Diffusion

Diffusion is the process behind Brownian motion. It was described by Albert Einstein in one of his annus mirabilis papers of 1905. The diffusion equation that describes this process is

${\frac {\partial P(r,t)}{\partial t}}=D\nabla ^{2}P(z,t),$

where $D$ is the (self-)diffusion coefficient. For initial conditions for a Dirac delta function at the origin, and boundary conditions that force the vanishing of $P(r,t)$ and its gradient at large distances, the solution factorizes as $P(r,t)=P(x,t)P(y,t)P(z,t)$, with a spreading Gaussian for each of the Cartesian components:

$P(x,t)={\frac {1}{\sqrt {4\pi Dt}}}\exp \left[-{\frac {x^{2}}{4Dt}}\right].$

Einstein relation

For a homogeneous system,

$D=\lim _{t\rightarrow \infty }{\frac {1}{6}}\langle \vert \mathbf {r} _{i}(t)\cdot \mathbf {r} _{i}(0)\vert ^{2}\rangle$

Green-Kubo relation

$D={\frac {1}{3}}\int _{0}^{\infty }\langle v_{i}(t)\cdot v_{i}(0)\rangle ~dt$

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References

1. G. L. Aranovich and M. D. Donohue "Limitations and generalizations of the classical phenomenological model for diffusion in fluids", Molecular Physics 105 1085-1093 (2007)