content_image/405911.jpg content_image/405910.jpg content_image/405908.jpg content_image/405909.jpg content_image/405902.jpg content_image/405903.jpg content_image/405907.jpg content_image/405900.jpg content_image/405906.jpg content_image/405905.jpg content_image/405899.jpg content_image/405901.jpg
Hence, the volume of the N -dimensional pyramid for which the i th coordinate is between x$_{i}$ and x$_{i}$ + dx$_{i}$ is V$_{N}$ − 1 ( E − x$_{i}$ ) dx$_{i}$ . We normalize it to satisfy Eq. (3), and obtain 

whose final form, after some calculation is 

If we call ϵ the mean wealth per agent, E = Nϵ , then in the limit of large N we have 

The Boltzmann factor e − x i /ϵ is found when N ≫ 1 but, even for small N , it can be a good approximation for agents with low wealth. After substituting Eq. (8) into Eq. (7), we obtain the Maxwell-Boltzmann distri- bution in the asymptotic regime N → ∞ (which also implies E → ∞ ): 

where the index i has been removed because the distri- bution is the same for each agent, and thus the wealth distribution can be obtained by averaging over all the agents. This distribution has been found to fit the real distribution of incomes in western societies [3]. 

This means that the geometrical image of the volume-based statistical ensemble allows us to re- cover the same result than that obtained from the mi- crocanonical ensemble [4] that we show in the next section. 

## 2.2 Multi-agent economic closed systems 

Here, we derive the Boltzmann-Gibbs distribution by considering the system in isolation, that is, a closed economy. Without loss of generality, let us assume N interacting economic agents, each one with coordinate x$_{i}$ , i = 1 ,... ,N , with x$_{i}$ ≥ 0 , and where x$_{i}$ repre- sents an amount of money. If we suppose now that the total amount of money E is conserved, 

then this isolated system evolves on the positive part of an equilateral N -hyperplane. The surface area S$_{N}$ ( E ) of an equilateral N -hyperplane of side E is given by 

<img src='content_image/405911.jpg'>

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Different rules, deterministic or random, for the ex- change of money between agents can be given [2]. Depending on these rules, the system can visit the N -hyperplane in an equiprobable manner or not. If the ergodic hypothesis is assumed, each point on the N -hyperplane is equiprobable. Then the probability f ( x$_{i}$ ) dx$_{i}$ of finding agent i with money x$_{i}$ is pro- portional to the surface area formed by all the points on the N -hyperplane having the i th-coordinate equal to x$_{i}$ . We show that f ( x$_{i}$ ) is the Boltzmann factor (Boltzmann-Gibbs distribution), with the normaliza- tion condition (3). 

If the i th agent has coordinate x$_{i}$ , the N − 1 re- maining agents share the money $^{E}$−x$_{i}$ on the ( $^{N}$− 1) - hyperplane 

<img src='content_image/405907.jpg'>

whose surface area is S$_{N}$ − 1 ( $^{E}$− x$_{i}$ ) . If we define the coordinate θ$_{N}$ as satisfying 

it can be easily shown that 

Hence, the surface area of the N -hyperplane for which the i th coordinate is between x$_{i}$ and x$_{i}$ + dx$_{i}$ is proportional to S$_{N}$ − 1 ( E − x$_{i}$ ) dx$_{i}$/ sin θ$_{N}$ . If we take into account the normalization condition (3), we obtain 

whose form after some calculation is 

<img src='content_image/405900.jpg'>

If we call ϵ the mean wealth per agent, E = Nϵ , then in the limit of large N we have 

As in the former section, the Boltzmann factor e − x i /ϵ is found when N ≫ 1 but, even for small N , it can be a good approximation for agents with low wealth. After substituting Eq. (17) into Eq. (16), we obtain the Boltzmann distribution (9) in the limit N → ∞ (which also implies E → ∞ ). This asymptotic re- sult reproduces the distribution of real economic data [3] and also the results obtained in several models of 

<img src='content_image/405906.jpg'>

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