Spectral density = fluctuation spectrum

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

Since the magnetic interactions between the nuclei are fluctuating in step with the random molecular motions, the frequencies of the magnetic interactions are related to the frequencies of the molecular motion. For any random motion there is a whole spectrum of frequencies, and the variation of the intensity of the fluctuations with frequency (referred to as the spectral density 7general form shown in Fig. 1. 

Thus from a time-average of Ato 2 we can obtain the spectral density Ia(coo), and by tuning the filter through different values of co0 we can determine the spectrum of the fluctuation A. From its definition we see that lim A Ato 2)>r, and correspondingly... 

The actual fluctuation spectrum in a solid sample is very complex with a broad distribution of correlation times. A number of widely used models for spectral density functions in solids is summarized in Beckman s review paper, where exponential correlation is assumed. However, it is possible that there is a nonexponential correlation case, as recently confirmed by Spiess group with 3D and 4D methods. The BPP model which is the simplest, is mostly applied both in solid and solution NMR. 

Flicker noise has long been a nuisance in measurements of the spectral density of conductance fluctuations in experiments with biological membranes. In 1973, Fishman measured the differential voltage noise spectra of a native membrane and a membrane whose potassium conductance has been blocked with tetraethylammonium. The differential spectrum he obtained fitted well... 

The spectral density function of the fluctuation can be calculated from the autocorrelation function by the Wiener-Khintchine relation (Wiener, 1930 Khintchine, 1934). The original formulation of the theorem refers to stationary stochastic processes for a possible generalisation see, for example, Lampard, 1954. The relationship connects the autocorrelation function to the spectrum ... 

The frequency noise power spectral density of a SL typically exhibits a 1/f dependence below 100 kHz and is flat from 1 MHz to well above 100 MHz. Relaxation oscillations will induce a pronounced peak in the spectrum above 1 GHz. The "white" spectral component represents the phase fluctuations that are responsible for the Lorentzian linewidth and its intensity is equal to IT times the Lorentzian FWHM.20 xhe 1/f component represents a random walk of the center frequency of the field. This phase noise is responsible for a slight Gaussian rounding at the peak of the laser field spectrum and results in a power independent component in the linewidth. Figure 3 shows typical frequency noise spectra for a TJS laser at two power levels. 

Wiener-Khinchin theorem of statistical mechanics, the Fourier transform of the autocorrelation function of a fluctuating quantity is the power spectrum of the fluctuations, where the power at a given frequency is the mean square amplitude of the fluctuations at that frequency. To convert Mnmjkit) to a suitable function of frequency, Redfield [19] defined the spectral density function, J mjk(,co)> ... 

Relaxation measurements provide another way to study dynamical processes over a large dynamic range in both thermotropic and lyotropic liquid crystals (see Sec. 2.6 of Chap. Ill of Vol. 2A). The two basic relaxation times of a spin system are the spin-lattice or longitudinal relaxation time 7] and the spin-spin or transverse relaxation time T2. A detailed description, however, requires a more precise definition of the relaxation times. For spin 7=1, for instance, two types of spin-lattice relaxation must be distinguished, related to the relaxation of Zeeman and quadrupolar order with rates 7j"2 and Jfg. The relaxation rates depend on spectral density functions which describe the spectrum of fluctuating fields due to molecular motions. A detailed discussion of spin relaxation is beyond the scope of this... 

Rayleigh scattered light from dense transparent media with nonuniform density. If these nonuniformities are time-independent, there will be no frequency shift of the scattered light. If, however, time-dependent density fluctuations occur, as e. g. in fluids, due to thermal or mechanical processes, the frequency of the scattered light exhibits a spectrum characteristic of this time dependence. The type of information which can be obtained by determining the spectral line profile and frequency shift is described in an article by Mountain 235). 

In most detection schemes of saturation or polarization spectroscopy the intensity fluctuations of the probe laser represent the major contribution to the noise. Generally, the noise power spectrum PNoise(/) shows a frequency-dependence, where the spectral power density decreases with increasing frequency (e.g., l//-noise). It is therefore advantageous for a high S/N ratio to detect the signal S behind a lock-in amplifier at high frequencies /. 

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