  Brownian motion imposes a hard limit on the overall precision of a
nanomechanical measurement. Here, we present a combined experimental and
theoretical study of the Brownian dynamics of a quintessential nanomechanical
system, a doubly-clamped nanomechanical beam resonator, in a viscous fluid. Our
theoretical approach is based on the fluctuation-dissipation theorem of
statistical mechanics: We determine the dissipation from fluid dynamics; we
incorporate this dissipation into the proper elastic equation to obtain the
equation of motion; the fluctuation-dissipation theorem then directly provides
an analytical expression for the position-dependent power spectral density
(PSD) of the displacement fluctuations of the beam. We compare our theory to
experiments on nanomechanical beams immersed in air and water, and obtain
excellent agreement. Within our experimental parameter range, the Brownian
force noise driving the nanomechanical beam has a colored PSD due to the
``memory" of the fluid; the force noise remains mode-independent and
uncorrelated in space. These conclusions are not only important for
nanomechanical sensing but also provide insight into the fluctuations of
elastic systems at any length scale.
