Do electrons really move on the conductor? For my understanding:
Maxwell's equations combined with Poynting's theorem give us a model where electricity is energy carried along with the electromagnetic field (energy is stored in the field, not into kinetic energy of the electron in this model).
In around year 1900, the Drude model was created using kinetic gas theory and Boltzmann statistics. In this theory, electricity is carried by movement of "free electrons" that are accelerated by electric field and then "bounce" along the way. This is model I was taught in high school. We are able to get "Drift velocity" of the electron this way.
The theory was improved in 1933, combining the concepts of quantum mechanics and Fermi-Dirac statistics, resulting in Sommerfeld model that is still used in solid-state physics. Drift velocity is replaced with fermi velocity, and electrons that have fermi energy will take part in electric conductivity. With the quasi-particle model, semi conductors are explained pretty well and it's a very useful way to do it because even a single small grain of sand contains so many electrons that solving the exact Schrödinger partial differential equation would be practically impossible. Sometimes the real dynamics can be replaced by a quasi-particle model, which is based on the overall result of the interactions of a complex system.
So I'm now little bit confused about whole electricity. One model says we have some "moving electrons" on the wire/conductor, other says that we have probability for wave function to interact with neighbour wave function via photon field (or with a wave function more further away, the probability for this to happen would be smaller though), so the conductor/wire is just kind of tightly packed possibilities for electromagnetic interaction to happen by one way to look at it.
So my Question:
Can someone explain to me what electricity is and how it transports on the conductors? Basis of the Drude model seem so different to Maxwell's model.
I don't know how quantum field theory works so I will just leave it out for now. When I look at some QFT equations they just looks like algebraic mess for me... well defined mess, but mess anyway.
 A: I think the heart of your question is connected to how charge is distributed in steady state and during transients. The short answer is yes, both the models are useful, correct and consistent with each other.
It is not a trivial question there are a lot of sketchy explanations floating around. The American Journal of physics is a good place to find some good explanations.
Jacksons paper "Surface charges on circuit wires and resistors play three roles" probably answers your question most directly talking about both surface charge and Poynting vector. Preyer has two papers
Transient behavior of simple RC circuits and Surface charges and fields of simple circuits that have good explanations.
It is more complicated if your circuit is radiating like an antenna but a lot of circuits are slow and you can look at it from a quasistatic point of view thinking about how the electrons in the circuit respond to the other charges, and also considering that it takes time for a change in an electrons position to show up as a potential to an electron some distance away in the circuit.

In this figure from Preyer, at t=0 charge is put on the capacitor, and then for very small time steps the potential from all the electrons is computed and then the charge is moved according to the field. This is done by subdividing the circuit into very small cubes and using the Drude model and  $J=\sigma E$ for each little cube. This is a computationally hard way to do it since every cube of acts on every other cube.  It also has to consider the length of the circuit and the speed of light. As you can see it takes time for the fields to propagate and change the potential.  If you look closely at the figure the surface charge density is in gray scale, and you can see the charge density change at corners and along the length of the circuit.
In that example, the charge on the capacitor is supplying the potential, but you can also think about a potential applied to a long cylinder of conductor, and for a return path in the circuit imagine a larger coaxial cylinder far away, and find out the electric field with-in the conductor. In steady state the electric field goes a V=-Ez. Where z is the coordinate and E is parallel and uniform in the conductor.  So in steady state in long wire you have the potential drop along the length of the wire, with the electric field at the wire surface very nearly perpendicular to the wire, and the magnitude of the surface charge density proportional to the normal component of the electric field at that point.
If you think about current in the conductor you can use
$$J=\sigma E$$ where $\sigma$ is gotten from the Dude Model, $$J=env$$ where v is the drift velocity and n is the electron density. Thus we can have a connection between $\sigma$ and the drift velocity. In other words for a DC current we only care about the charge that moves through some cross-sectional area per unit time. Since the field is uniform and the electrons can't leave the circuit the drift velocity is what we care about.
That doesn't prevent us from also using the Poynting vector. We have a E field, we have a current from which we can compute the B field. In the cylindrical wire example we can compute the E and B fields everywhere and the power will flow according to the right hand rule. In the area outside the wire E is radial and B is coaxial to the wire, but the B field came from the electrons moving at the drift velocity through the wire.
Another minor point to consider is that is that there is a lot wrapped up in $\sigma$. Conductivity can vary over many orders of magnitude. This also means that the relaxation time can vary over many orders of magnitude. The relaxation time and is very short for metals ~ $10^{-19}$ seconds and can be very slow (days)for an insulator. Since most circuits, the electrons redistribute very quickly we don't don't worry about the relaxation time much in the wires, but for something might care about dielectric relaxation in capacitors.
