A Modified Gambler's Ruin Model of Polyethylene Chains in the Amorphous Region

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 differen...

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Bibliographic Details
Published inProceedings of the National Academy of Sciences - PNAS Vol. 93; no. 19; pp. 10007 - 10011
Main Authors Duan, Zhong-Hui, Howard, Louis N.
Format Journal Article
LanguageEnglish
Published United States National Academy of Sciences of the United States of America 17.09.1996
National Acad Sciences
National Academy of Sciences
Subjects
Applied mathematics
Arithmetic mean
Crystallography
Cubic lattices
Difference equations
Mathematical lattices
Mathematical models
Modeling
Polyethylenes
Random walk
Statistical variance
Statistics
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Summary: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 (2M - 2)/(2M + 1), the fraction of ties 3/(2M + 1), the average 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 28/45M$^{4}$ + O(M$^{3}$), and the variance of the walk length 2M$^{3}$ + O(M$^{2}$).
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ISSN:0027-8424
1091-6490
DOI:10.1073/pnas.93.19.10007