Dissipation operator correlation function

The present reduced density operator treatment allows for a general description of fluctuation and dissipation phenomena in an extended atomic system displaying both fast and slow motions, for a general case where the medium is evolving over time. It involves transient time-correlation functions of an active medium where its density operator depends on time. The treatment is based on a partition of the total system into coupled primary and secondary regions each with both electronic and atomic degrees of freedom, and can therefore be applied to many-atom systems as they arise in adsorbates or biomolecular systems. 

Another situation in which error bounds can be provided is in the calculation of a correlation function of a piuely dissipative operator (eigenvalues = —1 , > 0, on the negative imaginary axis), earlier discussed in this... 

Thus for the calculation of error bounds in the correlation function associated to a pure dissipative operator, we can exploit the properties summarized by Eq. (8.11). 

Clearly, a general theory able to naturally include other solvent modes in order to simulate a dissipative solute dynamics is still lacking. Our aim is not so ambitious, and we believe that an effective working theory, based on a self-consistent set of hypotheses of microscopic nature is still far off. Nevertheless, a mesoscopic approach in which one is not limited to the one-body model, can be very fruitful in providing a fairly accurate description of the experimental data, provided that a clever choice of the reduced set of coordinates is made, and careful analytical and computational treatments of the improved model are attained. In this paper, it is our purpose to consider a description of rotational relaxation in the formal context of a many-body Fokker-Planck-Kramers equation (MFPKE). We shall devote Section I to the analysis of the formal properties of multivariate FPK operators, with particular emphasis on systematic procedures to eliminate the non-essential parts of the collective modes in order to obtain manageable models. Detailed computation of correlation functions is reserved for Section II. A preliminary account of our approach has recently been presented in two Letters which address the specific questions of (1) the Hubbard-Einstein relation in a mesoscopic context [39] and (2) bifurcations in the rotational relaxation of viscous liquids [40]. 

Here the W are operators of the subsystem and the superscript dagger denote the Hermitian conjugate. The Redfield equation can be written in this form only when an additional symmetrization of the bath correlation functions is performed [48]. Note that this alternative equation also expresses the dissipative evolution of the density matrix in terms of N x N... 

The CODDE formulation of CS-QDT [Eqs. (2.21)] couples between p t) and a set of auxiliary operators K t) m > 0 that describe the effects of correlated driving and dissipation. The field-free dissipation action, TZs [Eq. (2.18)], can be evaluated relatively easily in terms of the causality spectral function Cab ( ) without going through the parameterization procedure of Eq. (2.24). The latter is required only for the correlated driving-dissipation effects described by the auxiliary operators. Methods of evaluating both the reduced dynamics p(t) and the reduced canonical density operator Peq(T) will be discussed in Sec. 3. 

We now focus on the comparison between the CODDE [Eq. (2.21)] and the POP-CS-QDT [Eqs. (2.17), (B.4) and (B.6)]. These two formulations share the identical long time and thermal equilibrium behaviors characterized by their common field-free dissipation superoperator TZsj but differ at their correlated driving-dissipation dynamics. With the parameterization expressions for the bath spectral functions (Sec. 2.3) the correlated driving-dissipation dynamics effects may be numerically studied in terms of equations of motion via a set of auxiliary operators, which are Ko, i i, in the CODDE [Eq. (2.21)], and o, in the POP-... 

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