The spectral density

The spectral density, Eqs (6.90) and (6.92) is seen to be a weighted density of modes that includes as weights the coupling strengths c lco). The harmonic frequencies atj 

This is a coarse-graining procedure that is valid if the spacing between frequencies a j is much smaller than the inverse time resolution of any conceivable observation. See Section 1.4.4 and Eq. (1.182). 

The spectral density (see also Sections (7-5.2) and (8-2.5)) plays a prominent role in models of thermal relaxation that use harmonic oscillators description of the thermal environment and where the system-bath coupling is taken linear in the bath coordinates and/or momenta. We will see (an explicit example is given in Section 8.2.5) that /(co) characterizes the dynamics of the thermal environment as seen by the relaxing system, and consequently determines the relaxation behavior of the system itself. Two simple models for this function are often used  

The Drude (sometimes called Debye) spectral density 

Problem 6.7. In Section 8.2.5 we will encounter the memory function Z(/) = (2/7rzz ) c/co(7(co)/co) cos(coZ). Calculate the memory functions associated with the Ohmic and the Drude models and compare their time evolutions. 

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

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. 

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. 

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. 

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 a posteriori probability P[A G] for having the spectral density A(uj), given the simulation data G, is... 

Among the dynamical properties the ones most frequently studied are the lateral diffusion coefficient for water motion parallel to the interface, re-orientational motion near the interface, and the residence time of water molecules near the interface. Occasionally the single particle dynamics is further analyzed on the basis of the spectral densities of motion. Benjamin studied the dynamics of ion transfer across liquid/liquid interfaces and calculated the parameters of a kinetic model for these processes [10]. Reaction rate constants for electron transfer reactions were also derived for electron transfer reactions [11-19]. More recently, systematic studies were performed concerning water and ion transport through cylindrical pores [20-24] and water mobility in disordered polymers [25,26]. 

Here, the J terms are the spectral densities with the resonance frequencies co of the and nuclei, respectively. It is now necessary to find an appropriate spectral density to describe the reorientational motions properly (cf [6, 7]). The simplest spectral density commonly used for interpretation of NMR relaxation data is the one introduced by Bloembergen, Purcell, and Pound [8]. 

Another way to describe deviations from the simple BPP spectral density is the so-called model-free approach of Lipari and Szabo [10]. This takes account of the reduction of the spectral density usually observed in NMR relaxation experiments. Although the model-free approach was first applied mainly to the interpretation of relaxation data of macromolecules, it is now also used for fast internal dynamics of small and middle-sized molecules. For very fast internal motions the spectral density is given by ... 

The measurement of correlation times in molten salts and ionic liquids has recently been reviewed [11] (for more recent references refer to Carper et al. [12]). We have measured the spin-lattice relaxation rates l/Tj and nuclear Overhauser factors p in temperature ranges in and outside the extreme narrowing region for the neat ionic liquid [BMIM][PFg], in order to observe the temperature dependence of the spectral density. Subsequently, the models for the description of the reorientation-al dynamics introduced in the theoretical section (Section 4.5.3) were fitted to the experimental relaxation data. The nuclei of the aliphatic chains can be assumed to relax only through the dipolar mechanism. This is in contrast to the aromatic nuclei, which can also relax to some extent through the chemical-shift anisotropy mechanism. The latter mechanism has to be taken into account to fit the models to the experimental relaxation data (cf [1] or [3] for more details). Preliminary results are shown in Figures 4.5-1 and 4.5-2, together with the curves for the fitted functions. 

Fig. 1.6. The spectral manifestation of intercollision anticorrelation. Curve I is the spectral density of the force F acting in liquid [45] (the accuracy of MD calculation is shown), curve II the same but in the approximation of independent collisions.

It is clear that the nature of the electromagnetic phenomena is the same for optics and radio wave, the only experimental differences being that radiowave photons are far below the spectral density of noise of actual detectors and that the temperature of the source is such that each mode is statistically populated by many photons in the radio wave domain whereas the probability of presence of photons is very small in the optical domain. 

The main source of noise of such a heterodyne detector is the photon noise that takes place at the splitting of the local oscillator. Quantum physicists see this noise as originating from vacuum fluctuation on the input arm. This gives directly the spectral density of noise at input hv/2. 

Wiener inverse-filter however yields, possibly, unphysical solution with negative values and ripples around sharp features (e.g. bright stars) as can be seen in Fig. 3b. Another drawback of Wiener inverse-filter is that spectral densities of noise and signal are usually unknown and must be guessed from the data. For instance, for white noise and assuming that the spectral density of object brightness distribution follows a simple parametric law, e.g. a power law, then ... 

In solids, as in atoms and molecules, the spectral density (k, co) contains much more information than simply the band peak position, i.e. the band dispersion. The... 

In the case of extreme narrowing in which fast isotropic molecular motions dominate as in the solution state, the spectral density is written by a single correlation time,... 

In Equation (5), we can first notice (i) the factor 1/r6 which makes the spectral density very sensitive to the interatomic distance, and (ii) the dynamical part which is the Fourier transform of a correlation function involving the Legendre polynomial. We shall denote this Fourier transform by (co) (we shall dub this quantity "normalized spectral density"). For calculating the relevant longitudinal relaxation rate, one has to take into account the transition probabilities in the energy diagram of a two-spin system. In the expression below, the first term corresponds to the double quantum (DQ) transition, the second term to single quantum (IQ) transitions and the third term to the zero quantum (ZQ) transition. 

The various symbols have the same meaning as before while the spectral density /csa(A),d(ffla) will be discussed in Section 4. For the moment, let us state that these cross-correlation rates can play a role only if the csa mechanism is important (i.e. for non-aliphatic carbons but certainly not for protons) and if measurements are performed at high... 

Elastomers are solids, even if they are soft. Their atoms have distinct mean positions, which enables one to use the well-established theory of solids to make some statements about their properties in the linear portion of the stress-strain relation. For example, in the theory of solids the Debye or macroscopic theory is made compatible with lattice dynamics by equating the spectral density of states calculated from either theory in the long wavelength limit. The relation between the two macroscopic parameters, Young s modulus and Poisson s ratio, and the microscopic parameters, atomic mass and force constant, is established by this procedure. The only differences between this theory and the one which may be applied to elastomers is that (i) the elastomer does not have crystallographic symmetry, and (ii) dissipation terms must be included in the equations of motion. 

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