Wiener-Khintchine theorem

It is a well known fact, called the Wiener-Khintchine Theorem [gardi85], that the correlation function and power spectrum are Fourier Transforms of one another ... 

This is the autocorrelation and by the Wiener-Khintchine theorem the power spectrum of the disturbance is given by its Fourier transform,... 

Wiener-Khintchine theorem). The right-hand side of this equation is often called the power spectrum. It is given by the autocorrelation function, Eq. 2.55. The Fourier transform of the autocorrelation function is related to the spectral moments,... 

The velocity, which equilibrates at large times, is not an aging variable. Thus, Fourier analysis and the Wiener-Khintchine theorem can be used equivalently to obtain the equilibrium correlation function Cvv(t — t2). 

As stated above, the Langevin force F(t) can be viewed as corresponding to a stationary random process. Clearly, the same is true of the solution v(f) of the generalized Langevin equation (22), an equation which is valid once the limit ti —> —oo has been taken. Thus, Fourier analysis and the Wiener-Khintchine theorem can be used to obtain the velocity correlation function, which only depends on the observation time Cvv(t, t2) = Cvv(t —12). As in the classical case, the velocity does not age. 

Applying the Wiener-Khintchine theorem, one obtains the velocity correlation function as the inverse Fourier transform of Cvv ( ), that is, in terms of the noise spectral density CFF([Pg.299]

The power sjjectrum is merely the Fourier transform of the VAF via the Wiener-Khintchine theorem. The integration is carried out as a discrete sum over the jjeriod of time in which the VAF decays to a zero value. This quantity gives the number of oscillators at a given frequency and is a very informative indicator of the transition from rigid, quasiperiodic motion to nonrigid, chaotic motion. Note that I(co = 0) is proportional to the diffusion constant. This quantity was calculated by Dickey and Paskin - in the study of phonon frequencies in solids and also by Kristensen et al. in simulations of cluster melting. 

This result is known as the Wiener-Khintchine theorem (see also Section 7.5.2). From the obvious inequality 0 it follows that... 

An important relationship between /(co) and the time correlation fimction C(t) = + r)) = x O jxlt ) ofxlt) is the Wiener-Khintchine theorem, yNblchslsAes... 

Once we evaluate (t) we can obtain the power spectrum of the randomly modulated harmonic oscillator using the Wiener-Khintchine theorem (7.76)... 

If 7 (Z) satisfies the Markovian property (8.20), it follows from the Wiener-Khintchine theorem (7.76) that its spectral density is constant... 

Equation (41a) means that the function B( r) is equivalent to the volume integral of the density matrix y(ri, ri) under the condition of r = r - r, and Eq. (41b) means that B(r) is the autocorrelation function of the position wave function (r). The latter is an application of the Wiener-Khintchin theorem (Jennison, 1961 Bracewell, 1965 Champeney, 1973), which states that the Fourier transform of the power spectrum is equal to the autocorrelation function of a function. Equation (41c) implies not only that B(r) is simply the overlap integral of a wave function with itself separated by the distance r (Thulstrup, 1976 Weyrich et al., 1979), but also that the momentum density p(p) and the overlap integral S(r) are a pair of the Fourier transform. The one-dimensional distribution along the z axis, B(0, 0, z), for example, satisfies... 

Figure C3.5.6 compares the result of this ansatz to the numerical result from the Wiener-Khintchine theorem. They agree well and the ansatz exhibits the expected exponential energy-gap law (VER rate decreases exponentially with Q). The ansatz was used to determine the VER rate with no quantum correction Q = 1), with the Bader-Beme harmonic correction [61] and with a correction based [M, M] on EgelstafPs method [62]. The Egelstaff corrected results were within a factor of five of experiment, whereas other eorrections were off by orders of magnitude. This ealeulation represents the present state of the art in computing VER rates in such difficult systems, inasmuch as the authors used only a model potential and no adjustable parameters. However the ansatz proeedure is clearly not extendible to polyatomic molecules or to diatomic molecules in polyatomic solvents.

Now, we consider the important limit of weak laser intensity. In this limit, the Wiener-Khintchine theorem relating the line shape to the one-time correlation function holds. As we shall show now, a three-time correlation function is the central ingredient of the theory of fluctuations of SMS in this limit. In Appendix B, we perform a straightforward perturbation expansion with respect to the Rabi frequency Q in the Bloch equation, Eq. (4.6), to find... 

In this chapter, we developed a stochastic theory of single molecule fluorescence spectroscopy. Fluctuations described by Q are evaluated in terms of a three-time correlation function C iXi, X2, T3) related to the response function in nonlinear spectroscopy. This function depends on the characteristics of the spectral diffusion process. Important time-ordering properties of the three-time correlation function were investigated here in detail. Since the fluctuations (i.e., Q) depend on the three-time correlation function, necessarily they contain more information than the line shape that depends on the one-time correlation function Ci(ti) via the Wiener-Khintchine theorem. 

The standard assumption of Markovian processes (e.g., the Poissonian Kubo-Anderson processes considered here) fails to explain the statistical properties of emission for certain single molecular systems such as quantum dots [21-23]. Instead of the usual Poissonian processes, a power-law process has been found in those systems. For such highly non-Mar-kovian dynamics stationarity is never reached, and hence our approach as well as the Wiener-Khintchine theorem does not apply. This behavior is the topic of our recent work in [104]. 

TJie Fourier transform of the first-order correlation function G r) represents the normalized frequency spectrum of the incident light-wave intensity I (o>) (Wiener-Khintchine theorem) [930, 935]. 

These records have been transformed into the frequency domain by a "Fast-Fourier Transformation" (FFT) and ensemble averaged there. The Fourier Transformation of the resulting array back into the time domain gives the correlation function (4) by the Wiener Khintchine theorem. 

Both types of data analysis are equivalent because the spectral power function Pj (oj) is a Fourier transform of the corresponding autocorrelation function (Wiener-Khintchine-theorem Wiener 1930 Khintchine 1934). In the following, only PCS will be discussed. 

By applying the Wiener-Khintchine theorem (Wiener, 1930 Khintchine, 1934 see Chapter 15), the scattering can be transformed into the power spectrum just like the pair 5(q, t) and /(co) ... 

The cross correlation function given in Equation (F6) may be further modified to a form better suited for the two-dimensional correlation analysis with the help of the Wiener-Khintchine theorem [5]. This theorem conveniently relates the cross correlation function with the corresponding Fourier transforms. In the first step, the expression for the dynamic spectrum y(v2, t -I- t) in Equation (F6) is rewritten in terms of the inverse of Fourier transform of y(v2, s )-... 

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