# 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.