Projection operators Mori-Zwanzig

In terms of the Zwanzig-Mori [282, 283] projection operator formalism the equation of motion for the dynamic structure factor is given by ... 

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. 

Another correlation function that often appears in quantum statistical mechanics is the Kubo transformed correlation function (cf. Zwanzig, 1965). This function can be related to <(A(0)A(t))>s so that it is unnecessary to define new scalar products and projection operators, although the Kubo transform itself can be fit into this context (cf. Mori, 1965). 

The standard methods of obtaining equations of motion for reduced systems are based on the projection-operator techniques developed by Zwanzig and Mori in the late 1950s [23,24]. In this approach one defines an operator, that acts on the full density matrix p to project out a direct product of cr and the thermalized equilibrium density matrix of the... 

Since its introduction in the f960s by Zwanzig and Mori [21, 22, 23], the memory-function formalism based on projection operators has pervaded many theoretical approaches dealing with the dynamics of strongly interacting systems. Indeed, the idea of describing a many-body system by a limited number of relevant variables characterized by a relatively simple dynamics appears to be extremely appealing. 

The review by Berne and Forste/ covers most of the important aj lications prior to 1971. Particular attention should be given to the seminal papers of Zwanzig and Mori cited therein. Since the writing of this review, most of the important applications involve mode-mode coupling theories. These subjects are covered in Chapters 4 and 6 of this volume. tThe application of projection operators to kinetic equations is presented in an excellent didactic... 

Zimm model 123,130,182,193 Zwanzig-Mori projection operator formalism 165... 

Due to complexity of the real world, all QDT descriptions involve practically certain approximations or models. As theoretical construction is concerned, the infiuence functional path integral formulation of QDT may by far be the best [4]. The main obstacle of path integral formulation is however its formidable numerical implementation to multilevel systems. Alternative approach to QDT formulation is the reduced Liouville equation for p t). The formally exact reduced Liouville equation can in principle be constructed via Nakajima-Zwanzig-Mori projection operator techniques [5-14], resulting in general two prescriptions. One is the so-called chronological ordering prescription (COP), characterized by a time-ordered memory dissipation superoperator 7(t, r) and read as... 

Dual Lanczos transformation theory is a projection operator approach to nonequilibrium processes that was developed by the author to handle very general spectral and temporal problems. Unlike Mori s memory function formalism, dual Lanczos transformation theory does not impose symmetry restrictions on the Liouville operator and thus applies to both reversible and irreversible systems. Moreover, it can be used to determine the time evolution of equilibrium autocorrelation functions and crosscorrelation functions (time correlation functions not describing self-correlations) and their spectral transforms for both classical and quantum systems. In addition, dual Lanczos transformation theory provides a number of tools for determining the temporal evolution of the averages of dynamical variables. Several years ago, it was demonstrated that the projection operator theories of Mori and Zwanzig represent special limiting cases of dual Lanczos transformation theory. 

The generalized Langevin equation given by Eq. (554) in our discussion of the Mori-Zwanzig projection operator formalism is an equation of motion for the vectors (5 corresponding to the fluctuations S Oj (t) =... 

If the system is close to thermal equilibrium, Eq. (580) assumes the form of the non-Maikovian equation of motion given by Eq. (548) in our discusssion of the Mori-Zwanzig projection operator formalism. 

In dual Lanczos transformation theory, orthogonal projection operators are used to decompose the dynamics of a system into relevant and irrelevant parts in the same spirit as in the Mori-Zwanzig projection operator formalism, but with a differing motivation and decomposition. More specifically, orthogonal projection operators are used to decompose the retarded or advanced dynamics into relevant and irrelevant parts that are completely decoupled. The subdynamics for each of these parts is an independent and closed subdynamics of the system dynamics. With this decomposition, we are able to completely discard the irrelevant information and focus our attention solely on the closed subdynamics of the relevant part. 

In this section we will first refer to the formalism originally introduced by Zwanzig and Bixon [96, 97], which was then applied to polymer dynamics by Schweizer [98, 99] and others [100-108]. This sort of theory is based on the Zwanzig/Mori projection operator technique in connection with treatments of the Generalized Langevin Equation [109-115]. It should be noted that this equation can be considered as the microscopic basis of phenomenological approaches based on the memory function formalism [116-122]. 

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