Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T05:08:48.926Z Has data issue: false hasContentIssue false
  • English
  • Français

Statistical mechanics and partitions into non-integral powers of integers

Published online by Cambridge University Press:  24 October 2008

B. K. Agarwala  and
F. C. Auluck
B. K. Agarwala
Affiliation:
University of DelhiDelhiIndia
F. C. Auluck
Affiliation:
University of DelhiDelhiIndia
Get access Rights & Permissions [Opens in a new window]

Extract

1. The problem of the partition of numbers, first investigated in detail by Hardy and Ramanujan (1), has in recent years assumed importance on account of its application by Bohr and Kalckar (2) in evaluating the density of energy levels in heavy nuclei. A ‘physical approach’ to the partition theory has been made by Auluck and Kothari (3), who have studied the properties of quantal statistical assemblies corresponding to the partition functions familiar in the theory of numbers. The thermodynamical approach to the partition theory, apart from its intrinsic interest, draws attention to aspects and generalizations of the partition problem that would, otherwise, perhaps go unnoticed. Thus we are led to consider restricted partitions such as: partitions where the summands are repeated not more than a specified number of times; partitions where the summands are all different; partitions into summands which must not be less than a specified value; partitions into a prescribed number of summands, and so on. The generalization that seemed to us to be the most interesting is the extension of the partition concept to include partitions into non-integral powers of integers.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 47 , Issue 1 , January 1951 , pp. 207 - 216
Copyright
Copyright © Cambridge Philosophical Society 1951

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)Hardy, G. H. and Ramanujan, S.Proc. London Math. Soc. (2), 17 (1918), 75.CrossRefGoogle Scholar
(2)Bohr, N. and Kalckar, F.K. Danske Vidensk. Selskab. Math. Phys. Medd. 14, no. 10 (1937). See also Van Lier, C. and Uhlenbeck, G. E. Physica, 4 (1937), 531.Google Scholar
(3)Auluck, F. C. and Kothari, D. S.Proc. Cambridge Phil. Soc. 42 (1946), 272.CrossRefGoogle Scholar
(4)Sommerfeld, A.Z. Phys. 47 (1928), 1.CrossRefGoogle Scholar
(5)Bethe, H. A.Rev. Mod. Phys. 9 (1937), 80.CrossRefGoogle Scholar
(6)Auluck, F. C. and Kothari, D. S.Proc. Roy. Irish Acad., Minutes of Proceedings 1946–1947, p. 13.Google Scholar
(7)Temperley, H. N. V.Proc. Roy. Soc. A, 199 (1949), 361.Google Scholar