Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T13:11:35.452Z Has data issue: false hasContentIssue false
  • English
  • Français

Statistical mechanics and the partitions of numbers

Published online by Cambridge University Press:  24 October 2008

F. C. Auluck  and
D. S. Kothari
F. C. Auluck
Affiliation:
Department of PhysicsUniversity of DelhiDelhi
D. S. Kothari
Affiliation:
Department of PhysicsUniversity of DelhiDelhi
Get access Rights & Permissions [Opens in a new window]

Extract

1. The properties of partitions of numbers extensively investigated by Hardy and Ramanujan (1) have proved to be of outstanding mathematical interest. The first physical application known to us of the Hardy-Ramanujan asymptotic expression for the number of possible ways any integer can be written as the sum of smaller positive integers is due to Bohr and Kalckar (2) for estimating the density of energy levels for a heavy nucleus. The present paper is concerned with the study of thermodynamical assemblies corresponding to the partition functions familiar in the theory of numbers. Such a discussion is not only of intrinsic interest, but it also leads to some properties of partition functions, which, we believe, have not been explicitly noticed before. Here we shall only consider an assembly of identical (Bose-Einstein, and Fermi-Dirac) linear simple-harmonic oscillators. The discussion will be extended to assemblies of non-interacting particles in a subsequent paper.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 42 , Issue 3 , October 1946 , pp. 272 - 277
Copyright
Copyright © Cambridge Philosophical Society 1946

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.Kgl. Danske Vid. Selskab. Math. Phys. Medd. 14, no. 10 (1937). (See also Van Lier, C. and Uhlenbeck G. E. Physica, 4 (1937), 531.)Google Scholar
(3)Rademacher, H.Proc. Nat. Acad. Sci., Washington, 23 (1937), 78.CrossRefGoogle Scholar
(4)Bethe, H. A.Rev. Mod. Phys. 9 (1937), 81.CrossRefGoogle Scholar
(5)Erdos, P. and Lehner, J.Duke Math. J. 8 (1941), 335.CrossRefGoogle Scholar
(6)Auluck, F. C., Chowla, S. and Gupta, H.J. Indian Math. Soc. 6 (1942), 105.Google Scholar
(7)Auluck, F. C.J. Indian Math. Soc. 6 (1942), 112.Google Scholar