Spectral moment density function

Fig. 3.6. The spectral moment y as function of the product of densities, for various rare-gas mixtures at room temperature only one density was varied for each system the neon densities were fixed at 77, 31 and 46.5 amagats for the neon-argon, neon-krypton and neon-xenon mixtures, respectively and the krypton and xenon densities were fixed at 152 and 50 amagats, respectively, in their mixtures with argon. The departures from the straight lines seen at intermediate densities squared indicate the presence of many-body interactions. Reprinted with permission by Pergamon Press from [329].

Collision-induced absorption takes place by /c-body complexes of atoms, with k = 2,3,... Each of the resulting spectral components may perhaps be expected to show a characteristic variation ( Qk) with gas density q. It is, therefore, of interest to consider virial expansions of spectral moments of binary mixtures of monatomic gases, i.e., an expansion of the observed absorption in terms of powers of gas density [314], Van Kranendonk and associates [401, 403, 314] have argued that the virial expansion of the spectral moments is possible, because the induced dipole moments are short-ranged functions of the intermolecular separations, R, which decrease faster than R 3. We label the two components of a monatomic mixture a and b, and the atoms of species a and b are labeled 1, 2, N and 1, 2, N, respectively. A set of fc-body, irreducible dipole functions U 2, Us,..., Uk, is introduced (as in Eqs. 4.46), according to... 

Detailed balance. Classical line shapes are symmetric so that all classical, odd spectral moments M of the spectral function vanish. The odd moments of actual measurements are, however, non-vanishing because measured spectral density profiles satisfy the principle of detailed balance, Eq. 5.73. This problem of classical relationships may be largely alleviated by symmetrizing the measured profile prior to determining the moments, using the inverse Egelstaff procedure (P-4) discussed on p. 254 this generates a close approximation to the classical profile from the measurement and use of classical formulae is then justified. 

For some time it has been known that the spectral moments, which are static properties of the absorption spectra, may be written as a virial expansion in powers of density, q", so that the nth virial coefficient describes the n-body contributions (n = 2, 3. ..) [400]. That dynamical properties like the spectral density, J co), may also be expanded in terms of powers of density has been tacitly assumed by a number of authors who have reported low-density absorption spectra as a sum of two components proportional to q2 and q3, respectively [100, 99, 140]. It has recently been shown by Moraldi (1990) that the spectral components proportional to q2 and q3 may indeed be related to the two- and three-body dynamical processes, provided a condition on time is satisfied [318, 297]. The proof resorts to an extension of the static pair and triplet distribution functions to describe the time evolution of the initial configurations these permit an expansion in terms of powers of density that is analogous to that of the static distribution functions [135],... 

Starting from the assumption that the only known information concerning the line shape is a finite number of spectral moments, a quantity called information is computed from the probability for finding a given spectral component at a given frequency this quantity is then minimized. Alternatively, one may maximize the number of configurations of the various spectral components. This process yields an expression for the spectral density as function of frequency which contains a small number of parameters which are then related to the known spectral moments. If classical (i.e., Boltzmann) statistics are employed, information theory predicts a line shape of the form... 

Both Ti and T2 relaxations of water protons are mainly due to fluctuating dipole-dipole interactions between intra- and inter-molecular protons [62]. The fluctuating magnetic noise from all the magnetic moments in the sample (these moments are collectively tamed the lattice) includes a specific range of frequency components which depends on the rate of molecular motion. The molecular motion is usually represented by the correlation time, xc, i.e., the average lifetime staying in a certain state. A reciprocal of the correlation time corresponds to the relative frequency (or rate) of the molecular motion. The distribution of the motional frequencies is known as the spectral density function. 

According to the theory ofLipardi and Szabo (1982), values of the spin-lattice (1/Tj) and spin-spin (1/T2) relaxation rates are dependent on three important structural and dynamic parameters. The first parameter d is proportional to pjp/r3, where p and pj are magnetic moments of nuclei interacting through space, and r is the distance between the nuclei. The second parameter c is proportional to the anisotropy of the nuclear chemical shift. In the spin-lattice relaxation case, the third parameter is the spectral density function ... 

The quantitative interpretation of the data for systems such as these needs some care372 as the theoretical models seem to be ambiguous at the moment and the treatment of systems in the rigid lattice condition needs attention to the details of the spin physics if the data are not to be misinterpreted.373 However, the use of the technique considerably extends the possibilities for the use of proton NMR in food systems and should allow a more detailed description of dynamical processes and an improvement in comparative studies. An important advantage is that the temperature dependence of the spectral density function becomes a measurable variable, thus allowing a more detailed investigation of temperature effects. 

The family of the density-matrix spectral moments is defined as S = L"t] which are the expansion coefficients in the short-time evolution of the density-matrix response function. These moments are used to construct the main DSMA equations - °°... 

