Connected equations

Another advantage of the open-equation format is that simple connection equations can be used rather than eliminating variables and equations that are connected. For example, the connections between two heat exchangers can be formulated as... 

These methods can be part of a series connection (Figure 3) or of a parallel connection (Equation 4). In other words, the knowledge base consists of the chemical models that form the building blocks and the statistical models that form the network of connections. 

Much of the recent literature on RDM reconstruction functionals is couched in terms of cumulant decompositions [13, 27-38]. Insofar as the p-RDM represents a quantum mechanical probability distribution for p-electron subsystems of an M-electron supersystem, the RDM cumulant formalism bears much similarity to the cumulant formalism of classical statistical mechanics, as formalized long ago by by Kubo [39]. (Quantum mechanics introduces important differences, however, as we shall discuss.) Within the cumulant formalism, the p-RDM is decomposed into connected and unconnected contributions, with the latter obtained in a known way from the lower-order -RDMs, q < p. The connected part defines the pth-order RDM cumulant (p-RDMC). In contrast to the p-RDM, the p-RDMC is an extensive quantity, meaning that it is additively separable in the case of a composite system composed of noninteracting subsystems. (The p-RDM is multiphcatively separable in such cases [28, 32]). The implication is that the RDMCs, and the connected equations that they satisfy, behave correctly in the limit of noninteracting subsystems by construction, whereas a 2-RDM obtained by approximate solution of the CSE may fail to preserve extensivity, or in other words may not be size-consistent [40, 42]. 

In this work, we derive—via explicit cancellation of unconnected terms in the CSE—a pair of simultaneous, connected equations that together determine the 1- and 2-RDMCs, which in mrn determine the 2-RDM in a simple way. Because the cancellation of unconnected terms is exact, we have in a sense done nothing the connected equations are equivalent to the CSE and, given Al-representabUify boundary conditions, they are also equivalent to the electronic Schrodinger equation. The important difference is that the connected equations for the cumulants automatically yield a size-consistent 2-RDM, even when solved approximately, because every term in these equations is manifestly extensive. 

In the present context, the way to ensure extensivity is to reformulate the CSE so that the RDMCs and not the RDMs are the basic variables. One can always recover the RDMs from the cumulants, but only the cumulants satisfy connected equations that do not admit the possibility of mixing noninteracting subsystems. Connected equations are derived in Section V. Before introducing that material, we first provide a general formulation of the p-RDMC for arbitrary p. 

Moreover, as the system of Eq. (24) is a starting point for the cluster-type approach and the MC method, therefore in principle the two methods can be used in combination. It has been proposed by Hood et al. [297] to write down the kinetic equations for describing variations in the occupancy state of each lattice site, i.e., to abandon consideration of the lattice ensemble, and to solve the system of the equations with the dimension equal to the number of sites. The system of the connected equations has been solved numerically. In each time interval the desorption probability for a given molecule is determined by the random sampling and then the general adsorbate change found. The combination approach allows to trace the adlayer structure and to construct a correlation between the structural and the kinetic behavior of the process. Such an approach has been applied to the N2/Ru(001) system to obtain a qualitative agreement with experiment [298]. 

As we mentioned in Section II.C, one of the most important requirements, which is applicable to the overall structure of reaction schemes, is its thermodynamic consistency. In other words, the scheme, as written, must allow the process to proceed asymptotically to its equilibrium state at infinite time. This can be reached only if any elementary step is included into the overall scheme together with its reverse reaction. Let us consider the consequences for the description of the process kinetics. For the sake of simplicity, we assume that the reaction proceeds in the ideal gas mixture. The value of rate constants k( for forward and k( ) for reverse reactions must satisfy the connecting equation... 

Let s estimate the adequacy of Equation (2.50) in the results of the experiments in Figures 2.39 and 2.40. With this purpose, we shall create the structural diagram (Figure 2.45) similar the one shown in Figure 2.44. Figure 2.45 shows that structural diagrams are a little bit displaced from the beginning coordinates and on an axis of ordinates cut a piece c = 0.06. In this connection, Equation (2.50) is transformed ... 

Figure 3.2 illustrates the configuration of two infinite vertical plates at the air-liquid interface [1,12]. The plate thickness is neglected for simplicity. The presence of the plates distorts the air-liquid interface. The plates have contact angles and and are apart in a horizontal distance L. The three regions divided by the two plates are labeled as i = 1,2, and 3. Note that the liquid in the three regions is connected. Equation (3.4) can be simplified to a 2-dimensional equation [1]... 

---