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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

REFERENCES

(1)Pearson, K.Drapers' Company Research Memoirs, Biometric Series, 3 (1906).Google Scholar
(2)Lord, Rayleigh. Phil. Mag. (6), 37 (1919), 321–347.Google Scholar
(3)Watson, G. N.Theory of Bessel functions (Cambridge, 1922), 419.Google Scholar
(4)Henry, P. H. S.Proc. Roy. Soc., A, 171 (1939), 215.Google Scholar