Spectral density classical limits

The equilibrium FDT is usually written in a form that involves frequency-dependent quantities such as generalized susceptibilities and spectral densities [30-35]. We show below how this theorem can be formulated in the time domain. Our arguments do not reduce to a simple Fourier transformation of the usual frequency-domain formulation. Instead they are developed from the very beginning in the time domain, and they use only the various time-dependent quantities entering into play. The corresponding formulations of the FDT, which are established in the whole range of temperatures, allow in particular for a discussion of both the classical limit and the zero-temperature case [36]. 

An alternative way to obtain the spectral density is by numerical simulation. It is possible, at least in principle, to include the intramolecular modes in this case, although it is rarely done [198]. A standard approach [33-36,41] utilizes molecular dynamics (MD) trajectories to compute the classical real time correlation function of the reaction coordinate from which the spectral density is calculated by the cosine transformation [classical limit of Eq. (9.3)]. The correspondence between the quantum and the classical densities of states via J(co) is a key for the evaluation of the quantum rate constant, that is, one can use the quantum expression for /Cj2 with the classically computed J(co). This is true only for a purely harmonic system [199]. Real solvent modes are anharmonic, although the response may well be linear. The spectral density of the harmonic system is temperature independent. For real nonlinear systems, J co) can strongly depend on temperature [200]. Thus, in a classical simulation one cannot assess equilibrium quantum populations correctly, which may result in serious errors in the computed high-frequency part of the spectrum. Song and Marcus [37] compared the results of several simulations for water available at that time in the literature [34,201] with experimental data [190]. The comparison was not in favor of those simulations. In particular, they failed to predict... 

Here, the solvent frequency scale is approximated to be cojl in the Ohmie spectral density. In the nonadiabatic limit (g l) the Zusman equation (eqn (12.41)) prediets the classical FGR ... 

In the present contribution, the effective-mode decomposition of spectral densities is demonstrated for the case of oligomers of poly-phenylene-vinylene (PPV) type. Here, the relevant spectral density is constructed from a classical-statistical correlation function that is in turn obtained from molecular dynamics (MD) simulations. Together with the effective-mode decomposition, this provides a practicable procedure for characterizing and reducing spectral densities within the high-temperature limit. 

In this entry only two problems have been presented representation of the power spectral density (PSD) and correlation by means of complex fractional moments and filter equations. Regarding the first problem, it has been shown that both PSD and correlation may be represented by the so-called fractional spectral moments (FSMs). The latter are the extension of the classical spectral moments introduced by Vanmarcke (1972) to complex order. The appealing in using these FSM is that they are able to reconstruct both PSD and correlation. As a result, it may be stated that FSM functiOTi is another equivalent representation of PSD and correlation. Moreover it has been shown that with a limited number of informations (few FSM), the whole PSD and correlation may be restored, including the trend at infinity. Extension to multivariate seismic process has been also provided in both frequency... 

The most sophisticated deterministic dynamic analysis of structures requires that the load should be applied in time domain. This is one of the major challenges in the reliability analysis for seismic loading. The classical random vibration-based approaches were used in the past for this purpose however, they did not provide information acceptable to the deterministic community. The classical random vibration-based approaches have numerous limitations including the loads which are applied in the form of power spectral density functions essentially appropriate for linear structural behavior, the uncertainty in the linear or nonlinear structural behavior which may need to be incorporated in approximate ways, several performance-enhancing features currently introduced in structures which cannot be incorporated in appropriate ways, etc. The most severe weakness is that the seismic loading cannot be applied in time domain. 

In the case of coherent laser light, the pulses are characterized by well-defined phase relationships and slowly varying amplitudes (Haken, 1970). Such quasi-classical light pulses have spectral and temporal distributions that are also strictly related by a Fourier transformation, and are hence usually refered to as Fourier-transform-limited. They are required in the typical experiments of coherent optical spectroscopy, such as optical nutation, free induction decay, or photon echoes (Brewer, 1977). Here, the theoretical treatments generally adopt a semiclassical procedure, using a density matrix or Bloch formalism to describe the molecular system subject to a pulsed or continuous classical optical field, which generates a macroscopic sample polarization. In principle, a fully quantal description is possible if one represents the state of the field by the coherent or quasi-classical state vectors (Glauber, 1965 Freed and Villaeys, 1978). For our purpose, however. 

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