Diffusion equation, famous laws

If we let $u$ denote the concetration of a particle species, this results in Fick's law of diffusion. If it denotes the temperature gradient, we have Fourier'slaw of heat conduction and if it refers to the electrostatic potential we have Ohm's law of electrical conduction.

Coupling the rate of change (temporal dependence) of $u$ with the flux density we have $$\begin{equation*} \frac{\partial u}{\partial t} = -\mathrm{div}\mathbf{\rho}, \end{equation*}$$ which results in $$\begin{equation*} \frac{\partial u}{\partial t}= D \mathrm{div} \mathbf{\nabla} u = D \Delta u, \end{equation*}$$ the diffusion equation, or heat equation.

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