# 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 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

## 1 Answer

Consider a diffusion process for a single particle at known location. The probability is according to diffusion equation is roughly

$$p(r) = \frac{1}{\sqrt{T}} e^{-r^2/t}.$$

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).

A more technical detail is that (I don't know how it is done in your references) one has usually a finite trajectory of say 100ps. Now one might hope to improve statistical error by starting the MSD calculation at different times. However, lowering of the variance will be hindered by the autocorrelation. That is to say, If r(10ps)-r(0ps) happends to be very large, so will most likely be r(11ps)-r(1ps). In later words, they are statistically correlated and therefore reduce the variance less, than is the case for uncorrelated samples. Now, back to the finite length of the simulation: there is roughly 100 independent 1ps trajectories in a 100ps trajectory to reduce the statistical error at 1ps. However, there is only roughly two independent 50ps trajectories for larger time diffusion. This also makes the higher time msd more difficult to sample accurately.