Homophase Fluctuations in the isotropic Phase

Since homophase fluctuations in the isotropic phase involve only states close to Q(r) = 0, we can neglect the cubic and quartic terms in the expansion of If we also set H = E = 0, Eq. [43] reduces to 

As shown in Section 2.3, Q(r) can be expressed in terms of S(r) and n(r). If we were to make this substitution in Eq. [57], the resulting expression would contain several interaction terms coupling S(r), n(r), and their spatial derivatives. Thus, in general, the problem of calculating the fluctuation spectra of S(r) and h(r) can be quite complicated because of this coupling. In the latter part of this section we will show one way to handle the coupling between S(r) and n(r), but first we want to consider a simple system that illustrates both the physics of homophase fluctuation phenomena and the mathematics involved in manipulating the formalism developed in previous sections. 

The system we wish to consider is one in which we can neglect the spatial variation of fi(r) and treat only the fluctuations of S(r). Setting all spatial derivatives of n(r) to zero and limiting our discussion for the moment to nematic liquid crystals (go = 0), Eq. [57] becomes 

The difficulty in using Eq. [59] for actual calculations lies in the functional integral fDS f) and we therefore digress briefly to consider functional integration. 

It is customary to proceed by Fourier analyzing S(r) into its Fourier components 

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