The Density Cumulant

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). 

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 ... 

The same result can also be obtained directly from the virial equation of state given above and the low-density fonn of g(r). B2(T) is called the second virial coefficient and the expansion of P in powers of is known as the virial expansion, of which the leading non-ideal temi is deduced above. The higher-order temis in the virial expansion for P and in the density expansion of g(r) can be obtained using the methods of cluster expansion and cumulant expansion. 

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... 

Since one is only rarely interested in the density at a precise point on the z-axis, the cumulative probability (cumulative frequency) tables are more important in effect, the integral from -oo to +z over the probability density function for various z > 0 is tabulated again a few entries are given in Fig. 1.13. 

Figure 2.19. Intersection of two linear regression lines (schematic). In the intersection zone (gray area), at a given c-value two PD-curves of equal area exist that at a specific y-value yield the densities zi and Z2 depicted by the dashed and the full lines. The product zi Z2 is added over the whole y-range, giving the probability-of-intersection value for that x. The cumulative sum of such probabilities is displayed as a sigmoidal curve the r-values at which 5, respectively 95% of Z2) s reached are indicated by vertical arrows. These can be...

By dividing the cumulative potential function of a class by the number of samples contributing to it, one obtains the (mean) potential function of the class. In this way, the potential function assumes a probabilistic character and, therefore, the density method permits probabilistic classification. 

If the detection screen D is constructed so that the locations of individual photon impacts can be observed (with an array of scintillation counters, for example), then two features become apparent. The first is that only whole photons are detected each photon strikes the screen D at only one location. The second is that the interference pattern is slowly built up as the cumulative effect of very many individual photon impacts. The behavior of any particular photon is unpredictable it strikes the screen at a random location. The density of the impacts at each point on the screen D gives the interference fringes. Looking at it the other way around, the interference pattern is the probability distribution of the location of the photon impacts. 

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

Sum Distribution. A cumulative presentation of equivalent diameters, which converts the density distribution curve to a plot that represents percentages of particles which are smaller than a given equivalent diameter D. 

The density functional methods assessed in this study (B3LYP, BLYP, and LDA) all perform much worse for the enthalpies of formation of the larger molecules in the G3/99 set. This is due to a cumulative effect in the errors for the larger molecules in this test set. The errors are found to be approximately proportional to the number of pairs of electrons in the molecules but the methods are not improved significantly when a higher-level correction such as that used in G2 or G3 theory is added the DFT methods. Further correction schemes may be necessary to improve the performance of density functional methods for large molecules. 

The first moment of the distribution is Pt0T the total, cumulative molar concentration of polymeric material. As the molecular weight of polymeric species increases, branching and crosslinking reactions yield a thermoset resin. Chromatography analysis of epoxy resin extracts confirms the expected population density distribution described by Equation 4, as is shown in Figure 2. Formulations and cure cycles appear in Table II. 

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... 

EXAMPLE 2.2 Use of the Sedimentation Equation for Particle Size Determination. A titanium dioxide pigment of density 4.12 g cm 3 is suspended in water at 33°C. At this temperature, the density and viscosity of water are 0.9947 g cm-3 and 7.523 10-3 P, respectively. A particle size analyzer (SediGraph, Micromeritics Instruments Corp., Norcross, GA 30093) plots the following data for cumulative weight percent versus equivalent spherical radius ... 

The density of the prills is reduced substantially when much evaporation occurs with 0.2-0.5% water in the feed, ammonium nitrate prills have a specific gravity of 0.95, but with 3-5% water it falls to 0.75. Prilled granules usually are less dense than those made by layering growth in drum or fluidized bed granulators. The latter processes also can make larger prills economically. To make large prills, a tall tower is needed to ensure solidification before the bottom is reached. The size distribution depends very much on the character of the atomization but can be made moderately uniform. Some commercial data of cumulative % less than size are ... 

The endpoint cumulant is defined by replacing in Eq. (5.16) the total segment density p r) by the total endpoint density... 

The use of this expression for a variational determination of T is a complex problem because of the /V-representability requirement [15, 16, 17], Nevertheless, there is a renewed interest in this problem and a number of methods, including so called cumulant-based approximations [18, 19] are being put forth as solutions to the representability problem. Although some advances can be obtained for special cases there appears to be no systematic scheme of approximating the density matrix with a well-defined measure of the N-representability error. Obviously, the variational determination of density matrices that are not guaranteed to correspond to an antisymmetric electronic wavefunction can lead to non-physical results. 

Example 1.1 One of the applications of using Stokes s law to determine the particle size is the Sedigraph particle analyzer. Table El.l shows the relationship between the cumulative weight percentage of particles and the corresponding particle terminal velocities for a powder sample. The densities of the particle and the dispersing liquid are 2,200 and 745 kg/m3, respectively. The liquid viscosity is 1.156 x 10-3 kg/m s. Find out the relationship of the mass fraction distribution to the equivalent dynamic diameter. 

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