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 (1905) papers.
by [[Albert Einstein]] in one of his ''annus mirabilis''  papers of 1905.
The diffusion equation is that describes the 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>
where <math>D</math> is the (self-)'''diffusion coefficient'''.
where <math>D</math> is the (self-)'''diffusion coefficient'''.
For initial conditions for a Dirac delta function at the origin, and
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>
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>,
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:
with a spreading [[Gaussian distribution |Gaussian]] for each of the Cartesian components:
:<math> P(x,t)=\frac{1}{\sqrt{4\pi D t}} \exp
:<math> P(x,t)=\frac{1}{\sqrt{4\pi D t}} \exp
   \left[ - \frac{x^2}{4 D t} \right]. </math>
   \left[ - \frac{x^2}{4 D t} \right]. </math>

Revision as of 13:48, 13 November 2007

Diffusion is the process behind Brownian motion. It was described by Albert Einstein in one of his annus mirabilis papers of 1905. 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

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)