An ideal gas obeys Maxwell Boltzman statistics. Gas atoms have an average kinetic energy of 3kT/2 but the individual atoms have energies that vary from this average (Chi square distributed). A solid (metal) has an average kinetic energy of about 3kT. Does anybody know what statistical distribution the energy of individual atoms of the metal obey?
I would conjecture that they would also obey a distribution similar to, if not equivalent to, that of a ##\chi^2## distribution, though the formula would probably be a lot more involved. If you want to really know more about this kind of thing, statistical mechanics is the topic you want to look into.
After some scaling of the variables, the energy is a sum of six squares: ##E=x^2+y^2+z^2+p_x^2+p_y^2+p_z^2##, hence E has also a chisquare distribution but not with 3 but 6 degrees of freedom.