The origin of the Boltzmann factor is revisited. An alternative derivation from the microcanonical picture is given. The Maxwellian distribution in a one-dimensional ideal gas is obtained by following this derivation. We also note other possible applications such as the wealth distribution in human society.

1.
A similar derivation was given in hyper-spherical polar coordinates in
X.
Calbet
and
R.
Lopez-Ruiz
, “
Extremum complexity distribution of a monodimensional ideal gas out of equilibrium
,”
Physica A
(to be published), or in arXiv:nlin.CD/0609035.
2.
A. J.
Stam
, “
Limit theorems for uniform distributions on spheres in high-dimensional euclidean spaces
,”
J. Appl. Probab.
19
,
221
228
(
1982
). When we were writing this note we became aware of this reference (Ref. 2), published in the mathematical context of statistics, where our result is given in a more general form. The abstract of Ref. 2 paper states: “If $X=(X1,X2,…,Xn)$ has uniform distribution on the sphere or ball in $Rn$ with radius $a$, then the joint distribution of $n1∕2a−1Xi$, $i=1,2,…,k$, converges in total variation to the standard normal distribution on $Rk$.” The case shown here is $k=1$.
3.

The kinetic theory approach that gives the evolution of the velocity distribution of a gas of particles out of equilibrium is the Boltzmann equation. The Boltzmann equation takes into account binary collisions and gives the Maxwellian distribution as a stable fixed point. This result is called the H-theorem. (Ref. 8) It is remarkable that the symmetries imposed by two-body collisions in the asymptotic equilibrium state determine the final distribution, that is, the Maxwellian distribution. Our derivation suggests that binary collisions provide a sufficient dynamical mechanism for the randomization of the energy among the particles [condition (2)] and, hence, are sufficient to establish the Boltzmann hypothesis of “molecular chaos,” which causes a gas to evolve toward equilibrium. We can go further and suggest that mechanisms other than the usual collisions among particles can also randomize the velocities and bring them to the Gaussian distribution.

4.
Ya. G.
Sinai
, “
On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics
,”
Sov. Math. Dokl.
4
,
1818
1822
(
1963
).
5.
H.
Bateman
,
Higher Transcendental Functions
(
McGraw-Hill
, New York,
1953
), Vol.
2
, Chap. XI.
6.
R.
Brockbank
,
J. M.
Huntley
, and
R. C.
Ball
, “
Contact force distribution beneath a three-dimensional granular pile
,”
J. Phys. II
7
,
1521
1532
(
1997
).
7.
A.
Dragulescu
and
V. M.
Yakovenko
, “
Statistical mechanics of money
,”
Eur. Phys. J. B
17
,
723
729
(
2000
);
Evidence for the exponential distribution of income in the USA
,”
Eur. Phys. J. B
20
,
585
589
(
2001
).
8.
K.
Huang
,
Statistical Mechanics
(
John Wiley & Sons
,
1963
), Chapter 4, pp. 68–69.