Where M2 is the second moment of the NMR lineshape, J the spectral density function, with (Dq the Larmor frequency, and (0i the frequency of the spin-locking field. The spectral density can be written in terms of the molecular correlation time, x, and the overall shape of the Tjp - temperature dispersion and the relatively shallow minima arc due to the correlation time distribution, although the location of the minimum is unaffected by this distribution. We have examined several models for the distribution, all of which give essentially the same results. One of the more simple is the Cole-Davidson function (75), which has also been applied to the analysis of dielectric relaxations. The relevant expression for the spectral density in this case is given by Equation 4. 

This frequency response functions enter the modified Priestley s formulation of Sec. 2, resulting in evolutionary output power spectral density functions for deflection and moments, respectively. We... 

The current section of the chapter on numerical methods is devoted to an outline of the most frequently used numerical methods for solving the population balance equation either for the particle number distribution function or for a few moments of the number density function. The methods considered are the standard method of moments, the quadrature method of moments (QMOM), the direct quadrature method of moments (DQMOM), the sectional quadrature method of moments (SQMOM), the discrete fixed pivot method, the finite volume method, and the family of spectral weighted residual methods with emphasis on the least squares method. 

Cottone G, Di Paola M (2010) New representation of power spectral density function and correlation function by means of fractional spectral moments. Prob Eng Mech 25 348-353... 

In the last four decades, Eq. 8 has been used by several authors to define the spectrum-compatible power-spectral density function. The methods proposed in literature mainly differentiate one from another for the hypothesis adopted to define the peak factor and for the approximations involved in the evaluation of the response spectral moments. 

Another contribution has been provided by Kaul (1978) in order to provide a simplified expression of the power-spectral density function compatible with a given response spectmm, the author adopted the peak factor in Eq. 11 in conjunction with the hypothesis that the response spectral moments are determined under the hypothesis of white noise input process. Therefore, Nu is determined according to Eq. 15 and the zero-order response spectral moments is given as follows ... 

The approximate solution given in Eq. 30 provides accurate results for low and intermediate frequencies. Therefore the author considered the solution proposed by Vanmarcke and Gasparini (1977) given by Eq. 10 with the adjusted spread factor as further improvement of Eq. 19. As the peak factor evaluation depends on the whole power-spectral density function, which is intrinsically related to the spectral characteristics of the unknown response process (see Eqs. 9, 13, 14, and 20), the author developed an algorithm which enables to update the spectral characteristics of the response process every frequency step of definition of the frequency domain. The iterative scheme mainly differs from the procedure proposed by Sundararajan (1980) as not only the zeroth-order spectral moment is updated at each step but also the peak factor. To this aim the... 

Ideally by Eqs. 54, 55, and 56 and the definition of the nonstationary spectral moments of the response, the evolutionary power-spectral density function pertinent to the codes prescriptions could be derivable. However this approach not yet attempted, to the best knowledge of the authors, it could be computationally burdensome, the convergence is not assured, and moreover it may lead to physically unacceptable results therefore alternative approaches are usually preferred. 

Evolutionary frequency response function Evolutionary power spectral density function Gaussian zero-mean random models of seismic accelerations Non-geometric spectral moments Stochastic analysis... 

The main steps of the described approach are (i) the use of modal analysis to decouple the equation of motion (ii) the determination, in state variable, of the evolutionary frequency response vector functions and of the evolutionary power spectral density function matrix of the structural response and (iii) the evaluatiOTi of the nongeometric spectral moments as weU as the spectral characteristics of the stochastic response of linear systems subjected to stationary... 

Nayak [11] proposed the parameters mo, m2 and m4, spectral moments of the profile Power Spectral Density, PSD, function, to determine a large number of useful statistics for isotropic, gaussian surfaces. The PSD function is the Fourier transform of the profile autocorrelation function. According to McCool [5], these spectral moments may be calculated, for isotropic surfaces, by the expressions ... 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

The approach to calculating spectral densities which we shall describe in the present work is based on the rigorous equations of motion, and should therefore be classed as dynamical, rather than stochastic. However, we do not attempt a direct solution to the equations of motion. Rather we take the point of view that a spectral density is some function of unknown form, about which we can calculate certain features directly from the equations of motion. These simple features of the spectral density are its moments, averages of powers of the frequency, over the spectral density, which are discussed further in Section II. 

The most remarkable feature of these average properties is that they are determined to within rigorous error bounds just by the knowledge of the general properties of the spectral densities discussed in Section II. That is, if we calculate a certain finite number of moments of a spectral density, then averages such as Eqs. (14) (18) must lie between certain calculable limits, no matter what (positive) functional form 1(a) actually has, as long as 1(a) has the specified moments. 

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