PHYS6562 W2 Daily Question

Local conservation

a. The density of tin atoms \(\rho_{Sn}(\textbf{x})\) is locally conserved since there is no tin atom created or annihilated.

b. It depends. Under the constraints that one cannot change his/her political belief, yes, the density of democrats is locally conserved. However, if changing of political belief is allowed, the density of democrats is not locally conserved. There might be democrats changing to republicans or vise versa, which implies that the democrats or republicans can teleport, created or annihilated.

Density-dependent diffusion

Suppose there is no external force. We can write the current as

\[J = -D(\rho)\nabla\rho\]

And apply

\[\frac{\partial\rho}{\partial t} = -\nabla\cdot J\]

\[\frac{\partial\rho}{\partial t} = \frac{\partial D(\rho)}{\partial\rho} \nabla\rho\cdot\nabla\rho + D(\rho)\nabla^2\rho\]

\[\frac{\partial\rho}{\partial t} = \frac{\partial D(\rho)}{\partial\rho} (\nabla\rho)^2 + D(\rho)\nabla^2\rho\]

This is the diffusion equation with density-dependent diffusion constant.