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Mean square displacement

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The atoms of a solid, a liquid or a gas do move, they are subject to a displacement almost constant. This displacement can be particularly important in the case of a liquid to ensure the fluid properties, furthermore the displacement of a single atom does not follow a simple trajectory: the particles this atom may "meet" along his "path" avoid its trajectory to follow a straight line.

It appears that the path follow by this atom can be described by a random pedestrian 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 (Markovien process).

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 be chosen.

One can conclude that the distance this atom may have traveled is closed to zero. Nevertheless if we choose not to sum the displacements them-self (+/-) but the square of these displacement then we will add a positive quantity to the total distance which therefore increases at each step.

Consequently this allows to obtain a better evaluation of the real (square) distance traveled by this atom.

The Mean Square Displacement MSD is defined by the relation:

MSD(t) = 〈$\displaystyle \bf {r}^{{2}}_{}$(t)〉 = $\displaystyle \left\langle\vphantom{ \vert{\bf {r}}_{i}(t)-{\bf {r}}_{i}(0)\vert^{2} }\right.$|$\displaystyle \bf {r}_{{i}}^{}$(t) - $\displaystyle \bf {r}_{{i}}^{}$(0)|2$\displaystyle \left.\vphantom{ \vert{\bf {r}}_{i}(t)-{\bf {r}}_{i}(0)\vert^{2} }\right\rangle$ (1)

where $ \bf {r}_{{i}}^{}$(t) is the position of the atom i at the time t, and the 〈 〉 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 probable drift of the center of mass of the simulation box:

MSD(t) = $\displaystyle \left\langle\vphantom{ \left\vert{\bf {r}}_{i}(t)-{\bf {r}}_{i}(0) - \left[{\bf {r}}_{cm}(t)-{\bf {r}}_{cm}(0)\right]\right\vert^{2} }\right.$$\displaystyle \left\vert\vphantom{{\bf {r}}_{i}(t)-{\bf {r}}_{i}(0) - \left[{\bf {r}}_{cm}(t)-{\bf {r}}_{cm}(0)\right]}\right.$$\displaystyle \bf {r}_{{i}}^{}$(t) - $\displaystyle \bf {r}_{{i}}^{}$(0) - $\displaystyle \left[\vphantom{{\bf {r}}_{cm}(t)-{\bf {r}}_{cm}(0)}\right.$$\displaystyle \bf {r}_{{cm}}^{}$(t) - $\displaystyle \bf {r}_{{cm}}^{}$(0)$\displaystyle \left.\vphantom{{\bf {r}}_{cm}(t)-{\bf {r}}_{cm}(0)}\right]$$\displaystyle \left.\vphantom{{\bf {r}}_{i}(t)-{\bf {r}}_{i}(0) - \left[{\bf {r}}_{cm}(t)-{\bf {r}}_{cm}(0)\right]}\right\vert^{{2}}_{}$$\displaystyle \left.\vphantom{ \left\vert{\bf {r}}_{i}(t)-{\bf {r}}_{i}(0) - \left[{\bf {r}}_{cm}(t)-{\bf {r}}_{cm}(0)\right]\right\vert^{2} }\right\rangle$ (2)

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 the DCM towards to a value (saturation), the kinetic energy is not sufficient enough to reach a diffusive behavior. Nevertheless if the system is not frozen (ex: liquid) then the DCM will grow linearly in time. In such case it is interesting to characterized the behavior of the system compared to the slope of the MSD. The slope of the MSD or so called diffusion constant D is defined by:

D = $\displaystyle \lim_{{t \to \infty}}^{}$ $\displaystyle {\frac{{1}}{{6t}}}$$\displaystyle \bf {r}^{{2}}_{}$(t)〉 (3)
Last Updated on Monday, 08 June 2009 15:52  

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