Foundation · Explanatory preview
What the diffusion coefficient means
D measures the rate of growth of a coordinate’s mean square: on an unbounded line that growth is 2D per unit time.
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What changes when D changes?
The time rate of change of density is D times its spatial curvature.
At a local dip with positive curvature, the diffusion term increases density. At a local peak with negative curvature it decreases density. Boundary conditions matter: a closed box conserves its contents, but its long-time distribution is not the unbounded Gaussian.
Fick’s constitutive law sends flux down the density gradient.
Combining the flux law with conservation gives the diffusion equation for constant D. The flux law itself is a model premise in this route, not something conservation alone proves.
One worked example
Holding everything else fixed, doubling D doubles the mean square at a given time and multiplies RMS displacement by √2. Four times the elapsed time doubles RMS displacement. Neither comparison specifies a single particle’s path.
A stopping point: D has units of length squared per time; boundary conditions and model premises must be stated.
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