# Is the continuity equation valid for a diffusion current?

On the one hand, we have the diffusion equation: \begin{align} \frac{\partial\rho}{\partial t}&=D \nabla^2 \rho \end{align} and on the other hand, we have Fick's first law: \begin{align} \vec J = - D \nabla \rho \, . \end{align} If we apply $$\nabla$$ to Fick's law: \begin{align} \nabla \vec J = - D \nabla^2 \rho \end{align} and insert this into the diffusion equation, we find \begin{align} \frac{\partial\rho}{\partial t}&=\nabla \vec J \, . \end{align} If we now assume that the current $$\vec J$$ can be described in terms of a velocity field $$\vec u$$: $$\vec J \equiv \rho \vec u,$$ this yields exactly the continuity equation: \begin{align} \frac{\partial\rho}{\partial t}&=D \nabla (\rho \vec u) \, . \end{align} Is there any error in the steps above? I'm somewhat puzzled by the result because the continuity equation is typically associated with advection and not with diffusion.

Yes, indeed, Diffusion equation is essentially a continuity equation. More general Fokker-PLanck type equation (i.e. a diffusion equation with a drift term), $$\partial_t \rho(\mathbf{r},t) = \nabla \cdot[\mathbf{f}(\mathbf{r})\rho(\mathbf{r},t)] + D\nabla^2\rho(\mathbf{r},t)$$ can be written as a continuity equation $$\partial_t \rho(\mathbf{r},t) = -\nabla\mathbf{J}(\mathbf{r},t),$$ where the current is defined as $$\mathbf{J}(\mathbf{r},t) = -\mathbf{f}(\mathbf{r})\rho(\mathbf{r},t) - D\nabla\rho(\mathbf{r},t).$$ Thus, converting a diffusion-like equation to a continuity equation is a qquestion of correctly defining the current.