Diffusion

Diffusion is the process behind Brownian motion. It was described by Albert Einstein in one of his annus mirabilis papers of 1905. What follows applies to homogeneous systems, see diffusion at interfaces for a non-homogeneous case.

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

It follows from the previous equation that, for each of the Cartesian components, e.g. $x$:

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

for every particle $i$. Therefore, an average over all particles can be employed in order to improve statistics. The same applies to time averaging: in equilibrium the average from $0$ to $t$ must equal the average from $\tau$ to $t+\tau$, so several time segments from the same simulation may be averaged for a given interval [2]. Adding all components, the following also applies:

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

Green-Kubo relation

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

where $v_{i}(t)$ is the center of mass velovity of molecule $i$.

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)
2. Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications". Academic Press 2002