Equilibrium Distributions in Open and Closed Statistical Systems Abstract: In this communication, the derivation of the Boltzmann-Gibbs and the Maxwellian distributions is presented from a geometrical point of view under the hypothesis of equiprobability. It is shown that both distributions can be obtained by working out the properties of the volume or the surface of the respective geometries delimited in phase space by an additive constraint. That is, the asymptotic equilibrium distributions in the thermodynamic limit are independent of considering open or closed homogeneous statistical systems. Key–Words: Equiprobability, asymptotic equilibrium distributions, geometrical derivation In this paper, different classical results [1, 2] are ob- tained from a geometrical interpretation of different multi-agent systems evolving in phase space under the hypothesis of equiprobability. We start by deriving in section 2 the Boltzmann- Gibbs (exponential) distribution by means of the geometrical properties of the volume of an N - dimensional pyramid. The same result is obtained when the calculation is performed over the surface of a such N -dimensional body. In both cases, the moti- vation is a multi-agent economic system with an open or closed economy, respectively. Also, the Maxwellian (Gaussian) distribution is derived in section 3 from geometrical arguments over the volume or the surface of an N -sphere. Here, the motivation is a multi-particle gas system in contact with a heat reservoir (non-isolated or open system) or with a fixed energy (isolated or closed system), re- spectively. Last section contains our conclusions. Here we assume N agents, each one with coordinate x$_{i}$ , i = 1 ,... ,N , with x$_{i}$ ≥ 0 representing the wealth or money of the agent i , and a total available amount of money E : whose volume is V$_{N}$ − 1 ( E − x$_{i}$ ) . It can be easily proved that If the i th agent has coordinate x$_{i}$ , the N − 1 re- maining agents share, at most, the money E x$_{i}$ on the ( N − 1) -dimensional pyramid We suppose that each point on the N -dimensional pyramid is equiprobable, then the probability f ( x$_{i}$ ) dx$_{i}$ of finding the agent i with money x$_{i}$ is proportional to the volume formed by all the points into the ( $^{N}$− 1) -dimensional pyramid having the i th- coordinate equal to x$_{i}$ . We show now that f ( x$_{i}$ ) is the Boltzmann factor (or the Maxwell-Boltzmann distri- bution), with the normalization condition Under random or deterministic evolution rules for the exchanging of money among agents, let us suppose that this system evolves in the interior of the N - dimensional pyramid given by Eq. (1). The role of a heat reservoir, that in this model supplies money in- stead of energy, could be played by the state or by the bank system in western societies. The formula for the volume V$_{N}$ ( E ) of an equilateral N -dimensional pyramid formed by N + 1 vertices linked by N per- pendicular sides of length E is V$_{N}$ ( E ) = E N N ! . (2) ∫ E 0 f ( x$_{i}$ ) dx$_{i}$ = 1 . (3) x 1 + x 2 · · · + x$_{i}$ − 1 + x$_{i}$ +1 · · · + x$_{N}$ ≤ $^{E}$− x$_{i}$, (4) V$_{N}$ ( E ) = ∫ E 0 V$_{N}$ − 1 ( $^{E}$− x$_{i}$ ) dx$_{i}$. (5) x 1 + x 2 $^{+}$· · · + x$_{N}$ − 1 + x$_{N}$ ≤ E. (1) 2 Derivation of the Boltzmann- Gibbs Distribution 2.1 Multi-agent economic open systems 1 Introduction arXiv:1001.3327v1 [nlin.CD] 19 Jan 2010 RICARDO L ´ OPEZ-RUIZ Universidad de Zaragoza Department of Computer Science and BIFI Campus San Francisco, E-50009 Zaragoza SPAIN rilopez@unizar.es JAIME SA ˜ NUDO Universidad de Extremadura Department of Physics and BIFI Avda. de Elvas, E-06071 Badajoz SPAIN jsr@unex.es XAVIER CALBET Instituto de Biocomputaci´on y F´ısica de Sistemas Complejos (BIFI), Edificio Cervantes Corona de Arag´on 42, E-50009 Zaragoza SPAIN xcalbet@googlemail.com