A contradiction between the Drude Model and Ohm's Law? Using the Drude Model, we get that
$$\vec{J}=\frac{ne^2 \tau}{m_e}\vec{E}$$
where $\tau$ is the mean free time of each electron and $n$ is the free electron density. This is simply ohms law and hence the Drude model provides a microscopic explanation for Ohms Law. But vital in this formula is the fact that $\tau$ is independent of $\vec{E}$. But this assumption seems to tear the logic of the whole model into shreds. Just as a starter, If $\tau$ is independent of $\vec{E}$, then an increase in $\vec{E}$ will not change the amount of collisions that occur per unit time (because the amount of collisions per unit time is exclusively a function of $\tau$ and $n$). The same amount of collisions per unit time means that the amount of energy dissipated in the wire/resistor should be independent of the electric field. This is obviously absurd though.
Secondly, the fact that the Drude model predicts Ohms law means that it also shows that the energy dissipated in the collisions (within a resistor) is proportional to $E^2$ on average (since $P=V\cdot I=V^2/R$). Now if we examine the situation of a single electron in a vacuum in a constant electric field $\vec{E}$ and suppose that at time $t=0$ it has zero kinetic energy($K(0)=0$), then at time $t=t_f$, it will have a kinetic energy equal to
$$K(t)=\frac{1}{2m_e}(eEt_f)^2$$
From this we can see that for a given time interval $\Delta t=t_f-0$, the kinetic energy gained by the electron (that is, the work done on the electron) is directly proportional to the square of the electric field. The reason for this quadratic dependence is that if we double $\vec{E}$, then we necessarily also double the distance that the force acts over within a given time interval because the particle moves faster and hence further in that time interval. Thus doubling $\vec{E}$ leads to 4 times the work for a constant time interval. However, if we now go back to the drude model, the mean free time and the mean free distance are assumed to be independent of $\vec{E}$ and only dependent on T. So if we do in fact have a constant $\tau$ and a constant mean free distance that are both independent of $\vec{E}$, then doubling the electric field strength cannot possibly quadruple the power because the electric force is still being exerted over the same distance in any given time interval. The only way that we can explain the $E^2$ power dissipation in a resistor by the Drude model seems to be that we must assume that the mean free distance increases in direct proportion with the electric field. But no reputable sources seem to indicate this. For example, in Ashcroft and Mermin, they assume that that the mean free distance is solely determined by the temperature (the average thermal speed). So how can the Drude model make any sense whatsoever given what I've just described?
Any help on this issue would be most appreciated as this issue has been driving me mad!
 A: Good catch! You've identified the fundamental problem with the Drude model: you get different assumptions if you assume the electrons travel a constant distance before colliding, and if you assume they travel a constant time.
If you assume a constant time $\tau$, then the average velocity of the electrons is $v \propto E \tau \propto E$, which is the expected result. But if you assume a constant distance $\ell$, then the typical speed $v$ of the electrons obeys $mv^2 \sim E \ell$ which means $v \propto \sqrt{E}$, a completely different result! But of course, if we think in terms of what causes collisions in the first place, a constant distance sounds much more reasonable than a constant time, which means the Drude model doesn't seem to make sense.
The reason solid state textbooks don't mention this subtlety is because the Drude model is not remotely true anyway; it's just a stepping stone to learn the more accurate Sommerfeld model. In the Sommerfeld model, the conducting electrons have extremely high speeds because of the Pauli exclusion principle, even when the electric field is off. This means that the electric field has little effect on the typical speed, so a collision per constant distance traveled is the same as a collision per constant time, and there's no need to distinguish between the two.
