Nakajima-Zwanzig identity

Starting point for the derivation of the Nakajima-Zwanzig identity is the Lioville-von Neumann equation for the density operator a of the complete system, i.e. relevant system plus environment. As stated before h = 1 is used and thus the QME reads... 

So within a few steps it is possible to derive the Nakajima-Zwanzig identity starting from the Lioville-von Neumann equation. 

The Nakajima-Zwanzig identity is of course not the only way to proceed. Defining the operator D(t) using... 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

The identity (10.100) is the Nakajima-Zwanzig equation. It describes the time evolution of the relevant part Pp f) of the density operator. This time evolution is determined by the three terms on the right. Let us try to understand their physical contents. In what follows we refer to the relevant and irrelevant parts of the overall system as system and bath respectively. 

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. 

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