Diffusion equation
Assuming that the flux is proportional to the gradient
\mathbf{\nabla} u but pointing in the opposite direction
since the flow is from regions of high concetration to lower concentrations, we obtain
\begin{equation*}
\mathbf{\rho} = -D\mathbf{\nabla} u,
\end{equation*}
resulting in
\begin{equation*}
\mathrm{div} \mathbf{\nabla} u = D\Delta u = 0,
\end{equation*}
which is Laplace's equation. The constant
D can be coupled with various physical
constants, such as the diffusion constant or the specific heat and thermal conductivity discussed below.