Maxwell-Boltzmann distribution in conductors? In the kinetic theory of gases, the Maxwell-Boltzmann distribution is commonly used to describe the velocity distribution of gas molecules. My question is to what extent (if at all) can such a distribution be used to model the velocity of free charges in a conducting, or semiconducting, solid (I'm primarily interested in semiconductors)?
I believe the gas model relies on the assumption that collisions are few and far between, which I suppose couldn't be said for a solid with the charges undergoing more collisions. Is there another distribution to describe this or is it simply that the mass is replaced by the effective mass of the charge in the lattice?
 A: At the room temperature kinetic energy of electrons is approximately two order ($\simeq$ 100) times less, than the Fermi energy, which usually $\sim 10^4$ K. I think, in most cases, degenerate Fermi gas https://en.wikipedia.org/wiki/Fermi_gas would be a nice approximation.
A: The Maxwell-Boltzmann distribution is suitable for semiconductors with low densities of charge carriers (electrons, holes). The derived velocities can then describe the diffusion of for example photo-excited charge carriers.
This depends on whether Boltzmann statistics is a reasonable approximation. The criterion is that the average distance between charge carriers is large compared to their de-Broglie wavelength, so that they can be treated as distinguishable particles.
In metals, the electron density is far too high for that.
A: The following scheme of subsequent approximations can be used to treat the electronic properties in solids.
Classical gas approximation
You treat the electron as a classical gas of non interacting free particles, you use Maxwell-Boltzmann distribution. This approximation fails completely, for instance you get a specific heat $\sim 100$ times larger than the experimental value. You are not able to explain why some solids are metals and some are insulators, and many more inconsistencies arise.
Quantum gas approximation
You treat the electrons as a quantum gas of non interacting free fermions, that obey Fermi-Dirac statistics instead of Maxwell-Boltzmann. With this you get reasonable results for equilibrium properties of metals, but still you have troubles to describe insulators or semiconductors. You still have some inconsistencies in the behaviour of transport properties (electrical and thermal conductivity for instance).
Quantum gas in periodic potential approximation
A lot of improvement is obtained considering the effect of the underlying crystal made up of ions, that produces a periodic potential landscape for electrons. Still you neglect interaction between electrons and you treat the crystal as a static object. With this you formulate band theory, and you are able to explain the presence of insulators and semiconductors as well. Still you can't explain superconductivity and some low temperature behaviours.
Quantum interacting gas
If you finally consider the Coulomb repulsion between electrons, and you treat the lattice not as a static object, but as a dynamical object (phonons theory), you recover a lot of physics. For instance you can explain a lot of low temperature phenomena, such as superconductivity and magnetism in materials. This is however a really complicated subject, still of current research interest.
For details about all of this, take a look at the very famous book by Ashcroft and Mermin.
A: Maxwell–Boltzmann statistics are often useful for semiconductors. Strictly speaking, Fermi-Dirac statistics are the only correct statistics for electrons. However, the "tail" of the Fermi-Dirac distribution looks just like a Boltzmann distribution. As a rule of thumb, you can use a Boltzmann distribution for energy levels outside the range $\left[E_f-3kT, E_f+3kT\right]$, where $E_f$ is the "Fermi level" ("chemical potential" is a better term), $k$ is Boltzmann's constant, and $T$ is temperature. (See Pierret's Semiconductor Device Fundamentals Sec 2.4.2 for more information.)
So, if for example, your chemical potential is more than $3kT$ below the conduction band minimum, then the electrons in the conduction band will (to a good approximation) follow Maxwell-Boltzmann statistics. You can do something similar for holes.
Now, you asked about the Maxwell-Boltzmann distribution. That follows directly from using Maxwell-Boltzmann statistics (derivation); no assumptions about collision rates or other things are needed.** So, I suppose that you could use it. However, there isn't that much use in knowing the velocity distribution of electrons in equilibrium.
The above only works because semiconductors have a gap that the chemical potential can fall in. Since there are no electrons in the gap, you don't need to worry about their distribution. Metals have no gap, so you can't ignore electrons with energies near the chemical potential. So, to my knowledge, Maxwell-Boltzmann statistics don't have much use in metals.
** Well, that's not quite true. There are some hidden assumptions. For example, there is the assumption that the particles (electrons) have a mass, which is actually a tricky thing for electrons in solids (and kind of beyond the scope of this answer). Strictly speaking, the Maxwell-Boltzmann distribution would need to be modified if the electrons have an anisotropic effective mass. However, in many cases you can approximate electrons as having an isotropic effective mass --- even if they don't (more info). All that said, if you have particles with a well defined mass (kinetic energy proportional to velocity squared), then if your particles follow Maxwell-Boltzmann statistics, their velocities will also follow a Maxwell-Boltzmann distribution.
A: Maxwell-Boltzmann depends on the particles spending most of their time not interacting.
But charges do interact, they interact cumulatively, so that it's a decent approximation to suppose that they are each on the outer surface, repelled by all the others.
There's probably a good way to modify the distribution to account for that, but I don't know how yet.
