Chains with two-body interactions real space expansions

2 Chains with two-body interaction real space expansions 

Let us expand in powers of b. For this purpose, we choose a curve r (s) 

The interesting result is that these conditions lead to the elimination of the factor l/2pp . Now, we can give well-defined values to s, s /,.. ., s p, s and, for a while, we can also fix the position vectors lr(s 1), lr (s /)... Ir(s f ), r(sJJ). These interaction points cut the polymer line into (2p +1) successive segments with areas Si. s2p+, respectively. 

The position of the origin of segment j is defined by r and the position of its end point by rJ+,. Thus the terms of order p lead to diagrams of order p and each diagram corresponds to a peculiar setting of the interaction points on the polymer line (see Fig. 10.1). 

Actually, this function is the distribution probability of the end-to-end vector for a Brownian chain and the result can be proved again by proceeding as follows 

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