Formal Derivation of the SCF Equations

The derivation here is in the same spirit as Edwards approach in terms of functional integrals. Since Edwards wanted to obtain explicit solutions to the SCF equations to determine y of (6.4), he found it convenient to introduce other mathematical approximations first and then introduce the SCF approximation. The formal physical content is, however, best exhibited by initially employing the SCF approximation to obtain the SCF equations (6.21) and (6.24), (6.26), (6.29), etc. In our approach, the resulting equations then still remain to be solved self-consistcntly. 

In order to avoid getting lost in the interesting mathematical details of the derivation, the major steps are sketched and then the details are supplied. Any SCF approximation must be of the form (6.21), i.e., a diffusion ec nation with some external potential which represents the average volume exclusion field due to the other segments in 0 L. It turns out that the exact G of (6.12), or equivalently (6.16), can be expressed in terms of an auxiliary Green s function G([ ]) which satisfies 

Equation (fi.25) describes the Green s function for diffusion in the (real) external field / / (R), where (R) is a random field. [The imaginary unit i appears in (6.25) only for mathematical convenience.] Specifically, the correspondence between G and C([ ]) is written as 

Before discussing gaussian random fields in more detail and explicitly proving (6. 25)-(6.27), we examine the implications of these equations for the existence of a SCF approximation. 

Since (6.25) expresses G([ ]) as the Green s function for some external field icj)(R), the average (6.26) over all implies that if an SCF approximation FgoF is meaningfui, some particuiar , caii it / o(R), must dominate the average (6.26). Thus a reasonable approximation is obtained by just taking o so 

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