Mean squared displacement

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In statistical mechanics, the mean squared displacement (MSD, also mean square displacement, average squared displacement, or mean square fluctuation) is a measure of the deviation of the position of a particle with respect to a reference position over time. It is the most common measure of the spatial extent of random motion, and can be thought of as measuring the portion of the system "explored" by the random walker. In the realm of biophysics and environmental engineering, the Mean Squared Displacement is measured over time to determine if a particle is spreading slowly due to diffusion, or if an advective force is also contributing.^[1] Another relevant concept, the variance-related diameter (VRD, which is twice the square root of MSD), is also used in studying the transportation and mixing phenomena in the realm of environmental engineering.^[2] It prominently appears in the Debye–Waller factor (describing vibrations within the solid state) and in the Langevin equation (describing diffusion of a Brownian particle).

The MSD at time ${\displaystyle t}$ is defined as an ensemble average:

$${\displaystyle {\text{MSD}}\equiv \left\langle \left|\mathbf {x} (t)-\mathbf {x_{0}} \right|^{2}\right\rangle ={\frac {1}{N}}\sum _{i=1}^{N}\left|\mathbf {x^{(i)}} (t)-\mathbf {x^{(i)}} (0)\right|^{2}}$$

where N is the number of particles to be averaged, vector ${\displaystyle \mathbf {x^{(i)}} (0)=\mathbf {x_{0}^{(i)}} }$ is the reference position of the ${\displaystyle i}$-th particle, and vector ${\displaystyle \mathbf {x^{(i)}} (t)}$ is the position of the ${\displaystyle i}$-th particle at time t.^[3]