Spectral density continuous

In Fig. 4 we compare the adiabatic (dotted line) and the stabilized standard spectral densities (continuous line) for three values of the anharmonic coupling parameter and for the same damping parameter. Comparison shows that for a0 1, the adiabatic lineshapes are almost the same as those obtained by the exact approach. For aG = 1.5, this lineshape escapes from the exact one. That shows that for ac > 1, the adiabatic corrections becomes sensitive. However, it may be observed by inspection of the bottom spectra of Fig. 4, that if one takes for the adiabatic approach co0o = 165cm 1 and aG = 1.4, the adiabatic lineshape simulates sensitively the standard one obtained with go,, = 150 cm-1 and ac = 1.5. 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

More pertinent to the present topic is the indirect dissipation mechanism, when the reaction coordinate is coupled to one or several active modes which characterize the reaction complex and, in turn, are damped because of coupling to a continuous bath. The total effect of the active oscillators and bath may be represented by the effective spectral density For instance, in the... 

Cole and Davidson s continuous distribution of correlation times [9] has found broad application in the interpretation of relaxation data of viscous liquids and glassy solids. The corresponding spectral density is ... 

The first term in the bracket stands for the mean, and the second for the spectral density matrix. For the continuous formulation, the covariances for the model and observation errors are given as... 

To make further progress, it is standard practice to take this definition of the spectral density and replace it by a continuous form based on physical intuition. A form that is often used for the spectral density is a product of ohmic dissipation qco (which corresponds to Markovian dynamics) times an exponential cutoff (which reflects the fact that frequencies of the normal modes of a finite system have an upper cutoff) ... 

Often one expects on physical grounds that the spectral density for a problem should be a smooth, or at least continuous, function of frequency. However, the special spectral densities constructed in the previous section were very singular functions, sums of 5 functions (i.e., sharp lines). These special spectral densities were constructed uniquely to minimize or maximize various functionals of the spectral densities, subject to the constraints of having given values of a finite number of moments. One would like to be able to construct also some of the smoother spectral densities, which also have the correct moments, and thus will necessarily have functionals interpolating somewhere between the error limits. Of course such functions will not be unique unless other constraints are added. 

This approximate method has been generalized to cases other than a spectral density on a finite interval. For example, if a spectral density is known to exponentially fall off at Targe to, then the continued fraction coefficients grow asymptotically as n2 at large n. Then if enough moments are known so that one approaches this asymptotic behavior, a similar estimate of the spectral density can be made. 

It should be clear that the extrapolation methods suggested in this section do not have rigorous error bounds like those developed in Section III. However, the extrapolation methods do furnish estimates of the spectral density itself, rather than only averages of the spectral density. Furthermore, these estimates satisfy all known conditions on the spectral density discussed in Section II. They are (a) positive functions, (b) with correct moments, insofar as they are known, and (c) satisfy any known asymptotic behavior at the ends of the frequency intervals. In a number of test cases with known positive continuous functions with known asymptotic behavior, estimates generally were correct to within a few per cent, even when only a few (say 10) moments were given. A typical spectral density obtained in this way for a lattice vibration problem is plotted in Figure 4.34 The results are similar to those obtained numerically for the same problem, by solving a random sample of secular equations for the lattice vibrations.36... 

It is of interest here to compare one of these methods with Mori s continued fraction representation for spectral densities.40 Consider the case of Lorentzian broadening of the spectral density (which describes the response to a damped harmonic perturbation, Section III-A). If we set... 

The differential cross section of a transition into a continuous spectrum can be also expressed in terms of generalized oscillator strengths. However, in this case we must introduce the spectral density of generalized oscillator strengths df q) =/( , g)113 14 ... 

When studying the absorption of increasing photon energy by an atom or ion initially in a given bound state, to be gradually excited until it becomes ionised, and to have afterwards the free electron increase its kinetic energy, there is no discontinuity in the oscillator strength spectral density at the ionisation threshold. An adequate theoretical calculation must reproduce such continuity, which may also be exploited to interpolate a value for the threshold photoionisation cross section. 

Time- and space-resolved major component concentrations and temperature in a turbulent gas flow can be obtained by observation of Raman scattering from the gas. (1, 2) However, a continuous record of the fluctuations of these quantities is available only in those most favorable cases wherein high Raman scattering rate and/or slow rate of time variation of the gas allow many scattered photons (> 100) to be detected during a time resolution period which is sufficiently short to resolve the turbulent fluctuations. (2, 3 ) Fortunately, in other cases, time-resolved information still can be obtained in the forms of spectral densities, autocorrelation functions and probability density functions. (4 5j... 

The limiting case of a continuous distribution of the spectral component (Oj with a spectral density pia>j) gives, for the correlation function,... 

On approaching the transition point of a continuous-phase transition, a critical slowing down of the fluctuations occurs (t [a(T—Tc) + Dq2] x) and the amplitudes of the fluctuations increase (< q y [a(T— Tc) + Dq2] x) at wave vector q (Fig. 7a). It can be proved that apparent pseudodivergences of the relaxation rate (]/Tr)c (T Tc) 7 (with y = 1/2 in mean field theories) can be observed near a phase transition if Tqm <3 1 where cop is the probing frequency. In that case, the associated spectral density... 

Fig. 4. Reduced spectral densities)j versus cot for dipolar coupling continuous curve is translational diffusion and broken curve is rotational diffusion.

A noise that has a clearly distinct origin from noise discussed in previous sections is the electric noise that originates in modulation of ion transport by fluctuations in system conductance. These temporal fluctuations can be measured, at least in principle, even in systems at equilibrium. Such a measurement was conducted by Voss and Clark in continuous metal films (44). The idea of the Voss and Clark experiment was to measure low-frequency fluctuations of the mean-square Johnson noise of the object. In accordance with the Nyquist formula, fluctuations in the system conductance result in fluctuations in the spectral density of its equilibrium noise. Measurement of these fluctuations (that is, measurement of the noise of noise) yields information on conductance fluctuations of the system without the application of any external perturbations. The samples used in these experiments require rather large amplitude conductance fluctuations to be distinguished from Johnson noise fluctuations because of the intrinsic limitation of statistics. Voss and... 

We performed a calculation of the relaxation rates using the phonon Green s functions of the perfect (CsCdBr3) and locally perturbed (impurity dimer centers in CsCdBr3 Pr ) crystal lattices obtained in Ref. [8]. The formation of a dimer leads to a strong perturbation of the crystal lattice (mass defects in the three adjacent Cd sites and large changes of force constants). As it has been shown in Ref. [8], the local spectral density of phonon states essentially redistributes and several localized modes appear near the boundary of the continuous phonon spectrum of the... 

The total spectral density of quasi-equilibrium plasma emission in continuous spectrum consists of the bremsstrahlung and recombination components, which can be combined in one general expression ... 

Here, we briefly describe MFDA. Considering the usual situations where the trajectory is sampled with discrete time steps, we focus on discrete time and continuous frequency spaces. For simplicity, we set the sampling time interval of the trajectory data. At, to unity. We define the spectral density matrix S(f) as the inverse Fourier transform of a lagged variance-covariance matrix C(r) = (v(f)v (f+x)). 

We wish to consider random variables in a continuous domain. This can be done by using different descriptive functions. For our derivations we need the concepts of probability density function (PDF), autocorrelation function (ACF) and power spectral density (PSD). Moreover, the following system functions are used the weighting function h(t) and the complex frequency response H(ju)). 

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