Correlation functions higher-order, factorization

Section IX is devoted to the important task of revealing the physical reason of the breakdown of the equivalence between density and trajectory perspective in the non-Poisson case. First of all, in Section IX.A, we show that several theoretical approaches to the generalized fluctuation-dissipation process rest implicitly on the assumption that the higher-order correlation functions of a dichotomous noise are factorized. In Section IX.B we show that the non-Poisson condition violates this factorization property, thereby explaining the departure of the density from the trajectory approach in the non-Poisson case. 

To account for the true (hpole moment in the states i),), one must consider not only the first neighbors of the water molecule but also the small contribution of the higher-order neighbors (see ref. 9, p. 102), that is, one must face the same difficulties as those met when calculating the dipole static correlation function [the Kirkwood g (rii) factor]. 

For-inatiy practical applications, the power law relationship (8.1 requires an appropriate proportionality constant or coefficient. Consider the case of nionodisperse primaty particles of radius ft,o which fonn power law agglomerates according to a process whose statistical features are independent of r ,o. The statistical properties of the agglomerates produced by this process do not depend on the magnitude of fl o. That is, if were multiplied by a factor of 10, the value of Df would not be affected nor would all of the other higher-order particle correlation functions that are not considered in this analysis. This means that the system should scale as R/apo so that (8,3) becomes... 

Several comments are in order as to the form (36) associated with G,(/ ), and these comments can apply also to the higher-order terms in the expansion of X(R) as well. The factor f occurring in G2(R) is our expansion parameter in the present problem, so that the higher-order terms in the expansion of X(R) contain higher powers of f. In the reduced form (36) for G2(R), the k dependence (therefore, T and p dependence) occur only in the second factor 0 as a proportionality constant. A detailed investigation we made shows that this linear oecurrence of i is a common feature associated with all terms involved in the expansion. Therefore this term can be factored out of the expansion and used to reduce the cell-pair correlation function t (a ) which is only a function of the reduced distance x, and its corresponding expansion has the following form ... 

If the renormalization of the wave function is also taken into account, the (1 - al) quantity is divided by ai, and the corresponding correction is called the renormalized Davidson correction. The effect of higher order excitations is thus estimated from the correlation energy obtained at the CISD level times a factor that measures how important the single-determinant reference is at the CISD level. The Davidson correction does not yield zero for two-electron systems, where CISD is equivalent to full Cl, and it is likely that it overestimates the higher order corrections for systems with few electrons. More complicated correction schemes have also been proposed, but are rarely used. 

In Eq. 14.41, the singular terms associated with the self-correlation are included in the higher-order contributions with respect to 1/N. Removing the irrelevant 5 (pa -b pb -b Pc) factor that is ascribed to the symmetrization of coordinates, we And the form of the ternary distribution function of ideal Fermi gas as... 

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