Green’s functions hierarchy

Substituting these functions for the first two terms in eqn (13.57), until both of them have the common factor of CU i as the third term (if applicable), we obtain the hierarchy Green s function the formal type of expression ... 

For entangled iV-state transfer processes, the hierarchy Green s functions formalism involves tensors each having elements. As far as the kinetics regime is concerned, however, many interesting rate processes proceed practically in a step-wise manner. In this case the kinetics rate matrix could be determined by the individual rates between two states. Remarkably, the hierarchy Green s functions for the modified ZE/HEOM is analytically solvable for two-state systems. The resulting analytical expression of rate resolution, K/,j[s), for individual elementary a) b rate process can therefore be used to construct the NxN kinetics rate matrix. 

To evaluate the hierarchy Green s functions, of eqn (13.59), we exploit the Dyson equation technique and obtain for the three key elements in eqn (13.66) the expressions ... 

Having the analytical solutions to the hierarchy Green s functions, the comparison between the original ZE and the modified ZE can be readily explored over the entire parameter space. Evidently, the proposed modification is remarkably significant. The protocol of accuracy control presented in section 13.3.3 has also been verified extensively. 

The propagator technique, which attempts to solve the hierarchy of equations for the many-body Green s function, has been faciUtated by the use of inner projection techniques and the superoperator representation of Goscinski and Lukman. Ohrn and co-workers have applied these... 

The relationship between the Wiener integral (3.20) and the simple diffusion equation (3.21) suggests that it might be instructive to convert (6.12) to a differential equation. As also noted by Whittington, for the case of a discrete chain (6.12) can be expressed only in terms of the solution of a hierarchy of integro-differential equations. The derivation in the continuous case is presented in Appendix B for convenience, although the result is quoted here. Define the three-point Green s function as... 

In the general case the EOM approach yields an infinite hierarchy of coupled Green s functions with increasing number of Fermion operators involved. As in the diagrammatic approach discussed below it is useful to incorporate all many-body effects by defining the selfenergy Ey z) by... 

There exists a hierarchy of Green s functions (GF). These functions are defined as the //-electron ground-state expectation value of a time-ordered product of annihilation and creation operators. The simplest and probably most relevant member of the hierarchy is the one-particle GF... 

I lie interrelation between Wiener s integral (Equation 189) and the diffusion equation (EquatioTi 190) presupposes the existence of a differentied equation for G( 0, LG) (Gelfand and Yaglom, 1956). Special analysis (Freed, 1972) shows that the function from Equation 20.5 can be represented only by a hierarchy of integro-differential equations. For this purpose, Green s three-point function... 

The Born-Green-Yvon (BGY) hierarchy is truncated by introducinga super position approximation = correlation function 1234. The resulting pair of two simultaneous integral equations including the triplet- and pair-correlation functions 423 S i2> which hereafter... 

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