The Frequency Representation

As noted in Sect. 1.2, the right-hand side of (1.2.32a) has the form of a convolution integral, given by (A3.1.17), since n(t) is zero for negative values of t. Its Fourier transform (FT) is given by 

The form (1.5.2c) may be demonstrated by deriving from it, with the aid of a partial integration, the form (1.5.2b). The subtraction constant G(oo) has been introduced into the integrand for convergence. Relation (1.5.1) has the same form as the linear elastic constitutive relations, apart from the frequency dependence of the modulus. This fundamental observation will be extremely important in later chapters. We write 

There is no reason in general to believe that is zero, so that /i(cu) must be 

In the first place, let us discuss, in a more concrete manner, the physical significance of /i(cu). Consider the simple but very important sinusoidal strain history given by 

These results apply to the steady-state response which prevails after a long time. If we replace the lower limit in (1.2.32a) by some finite time, there will be other, transient, terms. 

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Fast Fourier transformation (FFT) of the noisy ac polarographic data to yield the Fourier spectrum which is the frequency representation of the data. 

These equations are usually Fourier transformed from the time domain to the frequency domain. In the frequency representation, the KS response function is written in terms of the unperturbed KS orbitals their occupation numbers e,-, and their orbital energies pj as ... 

Kinetic information can be obtained from equilibrium data in CCD models and in CDS models from the time representation of the fluctuations or the frequency representation of fluctuations (spectrum). 

From the theoretical point of view measurements based on fluctuation phenomena are better, since the equilibrium fluctuation may be interpreted as spontaneous perturbation and relaxation. Consequently, kinetic information is available without perturbing the system externally. In practice, chemical fluctuation measurements refer to small systems, therefore the problem is discussed in this section. Fluctuation measurements can be classified as follows (Romine 1976) indirect measurements, and direct measurements, which may be based on the time representation of fluctuations, or the frequency representations of fluctuations. 

An example of direct measurements based on the frequency representation of fluctuation is the study of the association-dissociation reaction of BeS04 in a 0.03 M solution, with conductance measurements (Feher Weismann, 1973). It is particularly interesting that they could increase the ratio of the reaction noise to Johnson noise of the circuit, since the former is a quadratic function of the applied direct voltage, while the latter is independent of the voltage. 

The most commonly occurring value in the distribution is the mode, and is the value at which the frequency representation is a maximum. The median divides the frequency curve into two equal parts, and equals the particle size at which the cumulative representation equals 50%. In a rigorous normal distribution, the mean, mode, and median have the same value. 

Problem 1.2.2 Show that in the non-aging case, operator multiplication as defined by (1.2.10) is commutative. In fact, this is probably perceived more easily by using the alternative form, akin to matrix multiplication, derived from (1.2.32). The easiest way to show it is to take into account a result of Sect. 1.5, namely that in the frequency representation, operator multiplication becomes simple multiplication, by virtue of the Faltung theorem (Sect. A3.1). 

The relaxation of the dielectric modulus can be mathematically represented equivalently as a function of time or of frequency. This approach develofjed for a single relaxation time for dielectric relaxations by Debye has been extended by Havriliak and Negami to a distribution of relaxation times." The frequency representation for the complex modulus as found by the empirical Havrihak-Negami model function as ... 

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