- #1

Derivator

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in self diffusion theory you can see in different books, different definitions of the mean square displacement:

[itex]<r(t)^2> = \frac{1}{N} \int d\vec{r} \ \vec{r}(t)^2 c( \vec{r} ,t)[/itex]

where c(r,t) is the particle concentration and N the number of particles.

or

[itex]<r(t)^2> = \frac{1}{N}\sum_i^N \vec{r}_i(t)^2 [/itex]

In all theoretical derivations, only the integral version is used whereas in practical / applied treatements of self diffusion, the sum version is used. This is why I assume, the sum version is an approximation to the integral version. But I don't see it.

derivator

Edit:

See for example:

the sum version (formula 2.2): http://ocw.mit.edu/courses/nuclear-...of-transport-fall-2003/lecture-notes/lec2.pdf

the integral version (formula 12.4): http://books.google.de/books?id=PTO...sion mean square&pg=PA124#v=onepage&q&f=false