Diffusion: Difference between revisions

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The '''diffusion coefficient''', is given by
Diffusion is the process behind Brownian motion. It was described
by [[Albert Einstein]] in one of his annus mirabilis (1905) papers.
The diffusion equation is that describes the 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 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 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==
For a homogeneous system,
:<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>
:<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 velovity of molecule <math>i</math>.
where <math>v_i(t)</math> is the center of mass velovity of molecule <math>i</math>.


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


==References==
==References==
#[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]]

Revision as of 13:41, 13 November 2007

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

For a homogeneous system,


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)