Self-Diffusion, two versions of mean square displacement

[Derivator]

Dear all,

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

$<r(t)^2> = \frac{1}{N} \int d\vec{r} \ \vec{r}(t)^2 c( \vec{r} ,t)$
where c(r,t) is the particle concentration and N the number of particles.
or

$<r(t)^2> = \frac{1}{N}\sum_i^N \vec{r}_i(t)^2$

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

 

[vanhees71]

This is known as the test-particle method. The idea behind this method is to describe the diffusion of the particles, e.g., in an suspension by the motion of test particles under the influence of a friction force (the average force from many collisions of the fluid molecules of with the particle) and a fluctuating random force. This is the Langevin equation. This you do many times in a Monte Carlo simulation which gives trajectories $(\vec{\xi}(t),\vec{\pi}(t))$ in phase space. The concentration is then given by the test-particle ansatz,

$$c(t,\vec{x},\vec{p})=\frac{N}{N_e} \sum_{i=1}^{N_e} \delta^{(3)}[\vec{x}-\vec{\xi}_i(t)] \delta^{(3)}[\vec{p}-\vec{\pi}_i(t)],$$

where $N$ is the number of suspended particles and $N_e$ is the number of test particles, i.e., the ensemble size.

Of course the concentration as function of space alone is given by

$$c(t,\vec{x})=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{p} c(t,\vec{x},\vec{p}).$$

Now, if you plug in the test-particle ansatz for $c(t,\vec{x})$ into the equation for average quantities, you get your second equation.

The Monte Carlo statistics given by the Langevin equation, is by the way equivalent to the Fokker-Planck equation (or diffusion equation) for the concentration, which is a deterministic partial differential equation (transport equation).

The whole trick of test particles is that you can put them on a computer to solve the transport equation.