# Mean square displacement of atoms

Atoms in solids, liquids and gases move constantly at any given temperature, i.e. they are subject to a "thermal" displacement from their average positions. This displacement is particularly important in the case of a liquids. Atomic displacement does not follow a simple trajectory: "collisions" with other atoms render atomic trajectories quite complex shaped in space.
The trajectory followed by an atom in a liquid resembles that of a pedestrian random walk. Mathematically this represents a sequence of steps done one after another where each step follows a random direction which does not depend on the one of the previous step (Markov's chain of events).
In the case of a one-dimensional system (straight line) the displacement of the atom will therefore be either a forward step (+) or a backward step (-). Furthermore it will be impossible to predict one or the other direction (forward or backward) since they have an equal probability to occur.
One can conclude that the distance an atom may travel is close to zero. Nevertheless if we choose not to sum the displacements themselves (+/-) but the square of these displacements then we will end up with a non-zero, positive quantity of the total squared distance traveled. Consequently this allows to obtain a better evaluation of the real (square) distance traveled by an atom.
The Mean Square Displacement MSD is defined by the relation: $\label{dyna1} MSD(t)\ =\ \langle {\bf{r}}^{2}(t)\rangle\ =\ \left\langle |{\bf{r}}_{i}(t)-{\bf{r}}_{i}(0)|^{2} \right\rangle$ where $${\bf{r}}_{i}$$(t) is the position of the atom $$i$$ at the time $$t$$, and the $$\rangle\ \langle$$ represent an average on the time steps and/or the particles.
However, during the analysis of the results of molecular dynamics simulations it is important to subtract the drift of the center of mass of the simulation box: $\label{dyna2} MSD(t)\ =\ \left\langle \left|{\bf{r}}_{i}(t)-{\bf{r}}_{i}(0) - \left[{\bf{r}}_{cm}(t)-{\bf{r}}_{cm}(0)\right]\right|^{2} \right\rangle$ where $${\bf{r}}_{cm}$$(t) represents the position of the center of mass of the system at the time $$t$$.
The MSD also contains information on the diffusion of atoms. If the system is solid (frozen) then MSD "saturate", and the kinetic energy is not sufficient enough to reach a diffusive behavior. Nevertheless if the system is not frozen (e.g. liquid) then the MSD will grow linearly in time. In such a case it is possible to investigate the behavior of the system looking at the slope of the MSD. The slope of the MSD or the so called diffusion constant D is defined by: $\label{dyna3} D\ =\ \lim_{t \to \infty}\ \frac{1}{6t}\ \langle {\bf{r}}^{2}(t) \rangle$