# What the diffusion coefficient means

Authored explanation; editorial review pending.

$$
\frac{\partial p}{\partial t}=D\frac{\partial^2p}{\partial x^2}
$$

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.

$$
J=-D\frac{\partial c}{\partial x}
$$

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.

## 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.

D has units of length squared per time; boundary conditions and model premises must be stated.