I Drude model with and without an electric field

[aaronll]

Why in the Drude model without e-field no energy is transfer by electrons to ions, but when there is an e-field electrons transfer energy to ions ?

 

[BvU]

Why in the Drude model without e-field no energy is transfer by electrons to ions

What, precisely, gives you that impression ?
One of the assumptions says

The electron is considered to be immediately at equilibrium with the local temperature after a collision.

i.e. the electrons pick up just as much energy as they lose.

$\$

 

[aaronll]

What, precisely, gives you that impression ?
One of the assumptions says i.e. the electrons pick up just as much energy as they lose.

##\

that is clear, right, so with an electric field electron gain energy between collision and they transfer energy (kinetic energy) to lattice ion, and that energy is equal to the work done by the e-field on the electron?
So in "reality" electron gains velocity between collision but they lose that energy in collision and on average the velocity is the drift velocity?

 

[vanhees71]

It's an effective description of an electron in the medium. The interaction with the medium is described by a linear friction term and thus the equation of motion for the electron reads
$$m \dot{\vec{v}}=-\frac{m}{\tau} \vec{v}-e \vec{E}.$$
If $\tau$ is small enough the velocity "relaxes" quickly to the equilibrium limit, where the force vanishes, i.e.,
$$\vec{v}=-\frac{e \tau}{m} \vec{E}.$$
Now for DC you assume the electrons within the conductor are all independent and then the relation to the elctric current is
$$\vec{j}=-e n \vec{v}=\frac{e^2 n \tau}{m} \vec{E} \; \Rightarrow \; \sigma=\frac{e^2 n \tau}{m}.$$