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.

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