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).
-th particle at time t.[3] The probability density function (PDF) for a particle in one dimension is found by solving the one-dimensional diffusion equation.
(This equation states that the position probability density diffuses out over time - this is the method used by Einstein to describe a Brownian particle.
The diffusion equation states that the speed at which the probability for finding the particle at
More specifically the full width at half maximum (FWHM)(technically/pedantically, this is actually the Full duration at half maximum as the independent variable is time) scales like
dropping the explicit time dependence notation for clarity.
To find the MSD, one can take one of two paths: one can explicitly calculate
, then plug the result back into the definition of the MSD; or one could find the moment-generating function, an extremely useful, and general function when dealing with probability densities.
The first moment of the displacement PDF shown above is simply the mean:
With these definitions accounted for one can investigate the moments of the Brownian particle PDF,
by completing the square and knowing the total area under a Gaussian one arrives at
Plugging the results for the first and second moments back, one finds the MSD,
For a Brownian particle in higher-dimension Euclidean space, its position is represented by a vector
The n-variable probability distribution function is the product of the fundamental solutions in each variable; i.e.,
For each coordinate, following the same derivation as in 1D scenario above, one obtains the MSD in that dimension as
Hence, the final result of mean squared displacement in n-dimensional Brownian motion is:
, representing a particle undergoing two-dimensional diffusion.
Assuming that the trajectory of a single particle measured at time points
can be defined as an average quantity over time lags:[4][5]
This technique allow us estimate the behavior of the whole ensembles by just measuring a single trajectory, but note that it's only valid for the systems with ergodicity, like classical Brownian motion (BM), fractional Brownian motion (fBM), and continuous-time random walk (CTRW) with limited distribution of waiting times, in these cases,
don't equal each other anymore, in order to get better asymptotics, introduce the averaged time MSD:
Also, one can easily derive the autocorrelation function from the MSD:
is so-called autocorrelation function for position of particles.
Experimental methods to determine MSDs include neutron scattering and photon correlation spectroscopy.
The linear relationship between the MSD and time t allows for graphical methods to determine the diffusivity constant D. This is especially useful for rough calculations of the diffusivity in environmental systems.
In some atmospheric dispersion models, the relationship between MSD and time t is not linear.
Instead, a series of power laws empirically representing the variation of the square root of MSD versus downwind distance are commonly used in studying the dispersion phenomenon.