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 | by [[Albert Einstein]] in one of his ''annus mirabilis'' papers of 1905. | ||
The diffusion equation | 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
- 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} .