Statistical models of elasticity in main chain and smectic liquid ...

2.C. COUNTING CONFIGURATIONS BY INDUCTION 43hairpins use up a negligible amount of the end-to-end distance) to meet theend-to-end distance constraint. Similarly the second inequality is obtainedby considering the case when the added segment goes up a distance z ′ andthe existing segment must then come down a distance z −z ′ . Then the twoquantities must be added with the correct sign to obtain the arc length whichmust be less that the total available arc length. Thus Eq. (2.124) can beevaluated as follows: first we assert the form of G n+ (z,N)G n+ (z,N) =1n(n−2)!! 2(N +z)n 2(N −z) n−22 . (2.129)Then use this, together with the previous result that G 0− (z,N) = δ(z +N)to obtain∫ ∫G (n+1)+ (z,N) = dz ′ dN ′ 1n(n−2)!! 2(N −N′ +z −z ′ ) n 2× (N −N ′ −z +z ′ ) n−22 δ(z ′ +N ′ )==∫ 0z−N2dz ′ 1n(n−2)!! 2(N +z)n 2 (N +2z ′ −z) n−221n!! 2(N2 −z 2 ) n 2 (2.130)Another defect can be added on in a similar way using the G that we havejust derived∫ ∫G (n+2)+ (z,N) = dz ′ dN ′ 1n!! 2(N −N′ −z +z ′ ) n 2× (N −N ′ +z −z ′ ) n 2 δ(z ′ −N ′ )==∫ z+N2dz ′ 10 n!! 2(N −z)n 2 (N −2z ′ +z) n 21+z)n+2 2(n+2)(n)!! 2(N (N −z) n 2 (2.131)This is now the original form for the G n+ (z,N) so if it is true for n then it isalso true for n +2. A similar analysis can be carried out for the G n+ (z,N)(Fig. 2.14 a))which goes through in the same way, except with z → −z. Thuson combining the two results for the even caseG n+ (z,N) =1n(n−2)!! 2(N 2(N +z)n −z) n−22G n− (z,N) =1n−2n(n−2)!! 2(N −z)n 2 (N +z) 2G n (z,N) = G n− (z,N)+G n+ (z,N)=2n(n−2)!! 2N(N2 −z 2 ) n−22 (2.132)