Density cumulants

The moment equations of the size distribution should be used to characterize bubble populations by evaluating such quantities as cumulative number density, cumulative interfacial area, cumulative volume, interrelationships among the various mean sizes of the population, and the effects of size distribution on the various transfer fluxes involved. If one now assumes that the particle-size distribution depends on only one internal coordinate a, the typical size of a population of spherical particles, the analytical solution is considerably simplified. One can define the th moment // of the particle-size distribution by... 

W. Kutzelnigg and D. Mukherjee, Irreducible Brillouin conditions and contracted Schrodinger equations for n-electron systems. I. The equations satisfied by the density cumulants. J. Chem. Phys. 114, 2047 (2001). 

The cumulants of the density matrices, for short density cumulants, have a few nice properties [17]. We note in particular the Hermiticity, for example. 

Generalized normal ordering is intimately linked to the cumulants Xk of the k-particle density matrices (for short, density cumulants). The contractions in the sense of the generalized Wick theorem involve the... 

Unlike the density cumulant expansion, which can in principle be exact for certain states (such as Slater determinants), the operator cumulant expansion is never exact, in the sense that we cannot reproduce the full spectrum of a three-particle operator faithfully by an operator of reduced particle rank. However, if the density cumulant expansion is good for the state of interest, we expect the operator cumulant expansion to also be good for that state and also for states nearby. 

The density cumulant therefore measures the correlation among local density fluctuations p(r) — (p(r). It is these fluctuations which lead to scattering. 

As a further important quantity we consider the density cumulant. Here we have to proceed in t wo steps. As intermediate quantity we define the subset (q) of diagrams which are irreducible with respect to cuts which... 

Equation (5-56) yields the scaling function J, (q) of the density cumulant in unrenormalized tree approximation ... 

Fig. 24 Simultaneous CRYSTAF analysis of five LLDPE resins of different density (cumulative curves). 40 mg/40 mL in TCB crystallization rate 0.2°C/min...

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