Configuration of a Brownian chain in an attractive external potential

2 Configuration of a Brownian chain in an attractive external potential 

In order to calculate the entropy S C of a real chain in terms of the local monomer concentration C(r), we set the corresponding Brownian chain in an external potential V(r), the result being that the mean link concentration at point r is C(r). In this case, the entropy of the chain is a function of the potential but, on the other hand, there is a relation between P (r) and C(r) and consequently the entropy can be re-expressed in terms of C(r). First, let us study the statistical state of the chain. The weight associated with a Brownian chain of area S is given by (see Chapter 2) 

When an attractive potential acts on this chain, the new weight is given by 

Let (x, y S) be the (renormalized) partition function of a chain for which x is the position vector of the origin and y the position vector of the end point. We 

When interactions are present, we can define 3 (x, y, S) as a solution of the equation 

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