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 over 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