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The Drude Model of Conduction

Why does a metal obey Ohm's law? Drude answered it in 1900 with a picture so crude it should not work, and yet it does: treat the conduction electrons as a gas of tiny balls rattling around inside the metal, colliding at random with a mean time $\tau$ between hits. Each collision wipes out the memory of the electron's previous motion and sends it off in a random thermal direction at high speed. Switch on an electric field and, between collisions, every electron picks up a small extra velocity along the force, only to have it erased at the next hit. The result is not a runaway acceleration but a steady average drift, $v_d = -eE\tau/m$, a tiny systematic bias riding on top of the furious random motion. The scene shows exactly this: electrons (blue) zigzagging off impurities (red) at thermal speeds, while the whole cloud creeps slowly in the direction the field pushes them. The drift is exaggerated here for visibility (in a real wire it is a snail's pace, microns per second, against thermal speeds of hundreds of kilometers per second). Sum the drift over all the electrons and you get a current proportional to the field, $j = \sigma E$ with conductivity $\sigma = ne^2\tau/m$: Ohm's law falls out, and the lower-left panel shows the simulated current landing right on that straight line. The lower-right panel adds the model's other prediction, the way the conductivity rolls off when the field oscillates faster than the electrons can respond, above $\omega = 1/\tau$.

Figure 1. The Drude model. Top: electrons (blue) scatter off impurities (red) at thermal speed with a slow drift (the field E points right, electrons drift left, conventional current j flows right). Bottom left: current against field, Ohm's law j = sigma E, with the simulated current marked. Bottom right: the AC conductivity magnitude rolling off above omega = 1/tau. Method: stochastic Drude dynamics plus the analytic relations. Source: Ashcroft and Mermin, Solid State Physics, Ch. 1.
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WHAT TO TRY

  • Raise the field $E$: the drift speeds up in proportion, and the current climbs along the Ohm's-law line.
  • Raise the scattering time $\tau$ (cleaner metal): both the drift and the conductivity grow, since $\sigma\propto\tau$.
  • Compare the green theory point and the yellow simulated current: the random dynamics reproduce $v_d = -E\tau$.
  • Watch the AC panel: longer $\tau$ moves the rolloff $\omega=1/\tau$ to lower frequency.