Editing 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
:<math>\frac{\partial P(r,t)}{\partial t}= D \nabla^2 P(z,t),</math>
where <math>D</math> is the (self-)'''diffusion coefficient'''.
For initial conditions for a [[Dirac delta distribution |Dirac delta function]] at the origin, and
boundary conditions that force the vanishing of <math>P(r,t)</math>
and its gradient at large distances, the solution factorizes as <math>P(r,t)=P(x,t)P(y,t)P(z,t)</math>,
with a spreading [[Gaussian distribution |Gaussian]] for each of the Cartesian components:
:<math> P(x,t)=\frac{1}{\sqrt{4\pi D t}} \exp
   \left[ - \frac{x^2}{4 D t} \right]. </math>

==Einstein relation==

It follows from the previous equation that, for each of the Cartesian components, e.g. <math>x</math>:
:<math>D = \lim_{t \rightarrow \infty} \frac{1}{2} \langle \vert x_i(t) \cdot x_i(0) \vert^2\rangle  </math>,
for every particle <math>i</math>. 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 <math>0</math> to <math>t</math> must equal the average from <math>\tau</math> to <math>t+\tau</math>,
so several time segments from the same simulation may be averaged for a given interval [2].
Adding all components, the following also applies:
:<math>D = \lim_{t \rightarrow \infty} \frac{1}{6} \langle \vert \mathbf{r}_i(t) \cdot \mathbf{r}_i(0) \vert^2\rangle  </math>


==Green-Kubo relation==
:<math>D = \frac{1}{3} \int_0^\infty \langle v_i(t) \cdot v_i(0)\rangle ~dt</math>

where <math>v_i(t)</math> is the center of mass velocity of molecule <math>i</math>. Note
that this connect the diffusion coefficient with the velocity [[autocorrelation]].

==References==
#[http://books.google.es/books?id=XmyO2oRUg0cC&dq=understanding+molecular+simulations&psp=1 Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications". Academic Press 2002]
#[http://dx.doi.org/10.1063/1.1786579 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)]
#[http://dx.doi.org/10.1080/00268970701348758 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)]
[[Category: Non-equilibrium thermodynamics]]