Diffusion Equation

The diffusion equation is a parabolic PDE, and a Stochastic Differential Equation. It describes the Brownian Motion of several particles, particularly useful in how they collide with each other and the emergent behavior thereof.

It is described usually as:

$$\frac{\partial \varphi (\mathbf{r},t)}{\partial t}=\nabla \cdot [D(\varphi ,\mathbf{r})\nabla \varphi (\mathbf{r},t)]$$

Where:

• $\varphi (\mathbf{r},t)$ - the density of the diffusing material at location $\mathbf{r}$ and time $t$
• $D(\varphi ,\mathbf{r})$ - the collective diffusion coefficient for density $\varphi$ at location $\mathbf{r}$.
• $\nabla$ - the vector differential operator del