  We first established the dynamic equations to describe the noisy circling
motion of a single particle and the corresponding probability conservation
equation in both two dimensions and three dimensions, and then developed the
evolution equation of density distribution of many circling particles with
overdamped circle center. For many circling particle system without any
external force, the density gradient in one direction can induce a flow
perpendicular to this direction. While for single circling particle, similar
phenomena occurs only for non-zero external force. We performed numerical
evolution of the density distribution of many circling particles, the density
distribution behaves as a decaying Gaussian distribution propagating along the
channel. We computed the particle flow field and the effective force field.
Vortex shows up in the high density region. The force field drive particles to
the transverse direction perpendicular to the density gradient.
  We applied this non-equilibrium evolution equation to understand the
diffusion phenomena of many sperms(J. Exp. Biol. 210, 3805-3820). Numerical
evolution gave us similar density distribution as experimental measurement. The
transverse flow we predicted provide a theoretical understanding to the bias
concentration of many sperms(J. Exp. Biol. 210. 3805-3820).
