Diffusion Simulator/Animator

Current Status:

• NumPy simulation :
  • Simulation works for N-D diffusion.
  • Animation works in 1-D and 2-D diffusion.
  • Checkpointing functionality added.
  • Experimental: 3-D diffusion animation.
• TensorFlow-v1 :
  • Simulation works for N-D diffusion.
  • Animation for 2-D diffusion in tensorflow-v1/diffusion_tenorflow1.ipynb.

---

Diffusion equation:

The continuity equation:

$$\nabla \cdot \vec{j} + \frac{\partial \phi}{\partial t} = 0$$

$\phi$ : Density of diffusing quantity

$\vec{j}$ : Current density

Fick's first law:

$$\vec{j} = - \overleftrightarrow{D}[\phi,\vec{r}] \nabla\phi[\vec{r},t]$$ $\overleftrightarrow{D}[\phi,\vec{r}]$ : Diffusion (tensor) coefficient for density $\phi$ at position $\vec{r}$

Combining the above equations gives the general diffusion equation: $$\frac{\partial\phi}{\partial t} = \nabla \Big( \overleftrightarrow{D}[\phi,\vec{r}] \nabla\phi \Big) = \partial^{\alpha} \big(D_{\alpha\beta}\partial^{\beta} \phi \big)$$

The typical diffusion:

When D is an scalar constant(isotropic and independent of density and position), the diffusion equation reduces to: $$\frac{\partial\phi}{\partial t} = D \nabla^2 \phi$$

This Implementation:

$$\overleftrightarrow{D}[\phi,\vec{r}] \equiv D[\vec{r}]\bigg(1-\frac{B[\vec{r}]}{|\nabla \phi|}\bigg)\Theta[|\nabla\phi|-B[\vec{r}]] $$ Here, the diffusion coefficient is isotropic, hence is a scalar quatity The simulation here evolves the diffusing quantity given:

1. The boundary conditions
2. The initial distribution
3. The diffusion coefficient, $D[\vec{r}]$, as function of position
4. Minimum gradient, $B[\vec{r}]$, required to drive current