It can be easily proved that Rdθ = dp$_{i}$ $^{(1}$− p 2 i R 2 ) 1 / 2 . (37) f ( p ) dp = √ 1 2 πϵ e − p $^{2}$/ 2 $^{ϵ}$dp, (32) f ( p$_{i}$ ) = 1 S$_{N}$ ( R ) S$_{N}$ − 1 ( √ R 2 − p 2 $_{i}$) $^{(1}$− p 2 i R 2 ) 1 / 2 , (39) f ( p$_{i}$ ) = C$_{N}$R − 1 ( $^{1}$− p 2 i R 2 )$^{N$_{−}$}$$^{3}$ 2 , (40) C$_{N}$ = 1 √$_{π}$ Γ( N 2 ) Γ( N − 1 2 ) . (41) lim N ≫ $_{1}$C$_{N}$ ≃ 1 √$_{π}$ √ N 2 . (42) lim N ≫ 1 ( $^{1}$− p 2 i R 2 )$^{N$_{−}$}$$^{3}$ 2 ≃ e − p 2 $_{i}$/ 2 $^{ϵ}$. (43) S$_{N}$ ( R ) = 2 π N 2 Γ( N 2 ) R N − $^{1}$, (34) p 2 1 + p 2 2 $^{+}$· · · + p 2 N − 1 + p 2 N = R $^{2}$. (33) p 2 $_{1}$+ p 2 2 · · · + p 2 i − $_{1}$+ p 2 i +1 · · · + p 2 N = R $^{2}$− p 2 $_{i}$, (35) R $^{2}$cos$^{2}$θ = R 2 − p 2 $_{i}$, (36) S ( E ) = 1 2 kN ln ( E N ) + 1 2 kN (ln(2 π $^{)}$− 1) . (44) As in the former section, the Boltzmann 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. (42)–(43) into Eq. (40), we ob- tain the Maxwellian distribution (32) in the asymp- totic regime N → ∞ (which also implies E → ∞ ). Depending on the physical situation the mean en- ergy per particle ϵ takes different expressions. For an isolated one-dimensional gas we can calculate the de- pendence of ϵ on the temperature, which in the micro- canonical ensemble is defined by differentiating the entropy with respect to the energy. The entropy can be written as S = − kN ∫ ∞ $_{−∞}$f ( p ) ln f ( p ) dp , where f ( p ) is given by Eq. (32) and k is the Boltzmann con- stant. If we recall that ϵ = E/N , we obtain whose surface area is S$_{N}$ − 1 ( √ R 2 − p 2 $_{i}$) . If we define the coordinate θ as satisfying If the i th particle has coordinate p$_{i}$ , the ( N − 1) remaining particles share the energy R 2 − p 2 i on the ( N − 1) -sphere where $^{Γ(}$·) is the gamma function. If the ergodic hypothesis is assumed, that is, each point on the N - sphere is equiprobable, then the probability f ( p$_{i}$ ) dp$_{i}$ of finding the particle i with coordinate p$_{i}$ (energy p 2 $_{i}$) is proportional to the surface area formed by all the points on the N -sphere having the i th-coordinate equal to p$_{i}$ . Our objective is to show that f ( p$_{i}$ ) is the Maxwellian distribution, with the normalization con- dition (24). If we call ϵ the mean energy per particle, E = R 2 = Nϵ , then in the limit of large N we have For N ≫ 1 , Stirling’s approximation can be applied to Eq. (41), leading to We see that the isolated system evolves on the sur- face of an N -sphere. The formula for the surface area S$_{N}$ ( R ) of an N -sphere of radius R is We start by assuming a one-dimensional ideal gas of N non-identical classical particles with masses m$_{i}$ , with i = 1 , . . . , N , and total energy E . If particle i has a momentum m$_{i}$v$_{i}$ , newly we define a kinetic en- ergy K$_{i}$ given by Eq. (21), where p$_{i}$ is the square root of K$_{i}$ . If the total energy is defined as E ≡ R $^{2}$, we have with whose final form, after some calculation is Hence, the surface area of the N -sphere for which the i th coordinate is between p$_{i}$ and p$_{i}$ + dp$_{i}$ is S$_{N}$ − 1 ( R cos θ ) Rdθ . We rewrite the surface area as a function of p$_{i}$ , normalize it to satisfy Eq. (24), and obtain This newly shows 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 [5] that it is presented in the next section. where the index i has been removed because the dis- tribution is the same for each particle, and thus the ve- locity distribution can be obtained by averaging over all the particles. then in the asymptotic regime N → ∞ (which also implies E → ∞ ): 3.2 Multi-particle closed systems S$_{N}$ ( R ) = ∫ π/ 2 − π/ 2 S$_{N}$ − 1 ( R cos θ ) Rdθ. (38)