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

2.C. COUNTING CONFIGURATIONS BY INDUCTION 412.C Induction method to count number of chainconfigurationsThe Laplace transform method can be used to calculate the number of configurationsof hairpins for the first ten terms. The results of this calculation areshown in table 2.1. From this table the general form for the number of config-Number of defects Number of Configurations Ω n (z)0 δ(z +N)+δ(z −N)1 22 N3 N 2 (1− ( z 2)/2N)4 N 3 (1− ( z 2)/8N)5 N 4 (1− ( z 2)N) 2 /326 N 5 (1− ( z 2)N) 2 /1927 N 6 (1− ( z 2)N) 3 /11528 N 7 (1− ( z 2)N) 3 /92169 N 8 (1− ( z 2)N) 4 /7372810 N 9 (1− ( z 2)N) 4 /737280Table 2.1: The number of configurations of arranging n hairpinson a polymer chain calculated as a function of end to enddistance for the first 10 hairpins.urations of n defects on a chain can be guessed and then proven by induction.For the odd case the guess isand for the even n caseG n (z,N) =G n (z,N) =2(n−1)!! 2(N2 −z 2 ) n−12 , (2.122)2n(n−2)!! 2N(N2 −z 2 ) n−22 , (2.123)where !! denotes the usual double factorial function. To use the method ofinduction we proceed as follows. First it is necessary to split the sums upinto parts: those which started by taking a step up the z-axis and those whichstarted bytakingastepdown. Startwithaneven numberof defects onachainthat started in the updirection anddenote the numberof configurations of thechain as a function of arc length N and end-to-end distance z as G n+ (z,N).Another hairpin can then be added in by combining this with another section