Zh. Duan et Ln. Howard, A MODIFIED GAMBLERS RUIN MODEL OF POLYETHYLENE CHAINS IN THE AMORPHOUS REGION, Proceedings of the National Academy of Sciences of the United Statesof America, 93(19), 1996, pp. 10007-10011
Polyethylene chains in the amorphous region between two crystalline la
mellae M unit apart are modeled as random walks with one-step memory o
n a cubic lattice between two absorbing boundaries. These walks avoid
the two preceding steps, though they are not true self-avoiding walks.
Systems of difference equations are introduced to calculate the stati
stics of the restricted random walks, They yield that the fraction of
loops is (2M - 2)/(2M + 1), the fraction of ties 3/(2M + 1), the avera
ge length of loops 2M - 0.5, the average length of ties 2/3M(2) + 2/3M
- 4/3, the average length of walks equals 3M - 3, the variance of the
loop length 16/15M(3) + O(M(2)), the variance of the tie length 28/45
M(4) + O(M(3)), and the variance of the walk length 2M(3) + O(M(2)).