Kubo transform

The important step of identifying the explicit dynamical motivation for employing centroid variables has thus been accomplished. It has proven possible to formally define their time evolution ( trajectories ) and to establish that the time correlations ofthese trajectories are exactly related to the Kubo-transformed time correlation function in the case that the operator 6 is a linear function of position and momentum. (Note that A may be a general operator.) The generalization of this concept to the case of nonlinear operators B has also recently been accomplished, but this topic is more complicated so the reader is left to study that work if so desired. Furthermore, by a generalization of linear response theory it is also possible to extract certain observables such as rate constants even if the operator 6 is linear. 

Here we have introduced the notation, ( ), to indicate the Kubo transformed-correlation function. Rewriting the expression in the coordinate representation for the full system, Q = [Pg.401]

Here we have used the notation for the Kubo transform of an operator X defined by... 

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). 

An alternative justification for this relation is that C (l) is an approximation [4] to the Kubo-transformed position correlation function i (t) in Eq. (3.9) and that the relationship of the latter function to C(t) is always given by the expression in Eq. (3.13). 

In the early papers [4,8], the development of the CMD method was guided in part by the effective harmonic analysis and, in part, by physical reasoning. In Paper III, however, a mathematical justification of CMD was provided. In the latter analysis, it was shown that (1) CMD always yields a mathematically well-defined approximation to the quantum Kubo-transformed position or velocity correlation function, and (2) the equilibrium path centroid variable occupies an important role in the time correlation function because of the nature of the preaveraging procedure in CMD. Critical to the analysis of CMD and its justification was the phase-space centroid density formulation of Paper III, so that the momentum could be treated as an independent dynamical variable. The relationship between the centroid correlation function and the Kubo-transformed position correlation function was found to be unique if the centroid is taken as a dynamical variable. The analysis of Paper III will now be reviewed. For notational simplicity, the equations are restricted to a two-dimensional phase space, but they can readily be generalized. 

One can similarly expand the Kubo-transformed position correlation function, given by [5]... 

Upon inspection of the two expansions, it is seen that the centroid correlation function and the Kubo-transformed correlation function take a similar analytical form [cf. Eqs. (3.18) and (3.24)], the difference being between the terms and (if " ) in Eqs. (3.18) and (3.24),... 

Taking into account all of the preceding considerations and generalizing them to terms of higher order, the difference between the Kubo-transformed position correlation function [Eq. (3.22)] and the CMD correlation function [Eq. (3.16)] can be summed to give [5]... 

The CMD position correlation function C (t) has been shown to be an approximation to the exact Kubo-transformed position correlation function [5]. It is therefore relatively straightforward to obtain a CMD approximation for the quantum velocity and cross-correlation (position-velocity) functions. We first rewrite the Fourier relation in Eq. (3.13) as... 

Here (f) is the phase-space centroid trajectory which obeys the CMD equation of motion in Eq. (3.16), and the time-dependent Gaussian width matrix C O, q (f)) for the vector is given by the centroid-constrained correlation function matrix in Eq. (2.56) with the position centroid located at qc(0- As shown in the appendix of Paper II, the general centroid correlation function in Eq. (3.56) is an approximation to the Kubo-transformed version of the exact correlation function C g(t). Therefore, to calculate C g(t) one makes use of the Fourier relationship... 

In order to simplify the notation, we have made use of the where definition A = dXAQhX) of the Kubo transform... 

The vectors 8tOjp,) and (S/0/ appearing in Eqs. (559) and (560) correspond to the operators 8, Ojp(t) and 8tOk, where p(t) is the statistical density operator given by Eq. (341), and StOj and 8t0k are the operators defined by Eqs. (361) and (362), respectively. Unless stated otherwise, the tilde in the various expressions discussed here and in the remainder of our discussion of the maximum entropy approach to nonequilibrium processes should be interpreted as a generalized Kubo transform as defined by Eq. (363) or (450). As evident form Eq. (363), the generalized Kubo transform carries an additional time dependence. For the sake of notational simplicity, we have suppressed it. 

The generalized Kubo transform in 8,OiPi) 8t OiPf,) depends on t [ t ]. 

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