Chapter 1 · Animation 3 of 6 · Sections 1.3.3, 1.3.4 and 1.2.4
Diffusion: Random Walks Down a Gradient
No force pushes a diffusing carrier. Random motion plus more carriers on one side is enough for a net flow, and a field can hold that flow exactly in balance.
Density profile across the box (time averaged) and the theory curve
Flow across the dashed plane: diffusion, drift and the counted total
What to look for
Follow the tracked carrier: its walk has no preferred direction in every situation. Net flow appears only from the difference in how many carriers are on each side.
In the source-and-sink case the profile becomes a straight line, so the current is the same at every plane (no carriers are created or lost on the way).
In the closed box with a field the counted net flow is zero even though the density is far from uniform: diffusion and drift cancel, which is exactly the equilibrium of Section 1.3.4.
The physics behind it
Diffusion current: I = qD(bc)|dn/dx| (eq. 1.3.17). The Einstein relation D = μφt (eq. 1.3.18) ties it to mobility, because the same collisions limit both drift and diffusion.
Random walk: after a time t the spread is √(2Dt). The diffusion transit time across a length a to a perfect sink is a²/(2D) (Section 1.3.3).
Balance of drift and diffusion: μnℰ = D dn/dx gives n ∝ exp(ψ/φt), the Boltzmann relation of eq. 1.2.15. This is why every equilibrium profile in this course is exponential in potential.