Zwanzig approach

This same equation has been independently derived by Caceres [90] using van Kampen s lemma [91] and the Bourret-Frisch-Pouquet theorem [92], while the theory adopted by Annunziato et al. [87] rests essentially on the Zwanzig approach of Section III, namely, a Liouville-like perspective. 

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

A still more rigorous demonstration of the complete equivalence of the Mori and Nakajima-Zwanzig approach has been derived by Grabert. For simplicity we shall not illustrate his demonstration. We shall limit ourselves to remark that this clearly establishes that the two approaches are equivalent in that they are related in the same way as the Heisenberg and Schrddinger pictures are related. Note, however, that when the scalar product implies equilibrium as Eq. (3.5) does, the Mori theory appears... 

Although the Zwanzig and Mori techniques are closely related and, from a purely formal point of view, completely equivalent, the elegant properties of the Mori theory such as the generalized fluctuation-dissipation theorem imply the physical system under study to be linear, whereas this is not necessary in the Zwanzig approach. This is the main reason we shall be able to face nonlinear problems within the context of a Fokker-Planck approach (see also the discussion of the next section). An illuminating approach of this kind can be found in a paper by Zwanzig and Bixon, which has also to be considered an earlier example of the continued fraction technique iq>plied to a non-Hermitian case. This method has also been fruitfully applied to the field of polymer dynamics. 

Basically, we shall apply the AEP, derived by the Zwanzig approach, to explore the long-time behavior of nonlinear stochastic processes, whereas the CFP derived from the Mori approach will be used as a calculation technique, which sometimes will prove useful for application to the reduced equations of motion provided by the AEP itself. 

As in the Zwanzig approach to nonequilibrium processes, we partition the vector pt) correspondtag to the density operator p t) of some system into relevant and irrelevant parts by using orthogonal projection operators. However, the projection operators P t) and Q t) used to accomplish this task are time dependent rather than time independent. Nonetheless, P t) and Q t) possess the usual properties P(0+6(0=/, Htf = P t 6(O = 0(O, and Pp)QiO = QiOPiO = 0 of orthogonal projection operators, where 6 is the null operator. 

In this section, we take an approach that is characteristic of conventional perturbation theories, which involves an expansion of a desired quantity in a series with respect to a small parameter. To see how this works, we start with (2.8). The problem of expressing ln(exp (—tX)) as a power series is well known in probability theory and statistics. Here, we will not provide the detailed derivation of this expression, which relies on the expansions of the exponential and logarithmic functions in Taylor series. Instead, the reader is referred to the seminal paper of Zwanzig [3], or one of many books on probability theory - see for instance [7], The basic idea of the derivation consists of inserting... 

An alternative approach involves integrating out the elastic degrees of freedom located above the top layer in the simulation.76 The elimination of the degrees of freedom can be done within the context of Kubo theory, or more precisely the Zwanzig formalism, which leads to effective (potentially time-dependent) interactions between the atoms in the top layer.77-80 These effective interactions include those mediated by the degrees of freedom that have been integrated out. For periodic solids, a description in reciprocal space decouples different wave vectors q at least as far as the static properties are concerned. This description in turn implies that the computational effort also remains in the order of L2 InL, provided that use is made of the fast Fourier transform for the transformation between real and reciprocal space. The description is exact for purely harmonic solids, so that one can mimic the static contact mechanics between a purely elastic lattice and a substrate with one single layer only.81... 

The goal of studying the quantum Zwanzig Hamiltonian is to generalize these results to the case when excitations to higher doublets are possible. This detail changes the problem completely since there is no small parameter for a perturbative approach. 

In this work we shall follow the Langevin equation approach and in the spirit of Zwanzig s work we shall start from the following Hamiltonian ... 

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 quantum mechanical forms of the correlation function expressions for transport coefficients are well known and may be derived by invoking linear response theory [64] or the Mori-Zwanzig projection operator formalism [66,67], However, we would like to evaluate transport properties for quantum-classical systems. We thus take the quantum mechanical expression for a transport coefficient as a starting point and then consider a limit where the dynamics is approximated by quantum-classical dynamics [68-70], The advantage of this approach is that the full quantum equilibrium structure can be retained. 

A more general approach can be found in paper of Bixon and Zwanzig (1978). 

This section has the twofold purpose of illustrating a popular approach to GME by means of a contracted Liouville equation. This is the approach proposed by Zwanzig [15,16]. We plan also to outline some technical difficulties with this... 

The approach proposed by Zwanzig [15,16] for a formal derivation of the master equation proceeds as follows. We assume the whole Universe, namely, the system of interest and its environment, to obey the Liouville equation... 

This is a plausible way to prove that Eq. (162) is the diffusion equation that applies to the gaussian condition. It is important to point out that a more satisfactory derivation of this exact result can be obtained by using the Zwanzig projection approach of Section III [67,68]. Thus, Eq. (162) as well as Eq. (133) must be considered as generalized diffusion equations compatible with a Liouville origin. 

This equality seems to suggest a total equivalence between CTRW, from which the third equation is derived, and the Liouville approach, based on density. We note that the first equation was originally derived from the Zwanzig GME. Therefore, all this seems to be in line with the claimed equivalence between CTRW and GME. 

The simplest and most elegant theoretical technique operating in line with this leading idea is the Nakajima-Zwanzig projection method. By using this approach we are naturally led to replace the standard master equations. 

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