Connectivity diagram

Industrial Power Engineering and Applicalions Handbook Table 6.1 Connection diagrams for multispeed motors... 

The contribution from each of these connected diagrams is multiplied by iB 0 0>in, which consists of the diagrams indicated in Fig. 10-4. This fact, of course, is the counterpart of the statement that in higher order each of the connected diagrams appears together with the... 

In Eq. (11-120) Sc denotes the contribution to 8 from connected diagrams only, i.e., from the diagrams shown in Fig. 11-5. The energy E0 in this phase factor, exp (+iE0T), can be interpreted as the vacuum self-energy due to the interaction By redefining... 

The problem at hand is the evaluation of the activity coefficient defined in Eq. (76). It will be assumed that only pairwise interactions between the defects need be considered at the low defect concentrations we have in mind. (The theory can be extended to include non-pairwise forces.23) Then the cluster function R(n) previously defined in Eq. (78) is the sum of all multiply connected diagrams, in which each bond represents an /-function, which can be drawn among the set of n vertices, the /-function being defined by Eqs. (66), (56), and (43). The Helmholtz free energy of interaction of two defects appearing in this definition can be written as... 

It would be useful to be able to distinguish the connected diagrams from the non-connected diagrams. In general, one can, for a given contribution, separate the particles into "clusters ... 

Figure 5. Connection diagram of a 40-mgd (151,000 cu m/d) RO plant to be operated in conjunction with the proposed Mediterranean-Dead Sea hydroelectric project...

The first of these two terms cannot be considered a pure two-body term, therefore the A can only be considered as a connected diagram within the context of an antisymmetrized diagrammatical approach. 

These equations define the RDMCs in terms of the RDMs and do not depend on the validity of perturbative expansions of the RDMs, although insofar as perturbation theory is applicable, Ap is precisely the sum of connected diagrams in the expansion of Dp. 

Formulating conditions for the energy to be stationary with respect to variations of the wavefunction P in this generalized normal ordering, one is led to the irreducible Brillouin conditions and irreducible contracted Schrodinger equations, which are conditions on the one-particle density matrix and the fe-particle cumulants k, and which differ from their traditional counterparts (even after reconstruction [4]) in being strictly separable (size consistent) and describable in terms of connected diagrams only. 

We have already discussed the relations between the four stationarity conditions. In view of their separability, the two irreducible conditions are the right choice in the spirit of a many-body theory in terms of connected diagrams. 

There is a price to pay for the separability, or equivalently for the presence of only connected diagrams. Somewhat like in traditional many-body theory, one must be ready to accept so-caUed EPV (exclusion-principle violating) cumulants. Typical EPV cumulants are nonvanishing kk for k> n, while yj = 0 for k > n. 

Obviously, we gain precisely the same expressions as in the multi-root theory since we postulated the same form of the effective Hamiltonian. We recall that all matrix elements of the effective Hamiltonian are expressed by means of connected diagrams only in the case of diagonal elements just connected vacuum diagrams may come into consideration and in the case of off-diagonal elements at least one part of a disconnected diagram would correspond to an internal excitation. 

An important simplification which wc should introduce here, consists in the reduction of the Grccmsfunctions to their connected parts or cumulants7. By definition the cumulant G m . 7nT. equals the sum of all totally connected diagrams contributing to Tn Fig. 4.5, for example, only the last... 

H Pp is given by the sum of all topologically different connected diagrams obeying the following rules. 

For the diagrams of Fig. 5.3 we, for instance, find n (5.3a) = 1. nt(5.3 ) = 2. 71 (5.3c) = 1. We may show that with each independent internal momentum we may associate a dosed curve on the graph, through which that momentum flows. This explains the notion of loop . A connected diagram without loops is called a tree . 

The generating term J fj(r)p(r) is diagrammatically represented by a wiggly arrow (cf. Fig. 5.18) inserted into polymer lines. We then without any change can take ewer the discussion given above for Z[p7 to find that In Z[p.p, a] is given by the set of all connected diagrams with any number of (a - p) -insertions. Functional differentiation reduces a (u p) - insertion to a density insertion. The result, for I tC then is found from Eq. (A 5.8). 

We now have to analyze the behavior of the externally connected diagrams in the thermodynamic limit. In view of Eq. (A 5.37) we have to single out the contribution of order Q. and we have to show that no diagram increases stronger than fA This is an exercise in power counting. Let us address as (sjj) box a box containing s -legs distributed on j polymer lines. From Eqs. (A 5.40), (A 5.42) such a box carries an explicit factor Thus a... 

For the free energy, disconnected vertices first contribute in two loop order (Fig. 5.20c). For correlation functions, however, disconnected vertices may contribute even to one loop order, and a calculation ignoring such contributions by keeping only naively connected diagrams would yield wrong results. 

Electrical schematic wiring and/or pneumatic connection diagrams... 

---