Mean square displacement

Calculate the mean square displacements (MSD(\(\tau\))) of a given species in the system. The user needs to define the timestep between consecutive frames and set the maximum correlation depth \(\tau_{max}\) for the analysis. Generally, it is recommended to use a correlation depth of not larger than 1/3 of the total simulation time to assure enough sampling. During the analysis, every frame is set as the reference frame at \(\tau_{0}\) and the displacement is calculated for all frames until \(\tau_{max}\). The MSD is calculated normally, as well as independently for the x, y, and z components for each species studied. The raw data is saved in a CSV file, named msd_<species>.csv, and plotted in a separate figure msd_<species>.png. The MSD is calculated as:

\[\begin{split}\begin{aligned} MSD(\tau) &= \frac{1}{N} \sum_{i=1}^{N} \left( \mathbf{r}_i(\tau) - \mathbf{r}_i(0) \right)^2 \\ &= \frac{1}{N} \sum_{i=1}^{N} \left( (x_i(\tau) - x_i(0))^2 + (y_i(\tau) - y_i(0))^2 + (z_i(\tau) - z_i(0))^2 \right) \\ &= MSD_x(\tau) + MSD_y(\tau) + MSD_z(\tau) \end{aligned}\end{split}\]

where \(N\) is the number of atoms in the system, \(\mathbf{r}_i(\tau)\) is the position of atom \(i\) at time \(\tau\), and \(\mathbf{r}_i(0)\) is the position of atom \(i\) at time \(0\). The MSD is calculated for all liquid species in the system.

Note

Make sure the trajectory is not wrapped before the analysis. Any potential movement of the center of mass of the simulation box is removed/accounted for during the analysis.