Drude model

Remarks

For a constant field the steady state is $p = -e \tau E$, so $J = \sigma_0 E$ with the DC conductivity $\sigma_0 = n e^2 \tau / m$: this is Ohm's law. For a time-harmonic field $\mathrm{Re}(E_0 e^{-i\omega t})$ the steady state gives $J = \sigma(\omega) E$ with $$\sigma(\omega) = \frac{\sigma_0}{1 - i \omega \tau}.$$ The relative permittivity $\varepsilon_r = 1 + i \sigma / (\varepsilon_0 \omega)$ is then $$\varepsilon_r(\omega) = 1 - \frac{\omega_p^2}{\omega^2 + i \omega / \tau}, \qquad \omega_p^2 = \frac{n e^2}{\varepsilon_0 m}.$$ This is the Lorentz oscillator model with natural frequency $0$ and damping rate $1/\tau$. For $\omega \tau \gg 1$, $\varepsilon_r$ is close to $1 - \omega_p^2 / \omega^2$, negative for $\omega < \omega_p$, where the metal reflects, and positive for $\omega > \omega_p$, where it becomes transparent.

Example: in a constant field $E$, the function $p(t) = -e\tau E + (p(0) + e\tau E) e^{-t/\tau}$ solves the equation of motion and tends to the steady state $-e\tau E$.

Non-example: electrons bound to atoms by a restoring force do not fit the model. Their equation of motion has the additional term $-m \omega_0^2 x$ with $\omega_0 > 0$, and they are described by the Lorentz oscillator model instead.