# Self-Diffusion, two versions of mean square displacement

• Derivator
In summary, there are two versions of mean square displacement in self diffusion theory, one in the form of an integral and the other in the form of a sum. The sum version is often used in practical applications while the integral version is used in theoretical derivations. The sum version is an approximation to the integral version, which can be seen through the test-particle method. This method uses Monte Carlo simulations to solve the transport equation, which is equivalent to the Fokker-Planck equation for the concentration.
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

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.

## 1. What is self-diffusion?

Self-diffusion is the process by which particles move randomly and spontaneously within a substance, without the influence of an external force. This movement is caused by the thermal energy of the particles.

## 2. What is mean square displacement?

Mean square displacement is a measure of the average distance that particles have moved from their starting position over a given time interval. It is calculated by squaring the distance traveled by each particle and then taking the average of all the squared distances.

## 3. What are the two versions of mean square displacement?

The two versions of mean square displacement are the time-averaged mean square displacement and the ensemble-averaged mean square displacement. The time-averaged version looks at the displacement of a single particle over time, while the ensemble-averaged version considers the average displacement of all particles in a system at a given time.

## 4. How does mean square displacement relate to self-diffusion?

Mean square displacement can be used to measure the rate of self-diffusion in a substance. The higher the mean square displacement, the faster the particles are moving and the higher the rate of self-diffusion.

## 5. Why are there two versions of mean square displacement?

The two versions of mean square displacement allow for a more comprehensive understanding of self-diffusion. The time-averaged version provides information about the movement of individual particles, while the ensemble-averaged version gives an overall picture of the diffusion process in a system.

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