Mori-Zwanzig formalism

Equation [213] can be equivalently obtained using the Mori-Zwanzig formalism. " It is also seen that, in contrast to LRT developed for shear flow in bulk fluids, the one presented here has two coefficients, (which is similar to the shear viscosity q) and, which has no parallel in bulk fluids. should be interpreted as an average location at which hydrodynamics is found to be nominally invalid. Note that although the surface may have corrugations in the X as well as the y direction, the corrugation in the x direction alone matters to the frictional force in the planar Couette geometry. 

Several recent approaches, including Bloch optical equations and Mori -Zwanzig formalism, have been proposed to derive equations for the vibrational spectra of reacting molecules We briefly summarize here... 

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. 

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

In this article the memory function formalism has been used to compute time-correlation functions. It has been shown that a number of seemingly disparate attempts to account for the dynamical behavior of time correlation functions, such as those of Zwanzig,33,34 Mori,42,43 and Martin,16 are... 

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. 

In applying the Zwanzig-Mori formalism we follow the following prescription. 

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 generalized Langevin equation and the memory function equation simplify considerably when the set Ai, Am relaxes much more slowly than all other properties. If all such slowly relaxing variables are included in the set Ai,..., Am, the set is called a good set of variables. At the outset it is important to note that there are no rules by which a good set of variables can be chosen. Generally this is a matter of one s intuition. It is, however, the crucial step in the application of the Zwanzig-Mori formalism to specific problems. 

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. 

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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