Spectral function statistical distributions

Expressions for the medium modifications of the cluster distribution functions can be derived in a quantum statistical approach to the few-body states, starting from a Hamiltonian describing the nucleon-nucleon interaction by the potential V"(12, l/2/) (1 denoting momentum, spin and isospin). We first discuss the two-particle correlations which have been considered extensively in the literature [5,7], Results for different quantities such as the spectral function, the deuteron binding energy and wave function as well as the two-nucleon scattering phase shifts in the isospin singlet and triplet channel have been evaluated for different temperatures and densities. The composition as well as the phase instability was calculated. 

The relationship between spectroscopic and statistical functions has been exploited for a variety of phenomena related in different ways to the dynamical response of the medium. We cite as examples spectral line broadening, photon echo spectroscopy and phenomena related to TDFSS we are examining here. A variety of methods are used for these studies and we add here methods based on ab initio CS. The basic model is actually the same for all the methods in use ab initio CS has the feature, not yet implemented in other methods, of using a detailed QM description of the solute properties, allowing a description of effects due to specificities of the solute charge distribution. 

Plotting the ECDF versus TCDF, for a given probability, generates the linear version of a cumulative distribution function, a quantile-quantile (QQ) plot. Each cumulative probability value yields a pair of order statistics (one from each CDF) that form a point in the QQ plot. Quantile-quantile plots are valuable tools for distinguishing differences in shape, size, and location between spectral clusters. Two similar clusters of spectra will demonstrate a linear QQ plot (Fig. 1). Breaks and/ or curves in the QQ plot indicate that differences exist between the groups (Fig. 2). 

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. 

The Bayesian fast Fourier transform approach uses the statistical properties of discrete Fourier transforms, instead of the spectral density estimators, to construct the likelihood function and the updated PDF of the model parameters [292]. It does not rely on the approximation of the Wishart distributed spectrum. Expressions of the covariance matrix of the real and imaginary parts of the discrete Fourier transform were given. The only approximation was made on the independency of the discrete Fourier transforms at different frequencies. Therefore, the Bayesian fast Fourier transform approach is more accurate than the spectral density approach in the sense that one of the two approximations in the latter is released. However, since the fast Fourier transform approach considers the real and imaginary parts of the discrete Fourier transform, the corresponding covariance matrices are 2No x 2Nq, instead of No x No in the spectral density approach. Therefore, the spectral density approach is computationally more efficient than the fast Fourier transform approach. 

For a more explicit discussion of such a representation, we shall be guided by the physical view of the light beam as a statistical ensemble of wave trains emitted by the macroscopic source at some time and with a spectral distribution function which are both random quantities characterized by classical probability distribution functions. The individual wave packets have state vectors similar to (57) ... 

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. 

While service loading of offshore structures results from a variety of sources, as mentioned above, the primary contributor is generally wave action. Consequently, the input spectral density function is narrow banded and in the short term the process is statistically stationary with peaks conforming to a Raleigh distribution and with Individual cycles being identifiable. Several recent experimental programs have been performed based upon such a spectrum [8-10], and one of these has reported values for D which range from 0.5 to 2.0 [8]. 

Peterson dealt with the non-stationarity of the nonetheless random processes underlying the background motion of the earth by simply discarding time segments containing earthquakes or noise bursts. More recent models of ground motion spectra deal with the problem of non-stationarity in a more sophisticated way, using statistical approaches which evaluate the probability distribution of the power spectral density (PSD) as a function of frequency. 

---