Global order

With respect to the selected elemental and global orders of node numbering the elemental stiffness equations for elements ei, eu and em in Figure 6.4 are expressed as... 

In order to refer to the p-order correlation matrices with p > 2, the notation needs to become more precise. Thus a left superscript has to be introduced. As can be seen from the definition, Eq. (91), the first index of this left superscript denotes the global order of the matrix, and the following indices denote the order of the different TRDMs involved. The right subscripts, denoting the element labels, coincide with those of the 2-RDM from which they derive. 

Deposition of Au onto this surface leads to the nucleation of Au islands at the intersection of clean Cu stripes thus leading to a square island lattice with a period of 50 A [83,86-88]. The N-covered Cu(100) surface has also been used for the growth of so far less well-ordered lattices of Fe and Cu [89], Co [90-92], Ag [93,94], and Ni [95], We note that square lattices can in principle also be created on Au(f4,f5,f5) since this miscut leads to 70 A step distance, which is equal to the reconstruction period. However, the steps are already far apart reaching the limit of the elastic step repulsions which may render global order difficult. Finally we note that another interesting alternative square template, although with smaller lattice constant, is presented by the (3 /3 x 5)-phase of V-oxide on Rh(lll) [96]. 

The Gulnler analysis Is truly valid only for dilute systems of uniform size and shape, with no global order. However, the approximation has shown Its usefulness for a variety of other systems (34-36). In the case of oriented systems, data for the semi-log plot are taken as data slices at certain azimuthal angles to a reference axis. For densely packed systems application of the Gulnler analysis may or may not yield a physically significant value for , but Is normally useful In differentiating between samples that are composed of the same material, but processed In different manners. 

The premixed methanol flame [11, 12] does not show the Swan bands of C2, which are prominent in a methane flame [13]. The base of the flame shows strong emission from excited formaldehyde and further up the flame emission from OH and CH occurs. The burning velocity of a stoichiometric methanol—air flame [12] is about 45 cm. sec", and the global activation energy and global order are 43—47 kcal. mole" and unity, respectively [14(a)]. 

Mesoscopic non-equilibrium thermodynamics provides a description of activated processes. In the case considered here, when crystallization proceeds by the formation of spherical clusters, the process can be characterized by a coordinate y, which may represent for instance the number of monomers in a cluster, its radius or even a global-order parameter indicating the degree of crystallinity. Polymer crystallization can be viewed as a diffusion process through the free energy barrier that separates the melted phase from the crystalline phase. From mesoscopic non-equilibrium thermodynamics we can analyze the kinetic of the process. Before proceeding to discuss this point, we will illustrate how the theory applies to the study of general activated processes. 

Our MC results for the confined dipolar fluids indicate that spontaneous order does indeed occur over a certain range of wall separations. s. This can be seen from Figs. 6.6-6.8 for a system where = 7. Specifically, in Fig. 6.6, we have plotted the global order parameter... 

This eight-point tensor-product QMOM fixes the moment of order zero with respect to all the internal coordinates, nine pure moments plus four mixed moments, for a total of 14 moments. The highest-order moment accommodated by this particular choice is mi,i,i of global order 7 = 3. Comparison of this moment set with the optimal moment set reported in Table 3.7 for M = 3 and N = 8 clearly shows that is a subset of the 32 optimal moments. 

Table 7.2. A triangular table reporting the different moments of a bivariate NDF up to global order five...

The global order of the moment is defined as the summation of the order wifii respect to each internal coordinate. For example, mo.i is of global order one, whereas W23 is of global order five. 

Figure 7.4. Predictions of the evolution of m fl (normalized with its initial value) by the QBMM with A = 3 for a standard aggregation-and-breakage problem with different choices for the last three moments from among moments of global order three, global order four, and global order five, given the first six (mo.o, ri,o, wJo.i, wt2,o, and mo,2). Thin lines, QBMM thick line Monte Carlo method. Adapted from Zucca et al. (2007).

In the relation (8.19) a, p,. .. are partial orders of reaction. Their sum gives the global order of reaction n = a + P +. If the reference component is A then the reaction rate is expressed in general by the following relation ... 

For an irreversible reaction of global order m + n(m with respect to A, n with respect to S), the approach followed by Hikita and Asai [10] was very similar to that of Van Krevelen and Hoftijzer. The rate of reaction was written as ... 

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