Proceedings of the National Academy of Sciences

Volume 93, Number 19; Pages: 10007-10011

Applied Mathematics
### A modified gambler's ruin model of polyethylene chains in the amorphous
region

Zhong-Hui Duan,
Louis N. Howard

1996 by the National Academy of Sciences

**ABSTRACT **Polyethylene chains in the amorphous region
between two crystalline
lamellae *M* unit apart are modeled as random walks with
one-step memory on 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 statistics of the restricted random walks. They yield
that the fraction of loops is (2*M* -
2)/(2*M* + 1), the fraction of ties
3/(2*M* + 1), the average length of loops
2*M* - 0.5, the average length of ties
2/3*M*^{2} + 2/3*M* -
4/3, the average length of walks equals 3*M* -
3, the variance of the loop length
16/15*M*^{3} + *O(M*^{2}),
the variance of the tie length 28/45*M*^{4} +
*O(M*^{3}), and the variance of the walk length
2*M*^{3} + *O(M*^{2}).