# How do electrons in metals manage to have zero acceleration in constant $E$ field (as in a DC circuit)?

The simplest mathematical model would be a Newton's equation with viscous friction: $$m\ddot{x} = -\nu\dot{x} -eE,$$ which is written in terms of position and velocity as $$\dot{x} = v, \dot{v} = -\frac{\nu v}{m} -\frac{eE}{m},$$ and has a stationary solution for velocity: $$v = -\frac{eE}{\nu}.$$

A bit more realistic model is obtained by adding a deltta-correlated random force: $$m\ddot{x} = -\nu\dot{x} -eE + f(t),$$ where $\langle f(t) f(t')\rangle = D\delta(t-t').$ The average velocity, $\langle v\rangle$, here is given by the same solution as before, however one can also calculate the thermal fluctuations $\langle v^2\rangle$, which in a metal actually have higher velocity than the drift velocity of an electric current.

It is worth mentioning here the famous *Drude model*, although it is less suitable for description with Newton's equations: in this models the electrons do accelerate with a constant acceleration during the time between collisions. This model gives the same answer as discussed above, where the viscosity coefficient is replaced by $\nu = \frac{m}{\tau}$, where $\tau$ is the average time between collisions.