What is the difference between the diffusion equation and the heat equation?

I know that the diffusion equation is a more general version of the heat equation. But what is the exact difference informally?

• We have the laplace operator in the diffusion equation, and the second derivative in $x$ in the heat equation? Is that the difference? So, the heat equation brings it down to a single dimension? – onimoni Feb 19 '15 at 15:42

The difference is typically the diffusion coefficient: \begin{align} \frac{\partial \psi}{\partial t}&=\nabla\cdot\left(\kappa\nabla\psi\right)\tag{diffusion}\\ \frac{\partial \psi}{\partial t}&=\kappa\nabla^2\psi\tag{heat} \end{align} Under the diffusion equation, we typically take $\kappa$ to be a spatially-dependent variable whereas in the heat equation it is a uniform constant (allowing us to use the Laplacian on $\psi$).
There is no relation between the two equations and dimensionality. Both equations can be solved in one dimension, with a straight-forward substitution $\nabla\to\frac{\partial}{\partial x}$, or left in the multiple dimensions as I give above (which would likely require a numerical solver).