Electrons as a pinball gas, bouncing through a lattice with mean free time τ between collisions. Push with a field and the whole gas drifts — glacially — and out falls Ohm's law: σ = n e² τ / m, so J = σE. Down the center, data flows: the metal's numbers go in, the engine computes conductivity and drift, the proven law comes out. The blue team builds and defends it; the red team attacks it.
source P. Drude, Zur Elektronentheorie der Metalle, Ann. Phys. 306 (1900) 566–613 — doi:10.1002/andp.19003060312. Rendered, not quoted.
One idea, four moves. 1 Electrons are a free classical gas; ions are fixed scatterers. 2 Between collisions they fly ballistically; each collision randomizes velocity, memory erased after mean free time τ. 3 A field E accelerates them for ~τ, giving a steady drift v = −(eτ/m)E. 4 Current density J = n(−e)v = (n e²τ/m) E = σE — Ohm's law, derived not assumed.
For the current metal, the engine's live numbers:
| quantity | symbol | value |
|---|
Drude's free-electron gas (1900) is the classical floor everything later stands on. Keep σ = n e²τ/m and the mobility μ = eτ/m; then quantize the gas (Fermi–Dirac, Sommerfeld) and you get the-band-gap; add a magnetic field to the very same drift equation and you get the-hall-effect, which reads off n and its sign.
Each sphere is the next one's premise: the pinball picture is where "carrier density" and "scattering time" are born.
The blue team's live check: recompute σ from the law, confirm J is linear in E, confirm σ = n e μ, and confirm heavier electrons conduct worse. If red tampers with the mass term, this badge is where it shows.
A metal enters as four numbers: carrier density n (free electrons per m³), mean free time τ (seconds between collisions), applied field E (V/m), and temperature T (K). Fixed constants: electron charge e, mass m, Boltzmann kB.
Phonon scattering makes τ fall as T rises (modelled τ(T) = τ₀·300/T), so resistivity climbs with temperature — the hallmark of a metal. Feed these into the panel below.
Pinball gas: dots fly ballistically, scatter, and drift right with the field (drift is exaggerated ~10⁹× to be visible — the real ratio is in the readout).
Change any control — σ, ρ, μ, and the drift are computed from σ = n e²τ/m on the spot, never looked up.
What the machine proves: σ = n e² τ / m gives J = σE (current strictly linear in field), the mobility μ = eτ/m so σ = n e μ, the drift |vd| is a whisper next to the thermal speed yet carries all the current, and resistivity ρ = 1/σ rises with T because τ falls. Heavier electrons conduct worse (σ ∝ 1/m).
The blue team's witness (left) confirms these live; the red team (right) tries to invert the mass term.
So the free-electron picture is an idealization (AMBER): σ = n e²τ/m survives because τ and n are effective parameters, but any claim about where τ comes from or about heat capacity needs quantum mechanics and band structure.
"Electrons zip through the wire near light-speed to turn the light on." Cut. The drift speed is ~10⁻⁴ m/s — slower than an ant. The field propagates near c; the electrons barely crawl. The engine prints the real vd.
"Drude got the electronic specific heat right." Cut. Classical equipartition overpredicts it ~100×; only Fermi–Dirac fixes it.
"Heavier carriers carry more current." Cut. σ = n e²τ/m — conductivity falls as 1/m. This is exactly the tamper on the right.
The red team's move: flip the mass term, σ = n e²τ·m instead of ÷m, so heavier electrons wrongly conduct better. The blue team's witness (window 7) is watching.
Multiply by mass instead of dividing and σ inverts its mass dependence — the witness recomputes, finds heavier electrons "conducting better" and σ ≠ n e μ, and turns red. Nothing is faked; the attack is real and it is caught.