# Why Mean Square Displacement (MSD) always becomes chaotic at MD simulation end?

The MSD always becomes chaotic at MD simulation end, no matter how long the time is. So is it a algorithm defect, or is it because of statistical error? The problem bothers me a long time, thanks that if someone explains this to me. Some MSD data may look like this

or this

Ref: Macromolecule vol.31 no. 16, 1998

dx.doi.org/10.1021/jp501672t | J. Phys. Chem. C 2014, 118, 9841−9851

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 7 ( 2 0 1 2 ) 1 0 2 5 6 e1 0 2 6 4

$$p(r) = \frac{1}{\sqrt{T}} e^{-r^2/t}.$$
Now, one sees that the localization of particle is less defined and it is evident that sampling a random point at time T from this distribution has higher variance of $r^2$ and it grows with T. So, as the expectation value of $r^2$ grows linearly, so grows its variance (to some power not discussed here).