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economic agents with random exchange interactions [2]. 

Depending on the physical situation, the mean wealth per agent ϵ takes different expressions and in- terpretations. For instance, we can calculate the de- pendence of ϵ on the temperature, which in the micro- canonical ensemble is defined by the derivative of the entropy with respect to the energy. The entropy can be written as S = − kN ∫ ∞ 0 f ( x ) ln f ( x ) dx , where f ( x ) is given by Eq. (9) and k is Boltzmann’s constant. If we recall that ϵ = E/N , we obtain 

The calculation of the temperature T gives 

Thus ϵ = kT , and the Boltzmann distribution is ob- tained in its usual form: 

## 3 Derivation of the Maxwellian Dis- tribution 

## 3.1 Multi-particle open systems 

Let us suppose a one-dimensional ideal gas of N non- identical classical particles with masses m$_{i}$ , with i = 1 , . . . , N , and total maximum energy E . If particle i has a momentum m$_{i}$v$_{i}$ , we define a kinetic energy: 

where p$_{i}$ is the square root of the kinetic energy K$_{i}$ . If the total maximum energy is defined as E ≡ R $^{2}$, we have 

We see that the system has accessible states with dif- ferent energy, which can be supplied by a heat reser- voir. These states are all those enclosed into the vol- ume of the N -sphere given by Eq. (22). The formula for the volume V$_{N}$ ( R ) of an N -sphere of radius R is 

where $^{Γ(}$·) is the gamma function. If we suppose that each point into the N -sphere is equiprobable, then the 

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probability f ( p$_{i}$ ) dp$_{i}$ of finding the particle i with co- ordinate p$_{i}$ (energy p 2 $_{i}$) is proportional to the volume formed by all the points on the N -sphere having the i th-coordinate equal to p$_{i}$ . We proceed to show that f ( p$_{i}$ ) is the Maxwellian distribution, with the normal- ization condition 

If the i th particle has coordinate p$_{i}$ , the ( N − 1) remaining particles share an energy less than the max- imum energy R 2 − p 2 i on the ( N − 1) -sphere 

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whose volume is V$_{N}$ − 1 ( √ R 2 − p 2 $_{i}$) . It can be easily proved that 

Hence, the volume of the N -sphere for which the i th coordinate is between p$_{i}$ and p$_{i}$ + dp$_{i}$ is V$_{N}$ − 1 ( √ R 2 − p 2 $_{i}$) dp$_{i}$ . We normalize it to satisfy Eq. (24), and obtain 

whose final form, after some calculation is 

with 

For N ≫ 1 , Stirling’s approximation can be applied to Eq. (29), leading to 

If we call ϵ the mean energy per particle, E = R 2 = Nϵ , then in the limit of large N we have 

The factor e − p 2 $_{i}$/ 2 ϵ is found when N ≫ 1 but, even for small N , it can be a good approximation for particles with low energies. After substituting Eqs. (30)–(31) into Eq. (28), we obtain the Maxwellian distribution 

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