Limitations of the models we've developed

In solids, diffusion is not necessarily mediated by a pure random walk. In vacancy diffusion, there are correlations. Specifically, a vacancy is more likely to jump back to where it was one jump ago than to jump somewhere else. However, at the macro-level, this only affects the value for D, but not the diffusion equation.
In solids, there can also be chemical potential effects in which D is not independent of the concentration. In this case, the diffusion equation is written $\frac{\partial c}{\partial t}=\frac{\partial c}{\partial x}(D(c)\frac{\partial c}{\partial x})$ Concentration is of course a function of x, so in effect D is a function of x, i.e. diffusion speed depends on position in the material due to concentration differences across space. This is explored in a later matsci course.