Editing Diffusion equation (section)

== Derivation ==

The diffusion equation can be trivially derived from the [[continuity equation]], which states that a change in density in any part of the system is due to inflow and outflow of material into and out of that part of the system. Effectively, no material is created or destroyed:
<math display="block">\frac{\partial\phi}{\partial t}+\nabla\cdot\mathbf{j}=0,</math>
where '''j''' is the flux of the diffusing material. The diffusion equation can be obtained easily from this when combined with the phenomenological [[Fick's law|Fick's first law]], which states that the flux of the diffusing material in any part of the system is proportional to the local density gradient:
<math display="block">\mathbf{j}=-D(\phi,\mathbf{r})\,\nabla\phi(\mathbf{r},t).</math>

If drift must be taken into account, the [[Fokker–Planck equation]] provides an appropriate generalization.