Dirac delta function section

The phenomenological Langevin Eqs. (227) and (228) are only applicable to a very restricted class of physical processes. In particular, they are only valid when the stochastic forces and torques have infinitely short correlation times, i.e., their autocorrelation functions are proportional to Dirac delta functions. As was shown in the previous section, these restrictions can be removed by a suitable generalization of these Langevin equations. As we saw in the particular case of the velocity, the modified Langevin equation is... 

The Gaussian quadrature algorithm introduced in Section 3.1.1 is equivalent to approximating the univariate NDF by a sum of Dirac delta functions ... 

Even when multivariate EQMOM is used with kernel density functions, a Dirac delta function (dualquadrature) representation is employed to close the terms in the GPBE. See Section 3.3.4 for more details. 

The Kubo relation (25) of section 2.1 is obtained as the Fourier transform of (70). The term linear in V is the retarded two-time Green s function, first introduced in this context by Bogoliubov and Tyablikov [30]. The identification with Green s functions stems from the presence of the Heaviside step function that in part were introduced to allow integration over the full time interval and whose time derivative gives a Dirac delta function. For instance, t — to)U(tfo) is a solution of the inhomogeneous equation [31]... 

It is a property of Fourier transform mathematics that multiplication in one domain is equivalent to convolution in the other. (Convolution has already been introduced with regard to apodization in Section 2.3.) If we sample an analog interferogram at constant intervals of retardation, we have in effect multiplied the interferogram by a repetitive impulse function. The repetitive impulse function is in actuality an infinite series of Dirac delta functions spaced at an interval 1 jx. That is,... 

The Dirac delta function and its properties are related to various descriptions in Chapters 4 and 6. They are also essential in the description to be given in Section D.3.3. 

The Dirac delta comb and its Fourier transform are referred to in Section 4.4.2. An infinite train of the Dirac delta functions at intervals of a on the x axis is called the Dirac delta comb and denoted by dJ (x). This function is defined as... 

The procedure for obtaining the double-differential scattering cross-section is to now sum over possible final states f) of the scatterer, subject to the condition that the energy of the combined system of radiation -i- target is conserved. The latter is accomplished by attaching the Dirac delta function, d( — ), to each term in the sum. Finally, we can introduce... 

This result agrees with the qualitative analysis that we performed in Section 10.2.2.1. It could have been deduced without going through the formality of solving Eqn. (10-13). In a PFR, each and every element of fluid spends exactly the same time in the reactor. For a constant-density fluid, that time is V/i> = t. Therefore, if we inject a Dirac delta function of tracer at t = 0, a Dirac delta function wiU emerge at t = r. This is a necessary consequence of the fact that there is no mixing in the direction of flow in a PFR, and no gradients in the direction normal to flow. 

The corrections SPi to the scalar propagation constant are given in Table 14-1 in terms of /j and I2. In the numerator of each expression, the derivative d//d J is the Dirac delta function 3(R — 1), as explained in Section 14-6, and the integral in the denominator is given in Table 14-6. This leads to the expressions for SPi and the corresponding SUt in the same table. There is no correction for the TEo modes, whose fields satisfy the scalar wave equation exactly. 

Figure 3Ab illustrates the Dirac delta comb as a function of wavenumber, v. The function in Eq. 3.2 is an inbnite series. We multiply the analog interferogram with the Dirac delta comb of Eq. 3.1, and consequently, the Eourier transform of the interferogram (i.e., the spectrum) is convolved with the transformed comb (see Section 2.3). The effect of this convolution is to repeat the spectrum ad infinitum. If the spectrum covers the bandwidth 0 to v ax, the transformed Dirac delta comb must have a period of at least 2Vniax otherwise, the spectra will overlap as a result of the convolution. In other words. 

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