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.
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