  We investigate the dynamics of overdamped $D$-dimensional systems of
particles repulsively interacting through short-ranged power-law potentials,
$V(r)\sim r^{-\lambda}\;(\lambda/D>1)$. We show that such systems obey a
non-linear diffusion equation, and that their stationary state extremizes a
$q$-generalized nonadditive entropy. Here we focus on the dynamical evolution
of these systems. Our first-principle $D=1,2$ many-body numerical simulations
(based on Newton's law) confirm the predictions obtained from the
time-dependent solution of the non-linear diffusion equation, and show that the
one-particle space-distribution $P(x,t)$ appears to follow a compact-support
$q$-Gaussian form, with $q=1-\lambda/D$. We also calculate the velocity
distributions $P(v_x,t)$ and, interestingly enough, they follow the same
$q$-Gaussian form (apparently precisely for $D=1$, and nearly so for $D=2$).
The satisfactory match between the continuum description and the molecular
dynamics simulations in a more general, time-dependent, framework neatly
confirms the idea that the present dissipative systems indeed represent
suitable applications of the $q$-generalized thermostatistical theory.
