Electromagnetism: Conductors Even though the thermal velocity of electron in a conductor is comparatively high, the thermal velocity is not responsible for current flow? Why is this the case?
 A: that's because thermal motion is random in nature, you'll find almost same number of electrons moving in any direction at some specific time, so on average the net motion of charges(i.e., current) in any direction is zero, so no current due to thermal motion,
now, if you apply external field to the metal the overall random motion of charges starts drifting in the direction of field(for positive charges), now since this gives a net motion to the charges you get a current flowing in the metal,, this drifting of charges is the order of mm/s while thermal velocity is the order of km/s,, way higher than drift velocity !
in short,, current is zero due to thermal velocity because of the randomness.
A: Let's change that to "thermal velocity by itself is not responsible for current flow".
Thermal differences can produce current flow. And just like with electricity, a potential or change in thermal energy is required to convert the form of the energy to something else.
The reason heat cannot simply generate electric current is that in a material used to conduct electricity the construction is not designed to take advantage of temperature differentials. This is because no one can know exactly what kind of temperature differentials an object may be under prior to construction. If so than one would design the object with such heat conduction advantages. 
Also...
Conductive objects that are cooled, tend to conduct even better. This fact demonstrates that heat is not the source of conductivity. 
This is explained with the electron acting as a wave.
A: For something like a metallic crystal, if you apply an electric field then  the (Bloch) electrons just keep accelerating until they reach the end of the Brillouin zone (the momentum space box that they occupy), and then "wrap around" to the opposite end so that their average momentum is zero (Bloch oscillations). So a perfect crystal at 0 temperature would not conduct. What is needed is scattering by lattice defects or phonons (lattice vibrations) to break the symmetry and allow conductivity.
What mostly effects the conductivity is the mobility of the electrons.
