  We investigate the formation of spatial structure in dense, self-gravitating
particle systems such as Saturn's B-ring through local $N$-body simulations to
clarify the intrinsic physics based on individual particle motion. In such a
system, Salo (1995) showed that the formation of spatial structure such as
wake-like structure and particle grouping (clump) arises spontaneously due to
gravitational instability and the radial velocity dispersion increases as the
formation of the wake structure. However, intrinsic physics of the phenomena
has not been clarified. We performed local $N$-body simulations including
mutual gravitational forces between ring particles as well as direct
(inelastic) collisions with identical (up to $N\sim40000$) particles. In the
wake structure particles no longer move randomly but coherently. We found that
particle motion was similar to Keplerian motion even in the wake structure and
that the coherent motion was produced since the particles in a clump had
similar eccentricity and longitude of perihelion. This coherent motion causes
the increase and oscillation in the radial velocity dispersion. The mean
velocity dispersion is rather larger in a more dissipative case with a smaller
restitution coefficient and/or a larger surface density since the coherence is
stronger in the more dissipative case. Our simulations showed that the
wavelength of the wake structure was approximately given by the longest
wavelength $\hs{\lambda}{cr} = 4\pi^2 G\Sigma/\kappa^2$ in the linear theory of
axisymmetric gravitational instability in a thin disk, where $G$, $\Sigma$, and
$\kappa$ are the gravitational constant, surface density, and a epicyclic
frequency.
