I'm not very familiar with the subject matter, so it's possible that this is an easy question. Here goes: Let's say I have an ideal solid in which the atoms are unable to translate or rotate but have a certain distribution of vibrational energy related to the temperate. If there is a temperature gradient in the metal than this distribution will change along with it, but let's assume we are at equilibrium. Now since even solids can glow when heated, I'm assuming there is also a distribution function for the electronic states (with radiative lifetimes and quenching rates?) that is also, assuming equilibrium, related to the temperature. Correct me if I'm wrong so far. So it seems to me that since metals share electrons more willingly, the equilibrate much more quickly. Is this why they make good thermal conductors? Now for the real question: If I pass a current through a wire at steady state, is there a way to calculate these distributions? Will the excited states be uniformly distributed, in agreement with any temperature gradient, or mostly along the surface?