Mori theory

The well-known Mori theory can be interpreted as the choice of the best basis set /o>, /x), I/2), for expanding the operator F. Since we are interested in describing the time evolution of the state 1 4), it is evident that this state has to be included in this basis set. Therefore,... 

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... 

This agrees with the general structure allowed by the Mori theory. In other words, we are using the first two states of the Mori basis set defined by Eqs (3.8) to (3.19). Note, however, that whereas the standard Mori theory results in... 

As stressed in refs. 7 and 28b, the standard Mori theory is deeply different in natiu-e from the approach we have developed, which includes in the set of relevant variables both the variable /q (in the example under discussion, the velodty i>) and a few Mori auxiliary variables (in this example only one auxiliary variable, / => w). The auxiliary variable w plays the double role of variable of interest (being coupled to an irrelevant thermal bath and undergoing the influence of a standard fluctuation-dissipation process) and that of simulating the thermal bath of the real variable of interest. This makes it possible to study exdtation and preparation processes within the framework of the Mori theory, the range of validity of which is, however, limited to the case of linear systems. 

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. 

This disagreement can be accounted for by noticing that the rate provided by the Mori theory is determined from the value of the area below the curve whereas the AEP is equivalent to fitting with an exponential decay the long-time decay behavior of (see Fig. 2). The exact time revolution of 9 (t) can be obtained by means of the diagonalization of the matrix A given... 

The use of a basis set reminiscent of that provided by Mori theory can be found in the work of Baram. ... 

This basis is obtained by the generalized Mori theory (see Chapter I) without requiring iT to be Hemdtian. This leads us to show that the th-order correlation function... 

This expression is precisely analogous to that obtained by the original Mori theory, in which the Hermitian nature of F makes purely imaginaiy and purely real. In the present approach, both X and A are complex quantities, resulting in faster convergence than in ref. 33, where only the last step of the chain has a complex X introduced to simulate (i.e., take account of) the rest of the continued fraction. The real part of X represent the dissipative difTusional terms. 

We have thus shown that (z) can be expressed in a continued fraction form, whose expression can be given in terms of the moments s . The only theoretical tool we used to arrive at this important result is the generalized version of the celebrated Mori theory. We now have the problem of evaluating the parameters s to obtain the spectra of interest to EPR spectroscopy. This can be done as follows. First, let us define the nth-order state... 

It appears clear from Chapters I, III, and IV that the Mori theory is the major theoretical tool behind the algorithm illustrated in Section II, which derives the expansion parameters X, and from the moments s . This theory also affords us with a second straightforward way of determining these parameters that of deriving them directly from the biorthogonal basis set of states fi) and j/-) (Eqs. 2.15). As discussed at length in Chapter IV, this is an especially stable way of building up X, and A. The Lanczos method fol-... 

Given this background we may now turn to Onsager s theory [1,3,4], including its entension to finite memory due to Mori [5]. We both develop some of the basic equations of the Onsager-Mori theory, and also critically evaluate these equations in light of the physical assumptions which underlie them. 

The parameters Xi2> X2> Xa etc. have constant values for a given temperature. Since x depends on concentration, its value at infinite dilution obtained by GLC will usually differ from that obtained for other concentrations by conventional methods. The GLC value of x <1 (5.13) and (5.14) is in fact the entire non-combinatorial contribution to In ax at infinite dilution and is independent of any assumed theoretical form of In aj. However, the value, of In a" depends on the model chosen to estimate the configurational entropy. As an alternative to the expression given by the Tlory-Huggins theory, which leads to eqn (5.14), the Mory theory of equations of state can be used for Xjg [12,15—21] (see Section 3.2.6) according to this theory, the volume fractions segment fractions O, and the specific volumes by the specific core volumes v. The new volumes v correspond to the specific volumes of the close-packed hypothetical liquids at 0 K and are independent of... 

As a result of the transformation from the original Hamiltonian equations 15.1-15.4 to the effective-mode Hamiltonian equations 15.6-15.11, the spectral density has to be re-written in terms of the transformed quantities. As shown in Ref. [32], J (o) then takes a continued fraction form which is close to the results obtained in Mori theory [29-31] or the Rubin model [12,46]. 

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