Diffusion: Difference between revisions

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'''Diffusion''' is the process behind [[Brownian motion]]. It was described
'''Diffusion''' is the process behind [[Brownian motion]]. It was described
by [[Albert Einstein]] in one of his ''annus mirabilis'' papers of 1905.
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
The diffusion equation that describes this process is
:<math>\frac{\partial P(r,t)}{\partial t}= D \nabla^2 P(z,t),</math>
:<math>\frac{\partial P(r,t)}{\partial t}= D \nabla^2 P(z,t),</math>
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==Einstein relation==
==Einstein relation==


For a homogeneous system,
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>
:<math>D = \lim_{t \rightarrow \infty} \frac{1}{6} \langle \vert \mathbf{r}_i(t) \cdot \mathbf{r}_i(0) \vert^2\rangle  </math>


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#[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)]
#[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]]
[[Category: Non-equilibrium thermodynamics]]
# [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]

Revision as of 10:34, 4 December 2007

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 velovity of molecule .


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