# Root mean square fluctuation

The mean square fluctuation (MSF) is a measure of the deviation between the position of particle i and some reference position.

$MSF=\sqrt{\frac{1}{T}\sum_{t_j=1}^{T}(x_i(t_j) - \tilde{x}_i)^2}$,

where T is the time over which one wants to average, and $\tilde{x}_i$ is the reference position of particle i. Typically this reference position will be the time-averaged position of the same particle i, i.e. $\overline{x}_i$

Note that the difference between RMSD and RMSF is that with the latter the average is taken over time, giving a value for each particle i. With RMSD the average is taken over the particles, giving time resolved values.