Spectral density/

Goldfisher, Autocorrelation function and power spectral density of laser-produced speckle pattern . J. Opt. Soc. Am., vol.55, p.247(1965). 

The quantity introduced above is the spectral density defined as the energy per unit volume per unit frequency range and is... 

This is known as the Planck radiation law. Figure A2.2.3 shows this spectral density fiinction. The surface temperature of a hot body such as a star can be estimated by approximating it by a black body and measuring the frequency at which the maximum emission of radiant energy occurs. It can be shown that the maximum of the Planck spectral density occurs at 2.82. So a measurement of yields an estimate of the... 

Figure A2.2.3. Planck spectral density fimction as a fimction of the dimensionless frequency /)oi/(/rj 7). A2.2.4.7 APPLICATION TO IDEAL SYSTEMS ELASTIC WAVES IN A SOLID...

Ulness D J and Albrecht A C 1996 Four-wave mixing in a Bloch two-level system with incoherent laser light having a Lorentzian spectral density analytic solution and a diagrammatic approach Rhys. Rev. A 53 1081-95... 

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... 

In order to obtain a more realistic description of reorientational motion of intemuclear axes in real molecules in solution, many improvements of the tcf of equation Bl.13.11 have been proposed [6]. Some of these models are characterized in table Bl.13.1. The entry number of tenns refers to the number of exponential fiinctions in the relevant tcf or, correspondingly, the number of Lorentzian temis in the spectral density fiinction. 

Table Bl.13.1 Selected dynamic models used to calculate spectral densities.

McCain D 0 and Markley J L 1986 Rotational spectral density functions for aqueous sucrose ... 

This discussion suggests that even the reference trajectories used by symplectic integrators such as Verlet may not be sufficiently accurate in this more rigorous sense. They are quite reasonable, however, if one requires, for example, that trajectories capture the spectral densities associated with the fastest motions in accord to the governing model [13, 15]. Furthermore, other approaches, including nonsymplectic integrators and trajectories based on stochastic differential equations, can also be suitable in this case when carefully formulated. 

Since the stochastic Langevin force mimics collisions among solvent molecules and the biomolecule (the solute), the characteristic vibrational frequencies of a molecule in vacuum are dampened. In particular, the low-frequency vibrational modes are overdamped, and various correlation functions are smoothed (see Case [35] for a review and further references). The magnitude of such disturbances with respect to Newtonian behavior depends on 7, as can be seen from Fig. 8 showing computed spectral densities of the protein BPTI for three 7 values. Overall, this effect can certainly alter the dynamics of a system, and it remains to study these consequences in connection with biomolecular dynamics. 

BPTI spectral densities Cosine Fourier transforms of the velocity autocorrelation function... 

Fig. 8. Spectral densities for BPTI as computed by cosine Fourier transforms of the velocity autocorrelation function by Verlet (7 = 0) and LN (7 = 5 and 20 ps ). Data are from [88].

Another view of this theme was our analysis of spectral densities. A comparison of LN spectral densities, as computed for BPTI and lysozyme from cosine Fourier transforms of the velocity autocorrelation functions, revealed excellent agreement between LN and the explicit Langevin trajectories (see Fig, 5 in [88]). Here we only compare the spectral densities for different 7 Fig. 8 shows that the Langevin patterns become closer to the Verlet densities (7 = 0) as 7 in the Langevin integrator (be it BBK or LN) is decreased. 

Blackbody Emittance. Representative blackbody emittance (9,10), calculated as a power spectral density, is shown in Figure 2. The wavelength, X, of peak power density for a blackbody at temperature, T, is given by Wien s displacement law ... 

The generation of photons obeys Poisson statistics where the variance is N and the deviation or noise is. The noise spectral density, N/, is obtained by a Fourier transform of the deviation yielding the following at sampling frequency,... 

Moreover, in this linear-response (weak-coupling) limit any reservoir may be thought of as an infinite number of oscillators qj with an appropriately chosen spectral density, each coupled linearly in qj to the particle coordinates. The coordinates qj may not have a direct physical sense they may be just unobservable variables whose role is to provide the correct response properties of the reservoir. In a chemical reaction the role of a particle is played by the reaction complex, which itself includes many degrees of freedom. Therefore the separation of reservoir and particle does not suffice to make the problem manageable, and a subsequent reduction of the internal degrees of freedom in the reaction complex is required. The possible ways to arrive at such a reduction are summarized in table 1. 

Other spectral densities correspond to memory effects in the generalized Langevin equation, which will be considered in section 5. It is the equivalence between the friction force and the influence of the oscillator bath that allows one to extend (2.21) to the quantum region there the friction coefficient rj and f t) are related by the fluctuation-dissipation theorem (FDT),... 

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... 

We just quote the main results of [Leggett et al. 1987], which cover most of the possible situations. The spectral density is taken in the form... 

The interdoublet transitions may prevail over the intradoublet ones if the spectral density J a>) grows with oj faster than... 

Two most appealing features of this model draw so much attention to it. First, although microscopically one has very little information about the parameters entering into (5.24), it is known [Caldeira and Leggett 1983] that when the bath responds linearly to the particle motion, the operators q and p satisfying (5.24) can always be constructed, and the only quantity entering into the various observables obtained from the model (5.24) is the spectral density... 

The formal structure of (5.77) suggests that the reaction coordinate Q can be combined with the bath coordinates to form a new fictitious bath , so that the Hamiltonian takes the standard form of dissipative TLS (5.55). Suppose that the original spectrum of the bath is ohmic, with friction coefficient q. Then diagonalization of the total system (Q, qj ) gives the new effective spectral density [Garg et al. 1985]... 

If the surface (i.e., the best fit plane) is in the x—y plane, and Z x,y) is the surface height variation (surface roughness) relative to that plane, the power spectral density is given by... 

Although the power spectral density contains information about the surface roughness, it is often convenient to describe the surface roughness in terms of a single number or quantity. The most commonly used surface-finish parameter is the root-mean-squared (rms) roughness a. The rms roughness is given in terms of the instrument s band width and modulation transfer function, M(p, q) as... 

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