Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-07T11:23:57.640Z Has data issue: false hasContentIssue false

The probability distribution of the extent of a random chain

Published online by Cambridge University Press:  24 October 2008

H. E. Daniels
Affiliation:
Wool Industries Research AssociationLeeds

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
Copyright
Copyright © Cambridge Philosophical Society 1941

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Pearson, K.Drapers' Company Research Memoirs, Biometric Series, 3 (1906).Google Scholar
(2)Lord, Rayleigh. Phil. Mag. (6), 37 (1919), 321347.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