Diffusion: Difference between revisions
Carl McBride (talk | contribs) |
No edit summary |
||
| Line 1: | Line 1: | ||
The '''diffusion coefficient''', | 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>. | ||
==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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial P(r,t)}{\partial t}= D \nabla^2 P(z,t),}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} is the (self-)diffusion coefficient. For initial conditions for a Dirac delta function at the origin, and boundary conditions that force the vanishing of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(r,t)} and its gradient at large distances, the solution factorizes as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(r,t)=P(x,t)P(y,t)P(z,t)} , with a spreading Gaussian for each of the Cartesian components:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x,t)=\frac{1}{\sqrt{4\pi D t}} \exp \left[ - \frac{x^2}{4 D t} \right]. }
Einstein relation
For a homogeneous system,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D = \lim_{t \rightarrow \infty} \frac{1}{6} \langle \vert \mathbf{r}_i(t) \cdot \mathbf{r}_i(0) \vert^2\rangle }
Green-Kubo relation
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D = \frac{1}{3} \int_0^\infty \langle v_i(t) \cdot v_i(0)\rangle ~dt}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_i(t)} is the center of mass velovity of molecule Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} .