of infinite dilution limit, a binary mixmre, ionic solutions, and so on. These equations are supplemented by closure relations such as the Percus-Yevick (PY) and hypernetted chain (HNC) approximations. 

One should perhaps mention some other closures that are discussed in the literature. One possibility is to combine the PY approximation for the hard core part of the potential and then use the HNC approximation to compute the corrections due to the attractive forces. Such an approach is called the reference hypernetted chain or RHNC approximation [48,49]. Recently, some new closures for a mixture of hard spheres have been proposed. These include one by Rogers and Young [50] (RY) and the Martynov-Sarkisov [51] (MS) closure as modified by Ballone, Pastore, Galli and Gazzillo [52] (BPGG). The RY and MS/BPGG closure relations take the forms... 

To solve the replica OZ equations, they must be completed by closure relations. Several closures have been tested against computer simulations for various models of fluids adsorbed in disordered porous media. In particular, common Percus-Yevick (PY) and hypernetted chain approximations have been applied [20]. Eq. (21) for the matrix correlations can be solved using any approximation. However, it has been shown by Given and Stell [17-19] that the PY closure for the fluid-fluid correlations simplifies the ROZ equation, the blocking effects of the matrix structure are neglected in this... 

Finally, the closure relations for the inhomogeneous pair functions must be chosen. The PY approximation for the fluid-fluid direct correlation function presumes that its blocking part vanishes. This implies that c, ii(/,y) = 0, and... 

The summation can be simplified using an expression called the closure relation, to give finally... 

Closure Phases (III.9-10) Closure phases are obtained by triple products of the complex visibilities from the baselines of any subset of three apertures of a multi-element interferometer. Element-dependent phase errors cancel in these products, leaving baseline dependent errors which can be minimised by careful designs. Although there are many closure relations in a multi-element array, there are always fewer independent closure phases than baselines. Closure phases are essential for imaging if no referenced phases are available. 

This expression, known as the completeness relation and sometimes as the closure relation, is valid only if the set of eigenfunctions is complete, and may be used as a mathematical test for completeness. Notice that the completeness relation (3.31) is not related to the choice of the arbitrary function /, whereas the criterion (3.30) is related. 

When supplemented with a closure relation, Eq. (7) can be solved for h r) and c r). For example, the Percus-Yevick (PY) closure is given by [89]... 

The OZ (or PRISM) equation with closure relation can be solved using a Picard iteration procedure. One starts with a guess for the function y(r) = h(r) — c(r), either y(r) = 0 or the value of y(r) from some condition close to the condition of interest. Using the closure relation, c(r) is then obtained from y(r). With the PY closure, for example, we obtain... 

A graphical representation of the multilevel approach is shown in Fig. 4. All three models are now commonly accepted and are widely used by a number of research groups (both academic and industrial) around the world. In a recent paper, we have given an overview of the three models as they are employed at the University of Twente, together with some illustrative examples (Van der Hoef et al., 2004). In this chapter, we will focus on the technical details of each of the models, much of which has not been published elsewhere. The development of detailed closure relations from the simulations, as indicated in Fig. 3, is still ongoing. Some preliminary results for both the drag-force closures and solid pressure will be presented in the Sections II and III. In this chapter, we will... 

From the closure relation Z j j ) (j = 1 -1 g ) < g I, the sum over the product of transition matrix elements involving p,(r) and p (r )separates into two terms, one containing the ground-state expectation value of p (r) p (r ) and the other containing the product of the expectation values of p (r) and p (r ), both in the ground state. These terms can be further separated into those containing self interactions vs. those containing interactions between distinct electrons. Then... 

The "correlative" multi-scale CFD, here, refers to CFD with meso-scale models derived from DNS, which is the way that we normally follow when modeling turbulent single-phase flows. That is, to start from the Navier-Stokes equations and perform DNS to provide the closure relations of eddy viscosity for LES, and thereon, to obtain the larger scale stress for RANS simulations (Pope, 2000). There are a lot of reports about this correlative multi-scale CFD for single-phase turbulent flows. Normally, clear scale separation should first be distinguished for the correlative approach, since the finer scale simulation need clear specification of its boundary. In this regard, the correlative multi-scale CFD may be viewed as a "multilevel" approach, in the sense that each span of modeled scales is at comparatively independent level and the finer level output is interlinked with the coarser level input in succession. 

Here, H and C are symmetric matrices whose elements are the partial total hap(r) and direct cap,(r) pair correlation functions a,ft = A,B) W is the matrix of intramolecular correlation functions wap r) that characterize the conformation of a macromolecule and its sequence distribution and p is the average number density of units in the system. Equation 17 is complemented by the closure relation corresponding to the so-called molecular Percus-... 

This is a very important relation that we will be using over and over again in the following. The operator Pqi = qi) qi is called the projection operator for the ket qi). Equation (F.7), which is called the completeness relation, or closure relation, expresses the identity operator as a sum over projection operators. The relation is true for any orthonormal basis we may choose. 

Integral equations theories are another approach to incorporate higher order correlations, and consequently also lead to lowered osmotic coefficients. There are numerous variants of these theories around which differ in their used closure relations and accuracy of the treatment of correlations [36]. They work normally very well at high electrostatic coupling and high densities, and are able to account for overcharging, which was first predicted by Lozada-Cassou et al. [36] and also describe excluded volume effects very well, see Refs. [37] for recent comparisons to MD simulations. 

Answers to the following questions are sought. (1) Can a closure, or several closures, be found to satisfy theorems for simple liquids (2) If such closures exist, will they be improvements over conventional ones Do they give better thermodynamic and structural information (3) Will such closure relations render the IE method more competitive with respect of computer simulations and with other methods of investigation ... 

A Necessary Closure Relation Introduction of the Bridge Function... 

From what precedes, it is obvious that in order to determine the latter two correlation functions for a given pair potential u(r), Eq. (21) must be supplemented by an auxiliary closure relation [7, 17, 18, 27] derived from a cluster diagram analysis that reads... 

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