Irreducibility condition

In deriving Eqs. (454) and (455), we have explicitly used the factorization property of pf>aB for large separations the double prime in Eq. (455) indicates an irreducibility condition similar to the one used in Eq. (430), except that the words on its left have to be replaced by "on its right in the definition after Eq. (432). 

In Section IV, we develop the former results and we study the structure of the transport operator and of the generalized Boltzmann operator. We also analyse the irreducibility condition which appears in Prigogine s theory by using the graphs of equilibrium statistical mechanics. 

We study next the dynamical irreducibility condition which appeared in the definition of the transport operator. It eliminates from this quantity the reducible collision processes where the particles coming from infinity interact, recede to an infinite distance from one another, and then interact again. We define an extended transport operator from which the irreducibility condition is eliminated and which involves this time the reducible collisions. The relation between the transport operator and the extended transport operator is made explicit by means of a correspondence between the dynamical processes and the Mayer graphs for equilibrium. In this respect, we demonstrate, in these graphs, the importance of the role of the articulation points. 

Rule 1 No intermediate state k = 0 exists. This is the dynamical irreducibility condition of the diagonal fragment. 

Let us now specify the nature of the dynamical irreducibility condition in Eq. (56). The conservation rides of the wave vectors (Eq. 63) impose the condition that the k of certain particles is zero in certain intermediate states. For example, in Fig. 2a particles 2 and 3 have their k zero in the second state of propagation. It may be that the structure of the diagram is such that for one or many intermediate states the k of every particle is identically zero. The diagram is then reducible (see Fig. 2c) and is not contained in Eq. (56). This leads us to extend the definition of jijru. ) so as to include in it the reducible contributions. We shall define... 

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. 

In the parabolic model the equations for caustics are simply Q+ = Q, and Q- = <2-- The periodic orbits inside the well are not described by (4.46), but they run along the borders of the rectangle formed by caustics. It is these trajectories that correspond to topologically irreducible contours on a two-dimensional torus [Arnold 1978] and lead to the quantization condition (4.47). 

Inasmuch as sc results from fluctuations that cannot be eliminated so long as quanta are counted, this standard deviation is the irreducible minimum for x-ray emission spectrography under ideal conditions. Not only is it a minimum, but it is also a predictable minimum. When the standard deviation, s, significantly exceeds the standard counting error, sc, it is likely that errors resembling those the analytical chemist usually encounters are superimposed upon the random fluctuations associated with the emission process. 

Interpretation of NMR well logs is usually made with the assumption that the formation is water-wet such that water occupies the smaller pores and oil relaxes as the bulk fluid. Examination of crude oil, brine, rock systems show that a mixed-wet condition is more common than a water-wet condition, but the NMR interpretation may not be adversely affected [47]. Surfactants used in oil-based drilling fluids have a significant effect on wettability and the NMR response can be correlated with the Amott-Harvey wettability index [46]. These surfactants can have an effect on the estimation of the irreducible water saturation unless compensated by adjusting the T2 cut-off [48]. 

Conventionally, the sample is initially saturated with one fluid phase, perhaps including the other phase at the irreducible saturation. The second fluid phase is injected at a constant flow rate. The pressure drop and cumulative production are measured. A relatively high flow velocity is used to try to negate capillary pressure effects, so as to simplify the associated estimation problem. However, as relative permeability functions depend on capillary number, these functions should be determined under the conditions characteristic of reservoir or aquifer conditions [33]. Under these conditions, capillary pressure effects are important, and should be included within the mathematical model of the experiment used to obtain property estimates. 

Its irreducibility the prime means that the only transitions allowed in Eq. (429) are such that one starts with a single excited particle wave number (k 0) and arrives at the final state kf without any such intermediate state. As discussed previously in other contexts (see Section II-C), this condition implies that describes a dynamical process that is localized both in space and time ... 

The time dependence of the principal diagonal part —htMz indicates that this term corresponds to the one which is selected by the condition of dynamic irreducibility in Prigogine and Resibois s formalism. However, it is for the complete expression Tr[MzMz(t)] that one is allowed to retain in the development of Mz(t) only the part proportional to Mz and not for Q(/). 

Since we have not yet applied the condition of chronological separation, this expression still contains irreducible contributions —those in which, starting from the left, the first 8Lm) appears before the last 6L(n)—but they are easily eliminated so that the reducible contributions corresponding to our graph are given by the expression (see Eq. 58)... 

As discussed in Ref. [1], we describe the rotation of the molecule by means of a molecule-fixed axis system xyz defined in terms of Eckart and Sayvetz conditions (see Ref. [1] and references therein). The orientation of the xyz axis system relative to the XYZ system is defined by the three standard Euler angles (6, (j), %) [1]. To simplify equation (4), we must first express the space-fixed dipole moment components (p,x> Mz) in this equation in terms of the components (p. py, p along the molecule-fixed axes. This transformation is most easily done by rewriting the dipole moment components in terms of so-called irreducible spherical tensor operators. In the notation in Ref. [3], the space-fixed irreducible tensor operators are... 

Let 7T Z y be a proper projective morphism between algebraic varieties. Suppose that Y decomposes into a finite number of irreducible nonsingular subvarities, called strata Y = JyOy. Here y denotes a distinguished point in Oy. We further assume the following condition For each stratum Oy, the restriction of vr to K iOy) is a topological fiber bundle with base Oy and fiber... 

We have used for the row vectors of the respective entities, while we denote by ( ) and O the orbitals and many-electron functions, and by O and T(0) the two corresponding linear transformations, respectively. Various types of many-electron space for which such transformations may be carried out have been described by Malmqvist [34], In general, O may be non-unitary, possibly with subsidiary conditions imposed for ensuring that the corresponding transformation of the V-electron space exists e.g. a block-diagonal form according to orbital subsets or irreducible representations). 

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). 

GENERALIZED NORMAL ORDERING, IRREDUCIBLE BRILLOUIN CONDITIONS, AND CONTRACTED SCHRODINGER EQUATIONS... 

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. 

In terms of these conditions, a fc-particle hierarchy of approximations can be defined, with Hartree-Fock as the one-particle approximation for closed-shell states. Unfortunately, the stationarity conditions do not determine the fully, and for their constmction additional information is required, which essentially guarantees -representability. Nevertheless, the fe-particle hierarchy based on the irreducible stationarity conditions opens a promising way for the solution of the -electron problem. 

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