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 time between the position of a particle and some reference position. 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. 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 is defined as
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where N is the number of particles to be averaged,
Derivation of the MSD for a Brownian particle in 1D
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. Another method to describe the motion of a Brownian particle was described by Langevin, now known for its namesake as the Langevin equation.)
given the initial condition
It can be shown that the one-dimensional PDF is
This states that the probability of finding the particle at
Using the PDF one is able to derive the average of a given function,
where the average is taken over all space (or any applicable variable).
The Mean squared displacement is defined as
expanding out the ensemble average
dropping the explicit time dependence notation for clarity. To find the MSD, one can take one of two paths: one can explicitly calculate
So then, to find the moment-generating function it is convenient to introduce the characteristic function:
one can expand out the exponential in the above equation to give
By taking the natural log of the characteristic function, a new function is produced, the cumulant generating function,
where
by completing the square and knowing the total area under a Gaussian one arrives at
Taking the natural log, and comparing powers of
which is as expected, namely that the mean position is the Gaussian centre. The second cumulant is
the factor 2 comes from the factorial factor in the denominator of the cumulant generating function. From this, the second moment is calculated,
Plugging the results for the first and second moments back, one finds the MSD,
MSD in experiments
Experimental methods to determine MSDs include neutron scattering and photon correlation spectroscopy.