  The Stokes-Einstein relation for the self-diffusion coefficient of a
spherical particle suspended in an incompressible fluid is an asymptotic result
in the limit of large Schmidt number, that is, when momentum diffuses much
faster than the particle. When the Schmidt number is moderate, which happens in
most particle methods for hydrodynamics, deviations from the Stokes-Einstein
prediction are expected. We study these corrections computationally using a
recently-developed minimally-resolved method for coupling particles to an
incompressible fluctuating fluid in both two and three dimensions. We find that
for moderate Schmidt numbers the diffusion coefficient is reduced relative to
the Stokes-Einstein prediction by an amount inversely proportional to the
Schmidt number in both two and three dimensions. We find, however, that the
Einstein formula is obeyed at all Schmidt numbers, consistent with linear
response theory. The numerical data is in good agreement with an approximate
self-consistent theory, which can be used to estimate finite-Schmidt number
corrections in a variety of methods. Our results indicate that the corrections
to the Stokes-Einstein formula come primarily from the fact that the particle
itself diffuses together with the momentum. Our study separates effects coming
from corrections to no-slip hydrodynamics from those of finite separation of
time scales, allowing for a better understanding of widely observed deviations
from the Stokes-Einstein prediction in particle methods such as molecular
dynamics.
