softmatter:Mean Squared Displacement

From NSDL Materials Digital Library Soft Matter Wiki

The mean square displacement (MSD) is a measure of the average distance a given particle in a system travels. The MSD is definied as:

$MSD = \left \langle r^2(t) \right \rangle = \left \langle \frac{1}{N} \sum_{i=0}^{N}(r_i(t)-r_i(0))^2 \right \rangle$

Here, N is the number of particles, t corresponds to time, and $r i (t) - r i (0)$ is the vector distance traveled by a given particle over the time interval.

If the particle travels ballistically (i.e. it doesn't collide with any other particles), then the distance traveled is proportional to the time interval, and therefore the MSD would increase quadratically. In dense phases, this ballistic motion only occurs for very short time scales. After the ballistic regime, the MSD will increase linearly with time. This behavior for a Lennard-Jones system is outlined in the image below.

The slope of the MSD, considered for time long time intervals, is related to the self- diffusion constant D. Starting with the Einstein equation, $\left \langle r^2(t) \right \rangle =2dDt$ , where d is the dimensionality, for a three dimensional system we get: