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Diffusion is the process behind Brownian motion. It was described | |||
by [[Albert Einstein]] in one of his | by [[Albert Einstein]] in one of his annus mirabilis (1905) papers. | ||
The diffusion equation is that describes the process is | |||
The diffusion equation that describes | |||
:<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 | For initial conditions for a 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 | with a spreading 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> | ||
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==Einstein relation== | ==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> | |||
:<math>D = \lim_{t \rightarrow \infty} \frac{1}{ | |||
==Green-Kubo relation== | ==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 | where <math>v_i(t)</math> is the center of mass velovity of molecule <math>i</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)] | |||
[[Category: Non-equilibrium thermodynamics]] | [[Category: Non-equilibrium thermodynamics]] |