Entropy of an isolated chain

Let us consider on a lattice the self-avoiding chains with N links which start from the origin of the lattice. The entropy of these chains is defined in a very simple way. The energy of such chains is obviously zero. Consequently, the entropy is directly deduced from the partition function ZN 

the computer calculations made for chains on lattices, (described in Chapter 4) lead to the following asymptotic result 

Entropy can also be calculated by starting from the continuous model and, in this case, we expect a similar result. However, we must note that in the continuous case, the entropy of the system is really infinite and that a finite entropy can be obtained only after performing a subtractive renormalization. Actually, this question has been tackled several times, either indirectly with the help of a zero component field theory,3,4 or by using the direct renormalization method.2 We shall now describe in detail the latter approach. 

It has been shown in Chapter 10, Section 1, that the weight iT r associated with a chain defined by the function r(s) can be written in the form 

In this formula, s0 is a cut-off, and we saw in Chapter 9 that the introduction of this cut-off enables us to define a partition function + T(s) and the corresponding free energy 

---

To calculate the entropy of an isolated chain which is confined between two parallel plane surfaces, only the jc-component in the direction normal to the surfaces need be considered. The method of ima yields... 

---