Editing Drude model (section)

=== Time-varying analysis ===
[[File:DrudeResponse.gif|thumb|300 px|right|Drude response of current density to an AC electric field.]]

The dynamics may also be described by introducing an effective drag force. At time {{math|''t'' {{=}} ''t''<sub>0</sub> + ''dt''}} the electron's momentum will be:
<math display="block">\mathbf{p}(t_0+dt) = \left( 1 - \frac{dt}{\tau} \right) \left[\mathbf{p}(t_0) + \mathbf{f}(t) dt + O(dt^2)\right] + \frac{dt}{\tau} \left(\mathbf{g}(t_0) + \mathbf{f}(t) dt + O(dt^2)\right)</math>
where <math>\mathbf{f}(t)</math> can be interpreted as generic force (e.g. [[Lorentz force]]) on the carrier or more specifically on the electron. <math>\mathbf{g}(t_0)</math> is the momentum of the carrier with random direction after the collision (i.e. with a momentum <math>\langle\mathbf{g}(t_0)\rangle = 0</math>) and with absolute kinetic energy 
<math display="block">\frac{\langle|\mathbf{g}(t_0)|\rangle^2}{2m} = \frac{3}{2} KT.</math>

On average, a fraction of <math>\textstyle 1-\frac{dt}{\tau}</math> of the electrons will not have experienced another collision, the other fraction that had the collision on average will come out in a random direction and will contribute to the total momentum to only a factor <math>\textstyle \frac{dt}{\tau}\mathbf{f}(t)dt</math> which is of second order.<ref group="Ashcroft & Mermin" name=":2">{{harvnb|Ashcroft|Mermin|1976|p=11}}</ref>

With a bit of algebra and dropping terms of order <math>dt^2</math>, this results in the generic differential equation
<math display="block">\frac{d}{dt}\mathbf{p}(t) = \mathbf{f}(t) - \frac{\mathbf{p}(t)}{\tau}</math>

The second term is actually an extra drag force or damping term due to the Drude effects.