# 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

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

## Einstein relation

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

- ,

for every particle . 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 to must equal the average from to , so several time segments from the same simulation may be averaged for a given interval [2]. Adding all components, the following also applies:

## Green-Kubo relation

where is the center of mass velocity of molecule . Note that this connect the diffusion coefficient with the velocity autocorrelation.

## References

- Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications". Academic Press 2002
- Karsten Meier, Arno Laesecke, and Stephan Kabelac "Transport coefficients of the Lennard-Jones model fluid. II Self-diffusion" J. Chem. Phys.
**121**pp. 9526-9535 (2004) - 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)