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The probability distribution of the extent of a random chain

Published online by Cambridge University Press:  24 October 2008

H. E. Daniels  and
F. Smithies
H. E. Daniels
Affiliation:
Wool Industries Research AssociationLeeds
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Extract

1. Introduction and summary. A chain of N links is allowed to assume a random configuration in space. The extent of the chain in any direction is defined as the shortest distance between a pair of planes perpendicular to that direction, such that the chain is contained entirely between them. In the present paper the probability distribution of the extent is discussed, starting with a chain in one dimension for which formulae are derived for the probability and mean extent for all values of N. The limiting forms for large N are then considered. The results are extended to the case of a chain in three dimensions, and it is shown that the extents in two directions at right angles tend to be independently distributed when N is large. It is assumed that the links are infinitely thin, so that a point in space may be occupied by the chain any number of times.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 37 , Issue 3 , July 1941 , pp. 244 - 251
Copyright
Copyright © Cambridge Philosophical Society 1941

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References

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