# MeanSquareDisplacement¶

class MeanSquareDisplacement(md_trajectory, start_time=None, end_time=None, atom_selection=None, anisotropy=None, time_resolution=None, info_panel=None)

Constructor for the MeanSquareDisplacement object.

Parameters: md_trajectory (MDTrajectory | AtomicConfiguration) – The MDTrajectory or configuration to calculate the mean-square-displacement for. start_time (PhysicalQuantity of type time) – The start time. Default: 0.0 * fs end_time (PhysicalQuantity of type time) – The end time. Default: The last time frame atom_selection (PeriodicTableElement | str | list of ints) – Only include contributions from this selection. The atoms can be selected by element i.e. PeriodicTableElement, tag or a list of atomic indices. Default: all atoms anisotropy (list of type int | int | None) – The list of Cartesian directions (x=0, y=1, z=2) to calculate the anisotropic mean square displacement in or a single Cartesian direction. By default an isotropic calculation is performed. Default: None time_resolution (PhysicalQuantity of type time) – The time interval between snapshots in the MD trajectory that are included in the analysis. info_panel (InfoPanel (Plot2D)) – Info panel to show the calculation progress. Default: No info panel
data()

Return the mean-square-displacement values.

times()

Return the time values.

## Usage Examples¶

Load an MDTrajectory, and calculate the mean-square-displacement (MSD) of all aluminum atoms. Estimate the diffusion coefficient from the slope of the MSD-curve, according to $$MSD(t)=6 Dt$$:

md_trajectory = nlread('alumina_trajectory.nc')[-1]

msd = MeanSquareDisplacement(md_trajectory, atom_selection=Aluminum)

# Get the times in ps and the MSD values in Ang**2.
t        = msd.times().inUnitsOf(ps)
msd_data = msd.data().inUnitsOf(Angstrom**2)

# Plot the data using pylab.
import pylab

pylab.plot(t, msd_data, label='MSD of aluminum')
pylab.xlabel('t (ps)')
pylab.ylabel('MSD(t) (Ang**2)')
pylab.legend()

pylab.show()

# Fit the slope of the MSD to estimate the diffusion coefficient.
# If you discover non-linear behavior at small times, discard this initial part in the fit.
a = numpy.polyfit(t[5:], msd_data[5:], deg=1)

# Calculate the diffusion coefficient in Ang**2/ps.
diffusion_coefficient = a[0]/6.0


mean_square_displacement.py

## Notes¶

The MeanSquareDisplacement is calculated as:

$MSD(t) = \left \langle \left [ \mathbf{r}(t) - \mathbf{r}(0)\right ]^2 \right\rangle.$

In practice, the average $$<...>$$ runs over all selected atoms $$i$$ in the trajectory, and an additional average over simulation time is carried out to improve the statistical sampling. That means for a given time difference $$t$$ all image pairs that are separated by $$t$$ are taken into account in the average, as

$MSD(t) = \frac{1}{(t_{max} - t)N_{atoms}} \sum_{i=1}^{N_{atoms}} \sum_{t'=0}^{t_{max}-t} \left [ \mathbf{r}_i(t' + t) - \mathbf{r}_i(t')\right ]^2 .$

Note, that this requires a system which is equilibrated, i.e. its macroscopic properties do not change during the simulation.

By default, all elements are taken into account, but a specified selection can be given as well. The atom_selection parameter accepts an element, a tag name, or a list of indices to select atoms for the velocity distribution. This can be useful, e.g. in the presence of constraints as constrained atoms should be excluded in this analysis.