where the angle brackets indicate an ensemble average, and is the sphere's diffusion coefficient in the direction. In 1905, Einstein demonstrated that a sphere's diffusivity at temperature is simply related to its mobility through the now-familiar Stokes-Einstein relation

This was the first statement of the more general fluctuation-dissipation theorem which plays an important role in statistical mechanics. More generally, components of the -particle diffusivity tensor

parameterize generalized diffusion relations [59]

describing how particle 's motion in the direction influences particle 's in the direction. Off-diagonal terms in thus encode spheres' hydrodynamic interactions.

In general, can be diagonalized by projection onto a system of coordinates consisting of linear combinations of the . These normal modes evolve independently over time; they offer natural experimental probes of the particles' dynamics.