Gaussian Fluctuations and Random Phase Approximation

As long as fluctuations are weak, they can be treated within the Gaussian approximation. One famous example of such a treatment is the random phase approximation (RPA) which describes Gaussian fluctuations in homogeneous phases. The RPA has been extended to inhomogeneous saddle points by Shi, Noolandi, and coworkers [66,67]. In this section, we shall re-derive this generalized RPA theory within the formalism developed in the previous sections. 

Our starting point is the Hamiltonian g[l7, W] of Eq. 11. We begin with expanding it up to second order about the saddle point = [U, W ] of the SCF theory. To this end, we define the averaged single chain correlation functions [ 13,66,67 ] 

With these definitions, the quadratic expansion of 17, W] (Eq. 11) can be written as 

Since is an extremiun, the linear terms in (5t7 and 5W vanish. Next we expand this expression in 517 and 5W. To simplify the expressions, we follow 

Within the Gaussian approximation, the average (W(r)W(r )) can readily be calculated  

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