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Entropy 2009, 11(4), 1121-1122; https://doi.org/10.3390/e11041121

Discussion
Comment on “Equiprobability, Entropy, Gamma Distributions and Other Geometrical Questions in Multi-Agent Systems”, Entropy 2009, 11, 959-971
IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos, CSIC-UIB, Campus UIB, E-07122 Palma de Mallorca, Spain
Received: 11 December 2009 / Accepted: 18 December 2009 / Published: 22 December 2009

Abstract

:
The volume of the body enclosed by the n-dimensional Lamé curve defined by i = 1 n x i b = E is computed.
Keywords:
Lamé curves
A recent paper [1] derives asymptotic expressions for the volume of the n-dimensional body defined by 0 i = 1 n x i b E for b > 0 , x i 0 . This is the body enclosed by a Lamé curve in n dimensions. Here I compute exactly this volume by using a straightforward modification of the calculation that gives the volume of the n-dimensional sphere, the case b = 2 , see [2].
Writing E = R b , the volume V n ( R ) is
V n ( R ) = d x 1 d x n 0 i = 1 n x i b R b
By dimensional analysis V n ( R ) = C n R n . Let us now compute the integral
0 d x 1 0 d x n exp [ - ( x 1 b + + x n b ) ] = 0 d x exp [ - x b ] n = Γ 1 + 1 b n
by using the change of variables r = ( x 1 b + + x n b ) 1 / b and the volume element d V n ( r ) = n C n r n - 1 d r as
0 d V n ( r ) exp [ - r b ] = C n Γ 1 + n b
Equaling these two expressions one gets:
V n ( R ) = Γ 1 + 1 b n Γ 1 + n b R n
which is the desired formula. This validates the results in [1], since it coincides with the approximate calculation of that paper in the asymptotic limit n although, as proven here, it turns out to be valid for any value of n.

Acknowledgements

I acknowledge financial support by the MEC (Spain) and FEDER (EU) through project FIS2007-60327.

References

  1. López-Ruiz, R.; Sañudo, J.; Calbet, X. Entropy 2009, 11, 959–971.
  2. Pathria, R.K. Appendix C. In Statistical Mechanics, 2nd ed.; Butterwort-Heinemann: Stoneham, MA, USA, 1996. [Google Scholar]
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