  The behavior of non-spherical particles in a shear-flow is of significant
practical and theoretical interest. These systems have been the object of
numerous investigations since the pioneering work of Jeffery a century ago. His
eponymous orbits describe the deterministic motion of an isolated, rod-like
particle in a shear flow. Subsequently, the effect of adding noise was
investigated. The theory has been applied to colloidal particles,
macromolecules, anisometric granular particles and most recently to
microswimmers, for example bacteria. We study the Jeffery orbits of elongated
particles subject to noise using Langevin simulations and a Fokker-Planck
equation. We extend the analytical solution for infinitely thin needles
($\beta=1$) obtained by Doi and Edwards to particles with arbitrary shape
factor ($0\le \beta\le 1$) and validate the theory by comparing it with
simulations. We examine the rotation of the particle around the vorticity axis
and study the orientational order matrix. We use the latter to obtain scalar
order parameters $s$ and $r$ describing nematic ordering and biaxiality from
the orientational distribution function. The value of $s$ (nematic ordering)
increases monotonically with increasing P\'eclet number, while $r$ (measure of
biaxiality) displays a maximum value. From perturbation theory we obtain simple
expressions that provide accurate descriptions at low noise (or large P\'eclet
numbers). We also examine the orientational distribution in the v-grad v plane
and in the perpendicular direction. Finally we present the solution of the
Fokker-Planck equation for a strictly two-dimensional (2D) system. For the same
noise amplitude the average rotation speed of the particle in 3D is larger than
in 2D.
