Dynamical correlation, centroid density

One significant feature of Eq. (3.2) is the factorization of the expression into the centroid density (i.e., the centroid statistical distribution) and the dynamical part, which depends on the centroid frequency u>. It is not obvious that such a factorization should occur in general. For example, a rather different factorization occurs when the conventional formalism for computing time correlation functions is used [i.e., a double integration in terms of the off-diagonal elements of the thermal density matrix and the Heisenberg operator q t) is obtained]. This result sheds light on the dynamieal role of the centroid variable in real-time correlation functions (cf. Section III.B) [4,8]. 

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. 

Abstract The theoretical basis for the quantum time evolution of path integral centroid variables is described, as weU as the motivation for using these variables to study condensed phase quantum dynamics. The equihbrium centroid distribution is shown to be a well-defined distribution function in the canonical ensemble. A quantum mechanical quasi-density operator (QDO) can then be associated with each value of the distribution so that, upon the application of rigorous quantum mechanics, it can be used to provide an exact definition of both static and dynamical centroid variables. Various properties of the dynamical centroid variables can thus be defined and explored. Importantly, this perspective shows that the centroid constraint on the imaginary time paths introduces a non-stationarity in the equihbrium ensemble. This, in turn, can be proven to yield information on the correlations of spontaneous dynamical fluctuations. This exact formalism also leads to a derivation of Centroid Molecular Dynamics, as well as the basis for systematic improvements of that theory. 

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