Why is velocity normally distributed in a gas, but not energy? If one looks at a cubic box of gaseous atoms all initially flying in the same direction at the same speed (but flying at an angle to the walls, so as not to reflect up-and-down against the box walls forever), they will collide with the walls and each other, their previously uniform velocities becoming messed up randomly until they are distributed according to a Maxwell-Boltzmann distribution, which is a normal distribution. Looking at the set-up as a random-generator, this can be considered an application of the central limit theorem, which says that the mean of  a large number of random variables will end up being normally distributed.
My question: Why doesn't the same reasoning hold for energy? Aren't the energies of the atoms in such a box also random variables? However, the energies follows the Boltzmann distribution, which isn't a normal distribution.
Of course, the relationship between speed and energy prohibits both energy and speed being distributed normally, but doesn't this finding contradict the central limit theorem?
 A: The more natural relationship between the two distributions is the opposite one. The Boltzmann distribution
$$\exp(-E/kT)$$
is the more general one (connected with the microscopic definition of the temperature $T$ in any system in physics) and one may simply substitute the kinetic energy $mv^2/2$ for $E$ to get the Maxwell part of the distribution.
The non-normal distribution of the energy doesn't contradict the central limit theorem because the energy of a gas molecule after $N$ collisions isn't a simple sum of $N$ contributions to the energy. Instead, a molecule is likely to lose lots of energy in a collision if its initial energy before the collision (e.g. one accumulated from the previous collisions) was high to start with. So the previous history matters which makes the evolution of the energy non-Markovian or non-linear, if you wish.
The central limit theorem only talks about the distribution of a quantity that is a sum (or average) of many terms with a fixed distribution but this isn't the case for energy after many collisions. On the contrary, it is the case for the momentum (in non-relativistic physics, assuming elastic collisions).
It would be a clear inconsistency if e.g. kinetic energy were predicted to be normally distributed. In particular, the kinetic energy can't be negative but the normal distribution is nonzero for arbitrarily large positive and negative values of the variable.
