Why does electrical current start to flow? What happens microscopically when an electrical current starts to flow? I'd like to understand microscopically what happens in detail when electrons start moving (quasi-classically).
Electrons can have different velocity, they can produce electromagnetic fields, leads have free electrons and rigid atom cores and there exist electromagnetic fields. That's all the ingredients you should need?
Electrons only move due to EM fields, so basically this question boils down to what the EM fields look like and how they build up?! In steady state, what is the electric and magnetic field distribution in/around the lead? And what about the transient state?
What happens when you attack a battery to a lead? Are there EM fields between battery poles or why are electrons pushed? How do the EM field start to push electrons along an arbitraritly shaped long lead?
[EDIT: Ideally an explanation with the Drude model (which partly derives from Fermi model) or an explanation why that model isn't sufficient. Also stating the EM fields consistent with the electron density distribution would be important (i.e. $\vec{E}(r,\theta,z)$ and $\vec{B}(r,\theta,z)$) because otherwise it's hand-wavy arguments.]
(Please consider all remarks in this question. I know common arguments for parts of the question, but I've never seen a full microscopic in detail explanation.)
 A: The conduction electrons in a length of wire can be modelled as a (Fermi) gas. Just like molecules in a gas, individual electrons wander around randomly, and a local increase in density of electrons causes a locally higher pressure. You can even get electron density waves that are analogous to sound waves.
Using this analogy, suppose you have a tube full of air and you suddenly increase the pressure at one end. You will generate a pressure wave that travels down the tube causing gas molecules to start moving as it reaches them. Assuming you maintain the pressure at the end of the tube, the gas molecules will on average move down the tube. I say "on average" because any individual gas molecule diffuses at random, but when there is a pressure gradient the net flow is down the gradient.
All this pretty much transfers straight to the electrons in the wire. Your battery has an excess of electrons at the negative terminal and a deficit at the positive terminal (just like a charged capacitor in fact). When you connect it to the wire the excess electrons at the battery negative terminal start diffusing into the wire. Electrons in the wire then start diffusing away from the negative terminal along the wire and the end result is a voltage wave that travels down the wire at a few tenths of the speed of light. Because there is some resistance to the electron motion (assuming the wire isn't a superconductor!) you end up with a pressure, i.e. voltage, gradient along the wire.
It feels like I say this for every answer, but Wikipedia has articles on the electron gas model. See http://en.wikipedia.org/wiki/Free_electron_model and the links in that article like http://en.wikipedia.org/wiki/Nearly-free_electron_model
