# Ergodicity of the Drude model

The Drude model of electric conduction in solids deals with independent free electrons subject to random collisions with the crystal lattice (the direction where the electrons are scattered after a collision is random).

A simplified model is the Lorentz gas, where the collision are deterministic. If I understand correctly, it was shown by Sinai that this model is ergodic.

What about the original Drude model: is it ergodic? (Are there references on that?) A side question is whether this would have a physical significance?

• no need to go to the Drude model, if you simulate an ideal gas it's not ergodic either or the ergodicity property might depend on the shape of the box – gatsu Nov 5 '13 at 9:56
• It's tricky to classify the Drude model as ergodic or not, since it is a stochastic model. Ergodicity is mainly a subject of study with reversible dynamics, because it's very interesting if you can establish ergodicity in one of those systems. With the Drude model, after the typical time required for a particle to traverse the material, the system could be literally in any state due to all the randomness dumped into it. So in that sense it's "ergodic" however it's not a very interesting property. – Nanite Nov 11 '13 at 22:44

As far as I know, Gallavotti proved the ergodicity of the Lorentz gas, while Sinai proved that of a system of $N \leq 5$ rigid spheres. Anyway, this is a minor detail. For certain aspects, a more suitable model for the Drude model is the Boltzmann gas. Lanford has shown (in 1970s, I think) that the entropy for this model is always increasing, but anyone has proved that Boltzmann gas is ergodic. So the answer to your question is: if, at our level of accuracy, Lorentz gas was an appropriate mathematical scheme for the Drude model, than it would be ergodic. Else we can't conclude, since Sinai's result is very important but too limited. (at the present day.)

However, it is an interesting question from a mathematical point of view (for me, for example, it really is), but for a physicist it is not very important, since Drude model does not provide an appropriate level of precision for most of calculations carried out nowadays in solid state physics (at least, concerning what I have seen in my course of condensed matter, I'm not a specialist in solid state physics). Moreover, any of that models takes in account coulombian interactions between charged particles. (this remark restricts - in principle - a lot the domain of applicability of such schematizations.)

I think you could find very interesting the treatments of this subject (ergodic theory) by Halmos and Arnold in their classical monographs.

References. P. R. Halmos, Lectures on ergodic theory

V.I. Arnold, Ergodic problems of classical mechanics and Mathematical methods of classical mechanics

G. Gallavotti, Statistical mechanics and The elements of mechanics

• Thanks ! Indeed, the Drude model is not anymore used in condensed matter. When there is no (or low) disorder, a semi-classical treatment of Bloch electrons is the usual way to describe electronic transport, and it works very well. I wondered about the Drude model because it is fairly simple ; ergodic theory is fascinating, but it is very difficult to see the consequences it has or could have on the physics of a model (perhaps because I don't know much on the subject). – Georg Sievelson Nov 5 '13 at 1:57
• Essentially, if a system is ergodic, then temporal averages coincide with spatial averages. This turns out to be equivalent to the fact that there aren't invariant non-trivial sets under the action of the evolution operator and this leads to the important result that averages are independent on the starting point. Many authors see in it a "justification" for the statistical mechanics. (but such a view is not universally accepted.) – user91126 Nov 5 '13 at 18:54
• However, one usually carries out numerical calculations and these are rather independent on the existence of a rigorous theory. In my opinion, from a physical point of view, this is a situation similar to that of QFT: we can do incredibly accurate predictions by means of perturbation theory or even numerical non-perturbative calculations, but we don't even know if a complete theory exists. – user91126 Nov 5 '13 at 19:01