Zwanzig generalized master equation

III. The Generalized Master Equation and the Zwanzig Projection Method... 

III. THE GENERALIZED MASTER EQUATION, AND THE ZWANZIG PROJECTION METHOD... 

Equation (20) is the central result of the Zwanzig projection method, and it is one of the two theoretical tools under scrutiny in this chapter, the first being the Generalized Master Equation (GME), of which Eq. (20) is a remarkable example, and the second being the Continuous Time Random Walk (CTRW) [17]. It must be pointed out that to make Eq. (20) look like a master equation, it is necessary to make the third term on the right-hand side of it vanish. To do so, the easiest way is to make the following two assumptions ... 

To make this chapter as self-contained as possible, we shaU briefly review the Zwanzig approach to the generalized master equation. First of all. 

We end this discussion with two comments. First, we note that the Nakajima-Zwanzig equation (10.100) is exact no approximations whatever were made in its derivation. Second, this identity can be used in many ways, depending on the choice of the projection operator P. The thermal projector (10.87) is a physically motivated choice. In what follows we present a detailed derivation of the quantum master equation using this projector and following steps similar to those taken above, however, we will sacrifice generality in order to get practical usable results. 

The introduction of statistical features in the basic molecular models is considered in Section II,E,2. It is argued that, in most cases, at least one part of the nonradiant molecular manifold is unknown and should be treated under statistical assumptions. By means of a partially random representation of the zero-order Hamiltonian, of the kind introduced by Wigner and others in statistical nuclear theory (see Bloch, 1966, 1969), we define a general dynamical model in which both the quantal and statistical properties of the molecular excitations are combined. Special attention is given to the nature of the statistical limit and irreversible radiationless transitions for the molecular excited states. We also discuss the relationship between this concept and similar concepts in quantum statistical theory of relaxation and master equations (Zwanzig, 1961). 

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