[3] A. Dragulescu and V. M. Yakovenko, Statistical Me- chanics of Money, Eur. Phys. J. B17, 2000, pp. 723- 729; Evidence for the Exponential Distribution of In- come in the USA, Eur. Phys. J. B20, 2001, pp. 585- 589. [4] R. Lopez-Ruiz, J. Sa˜nudo, and X. Calbet, Geometri- cal Derivation of the Boltzmann Factor, Am. J. Phys. 76, 2008, 780-781. [5] R. Lopez-Ruiz and X. Calbet, Derivation of the Maxwellian Distribution from the Microcanonical Ensemble, Am. J. Phys. 75, 2007, pp. 752-753. [6] R. Lopez-Ruiz, J. Sa˜nudo, and X. Calbet, Equiproba- bility, Entropy, Gamma Distributions and Other Geo- metrical Questions in Multi-Agent Systems, Entropy 11, 2009, 959-971. [7] C. Pellicer-Lostao, R. Lopez-Ruiz, Economic Mod- els with Chaotic Money Exchange, Lect. Notes Comp. Sci. 5544, 2009, 43-52. [8] J. Gonzalez-Estevez, M.G. Cosenza, O. Alvarez- Llamoza, R. Lopez-Ruiz, Transition from Pareto to Boltzmann-Gibbs Behavior in a Deterministic Eco- nomic Model, Physica A 388, 2009, 3521-3526. [1] K. Huang, Statistical Mechanics , John Wiley & Sons, New York, 1987; A. Munster, Statistical Thermody- namics , Volume I, Springer-Verlag, Berlin, 1969. [2] V.M. Yakovenko, Econophysics, Statistical Mechan- ics Approach to, Encyclopedia of Complexity and System Science , Meyers, R.A. Ed., Springer, Berlin, 2009, pp. 2800-2826. References: 4 Conclusion The calculation of the temperature T gives Thus ϵ = kT/ 2 , consistent with the equipartition the- orem. If p 2 is replaced by 1 $_{2}$mv $^{2}$, the Maxwellian dis- tribution is a function of particle velocity, as it is usu- ally given in the literature: We have shown that the Boltzmann factor describes the general statistical behavior of each small part of a multi-component system whose components or parts are given by a set of random variables that satisfy an additive constraint, in the form of a conservation law (closed systems) or in the form of an upper limit (open systems). The derivation of this factor for open sys- tems in a general context has been presented in [6]. Let us remark that these calculations do not need the knowledge of the exact or microscopic random- ization mechanisms of the multi-agent system in or- der to reach the equiprobability. In some cases, it can be reached by random forces [3], in other cases by chaotic [7] or deterministic [8] causes. Evidently, the proof that these mechanisms generate equiprobability is not a trivial task and it remains as a typical chal- lenge in this kind of problems. In summary, this work has presented a straightfor- ward geometrical argument that recalls us the equiv- alence between canonical and microcanonical ensem- bles in the thermodynamic limit in the particular con- text of physical sciences. For the general context of homogeneous multi-agent systems, we conclude by highlighting the statistical equivalence of the volume- based and surface-based calculations in this type of systems. Acknowledgements: This research was supported by the spanish Grant with Ref. FIS2009-13364-C02- C01. g ( v ) dv = √ m 2 πkT e − mv $^{2}$/ 2 $^{kT}$dv. (46) T − 1 = ( ∂S ∂E ) N = kN 2 E = k 2 ϵ . (45)