Comment on “Equiprobability, Entropy, Gamma Distributions and Other Geometrical Questions in Multi-Agent Systems”, Entropy 2009, 11, 959-971

Raúl Toral

doi:10.3390/e11041121

Abstract

The volume of the body enclosed by the n-dimensional Lamé curve defined by

\sum_{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 \leq \sum_{i = 1}^{n} x_{i}^{b} \leq E

for

b > 0

,

x_{i} \geq 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

\begin{matrix} V_{n} (R) = & \int d x_{1} \dots \int d x_{n} \\ _{0 \leq \sum_{i = 1}^{n} x_{i}^{b} \leq R^{b}} \end{matrix}

(1)

By dimensional analysis

V_{n} (R) = C_{n} R^{n}

. Let us now compute the integral

\int_{0}^{\infty} d x_{1} \dots \int_{0}^{\infty} d x_{n} exp [- (x_{1}^{b} + \dots + x_{n}^{b})] = {[\int_{0}^{\infty} d x exp [- x^{b}]]}^{n} = {[Γ (1 + \frac{1}{b})]}^{n}

(2)

by using the change of variables

r = {(x_{1}^{b} + \dots + x_{n}^{b})}^{1 / b}

and the volume element

d V_{n} (r) = n C_{n} r^{n - 1} d r

as

\int_{0}^{\infty} d V_{n} (r) exp [- r^{b}] = C_{n} Γ (1 + \frac{n}{b})

(3)

Equaling these two expressions one gets:

V_{n} (R) = \frac{{[Γ (1 + \frac{1}{b})]}^{n}}{Γ (1 + \frac{n}{b})} R^{n}

(4)

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 \to \infty

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

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

© 2009 by the author; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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