See Stochastic process

See Brownian motion for derivations. Also Fick's laws of diffusion

Diffusion equation

$\frac{\partial P}{\partial t} = D \nabla^2 P$,

where $D$ is the diffusion coefficient.

Applications

Smoluchowski capture rate

Diffusion limit on rate of reaction between molecules.

Begin with an spherical particle, and assume stationary solution, $\partial_t P = 0$. Set the concentration to be fixed to $C_\infty$ far from particle, and $0$ on its surface, as we assume they are assumed to be captured.

To do general case where both particles are moving, one should use relative and center of mass coordinates (#trythis). The answer is:

$k_{ab}=4\pi(D_a+D_b)(R_a+R_b)$,

where $a$ and $b$ are the particle species, $D$ and $R$ are the diffusion constants and radii.

Phoretic mechanisms of colloids

• Diffusiophoresis

Diffusion on Manifolds, and graphs

see the generalized defintions of Laplacian

see this article