Renormalization of the Wilson type

Simple analysis of dimensions shows that this functional dependence in 3D space has the transformational property 

transformational property 1 relates the soiirc system to the cut-off a in a scales stipulated, remains the same  

Given N an l V2 are chosen properly, the function / in Kqiiation 2 will remain the same as in Kquation 1. 

Cicnerally, the relationships between JV and /V, V2 and 1)2 are very soj)histicaled how -ever, be N rather large, b can be chosen large enough to obtain a simple asymptotic relationship b tween the quantities with and without a tilde. 

For large N and i 2 0, U2 turns out to acquire a constant value insensitive to / , i.c. O2 approaches the fixed point vj. section 2.5) and, besides, /V Then, 

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Section 5.1 presents renormalization of the Wilson type following Oono and Freed (1981a), and renormalization by Gell-Mann and Low following Oono et al. (1981). In section 5.1, polymer chain renormalization in the momentum space of Kholodcnko and Freed (1983) will be discussed. 

The renormalization group transformation of the Wilson type for the macromolecular conformational space Ks is composed of the scaling transformation Ss and KsMlanoff s transformation /Cs -... 

In this chapter, we describe both types successively by showing how these techniques can be applied to the determination of properties of polymers in solution. However, one must be aware of the fact that this separation into two kinds of renormalization is somewhat arbitrary there are connections between the various points of view and bridges between the various methods, and all of them follow from the seminal work of Wilson.3,4... 

Oono, Ohta, and Freed have developed a renormalization method for conformational space of Gell-Mann and Low s (195 1) type for the excluded volume problem. In a number of cases, this approach proves to be more effective than Kadanoff-Wilson s approximation (see subsection 5.1.1) it provides a higher accurau y of the calculation of such quantities as the end-to-end vector distribution function, the scattering function, the conformation of a mricromolecule as a function of polymer concentration, etc. 

Here a z), < s z) and A(z) = zh(z) are the conventional expansion and second virial factors of the continuum Edwards model, and K and K2 are nonuniversal scale factors. Thus, the continuum Edwards model is a correct theory for a certain limiting regime in the molecular-weight/tempera-ture plane—but this regime is not the one previously thought. The explanation of eq. (2.104) relies on a Wilson-deGennes-type renormalization group see Ref. 222 for details, and Ref. 223 for further discussion. 

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