  We introduce a new method to study the velocity distribution of galaxy
systems, the Hellinger Distance (HD) - designed for detecting departures from a
Gaussian velocity distribution. We define a relaxed galactic system as the one
with unimodal velocity distribution and a normality deviation below a critical
value (HD<0.05). In this work, we study the gaussian nature of the velocity
distribution of the Berlind group sample, and of the FoF groups from the
Millennium simulation. For the Berlind group sample (z<0.1), 67% of the systems
are classified as relaxed, while for the Millennium sample we find 63% (z=0).
We verify that in multimodal groups the average mass of modes in high
multiplicity (N >= 20) systems are significantly larger than in low
multiplicity ones (N<20), suggesting that groups experience a mass growth at an
increasing virialization rate towards z=0, with larger systems accreting more
massive subunits. We also investigate the connection between galaxy properties
([Fe/H], Age, eClass, g-r, R_petro and <mu_petro>) and the gaussianity of the
velocity distribution of the groups. Bright galaxies (M_r <=-20.7) residing in
the inner and outer regions of groups, do not show significant differences in
the listed quantities regardless if the group has a Gaussian (G) or a
Non-Gaussian (NG) velocity distribution. However, the situation is
significantly different when we examine the faint galaxies (-20.7<M_r<=-17.9).
In G groups, there is a remarkable difference between the galaxy properties of
the inner and outer galaxy populations, testifying how the environment is
affecting the galaxies. Instead, in NG groups there is no segregation between
the properties of galaxies in the inner and outer regions, showing that the
properties of these galaxies still reflect the physical processes prevailing in
the environment where they were found earlier.
