Spectral function ensemble averaging

Finding the values of G allows the determination of the frequency-domain spectrum. The power-spectrum function, which may be closely approximated by a constant times the square of G f), is used to determine the amount of power in each frequency spectrum component. The function that results is a positive real quantity and has units of volts squared. From the power spectra, broadband noise may be attenuated so that primary spectral components may be identified. This attenuation is done by a digital process of ensemble averaging, which is a point-by-point average of a squared-spectra set. 

By assuming an Arrhenius type temperature relation for both the diffusional jumps and r, we can use the asymptotic behavior of /(to) and T, as a function of temperature to determine the activation energy of motion (an example is given in the next section). We furthermore note that the interpretation of an NMR experiment in terms of diffusional motion requires the assumption of a defined microscopic model of atomic motion (migration) in order to obtain the correct relationships between the ensemble average of the molecular motion of the nuclear magnetic dipoles and both the spectral density and the spin-lattice relaxation time Tt. There are other relaxation times, such as the spin-spin relaxation time T2, which describes the... 

Unlike Section V.E, now we consider directly the spectral function of an isotropic spatial ensemble, for which an average over the phase variable h and l should be found. In this case L z) = K (z) + 2K (z). Omitting again the first term, we have in view40 of Eqs. (3.62) and (3.71) in GT ... 

Noting that the spectral density of the ensemble-averaged velocity auto-correlation function is the diffusion tensor... 

An approach based on the sequential use of Monte Carlo simulation and Quantum Mechanics is suggested for the treatment of solvent effects with special attention to solvatochromic shifts. The basic idea is to treat the solute, the solvent and its interaction by quantum mechanics. This is a totally discrete model that avoids the use of a dielectric continuum. Statistical analysis is used to obtain uncorrelated structures. The radial distribution function is used to determine the solvation shells. Quantum mechanical calculations are then performed in supermolecular structures and the spectral shifts are obtained using ensemble average. Attention is also given to the case of specific hydrogen bond between the solute and solvent. 

This equation was used to calculate the absorption spectra as a function of the parameters a=Cl /fi= l/z and D=a where is the relaxation time of the ensemble average of and (jx y=k T ntCl. Examines of spectral shapes computed in this way are shown in Figure 6.12 for a fixed value of D and a range of values of ot. From Equation (6.17) it can be seen that for very smaD values of a, a Lorentzian spectrum of linewidth F+2k D, as expected for a freely diffusing particle, is obtained. For large a (a/F=QV F> 1)... 

The subscript i refers to a particular particle and the brackets represent an ensemble average. To calculate many spectral properties in a fluid it is necessary to compute collective correlation functions. 

In addition to corrections tising an appropriate spectral density function, in principle one also needs to consider an ensemble of structures. Bonvin et al. U993) used an ensemble iterative relaxation matrix approach in which the NOE is measured as an ensemble property. A relaxation matrix is built from an ensemble of structures, using averaging of contributions from different structures. The needed order parameters for fast motions were obtained fi um a 50-ps molecular dynamics calculation. The relaxation matrix is then used to refine individual structures. The new structures are used again to reconstruct the relaxation matrix, and a second new set of structures is defined. One repeats the process until the ensemble of structures is converged. The caveat espressed earlier that the accuracy of the result is limited by the accuracy of the spectral density function applies to all calculations of this typ . 

These functions play a central role in transport theory and in the theory of spectral line shapes.They measure the correlation between A (t) and B (0). The brackets ( ) in Eq. (11) indicate an average over any of the equilibrium ensembles. To be specific we shall use the canonical ensemble so that... 

The laser intensities required to exploit gaseous nonllnearltles in a short time period dictate employment of pulsed lasers. These are typically frequency-doubled neodymiumrYAG lasers at 532 nm which are ideally spectrally situated for CARS work from both a dye laser pumping and optical detection standpoint. These lasers operate at repetition rates in the 20-50 Hz range. The combustion medium cannot be followed in real time, but is statistically sampled by an ensemble of single shot measurements which form a probability distribution function (pdf). From the pdf, the parameter time average can be ascertained as well as the magnitude of the turbulent fluctuations. 

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