  In this work the non-additive entropy is examined. It appears in isolated
particle systems composed of few components. Therefore, the mixing of isolated
particle systems S=S1+S2 has been studied. Two cases are considered T1=T2 and
T1\leqT2, where T1,T2 are the initial temperatures of the system S1 and S2
respectively. The concept of similar systems containing interacting particles
is introduced. These systems are defined by a common temperature and an
identical time evolution process, i.e. the approach to the same thermodynamic
equilibrium. The main results are: 1) The properties of the similar particle
systems yield the non-additive entropy and free energy. The Gibbs Paradox is
not a paradox. 2) The relation between the initial temperatures T1 and T2
governs the mixing process. 3) In the two cases T1=T2, T1\leqT2 mixing of the
systems S1, S2 results in a uniform union system S=S1+S2. The systems S, S1, S2
are similar one to the other. 4) The mixing process is independent of the
extensive quantities (volume, particle number, energy) and of the particle
type. Only the mean energy plays an important role in the mixing of the systems
S1, S2. 5) Mixing in the case T1\leqT2 is in essence a thermalization process,
but mixing in the case T1=T2 is not a thermodynamic process. 6)Mixing is an
irreversible process. Keywords: Entropy; Similar systems of interacting
particles; Mixing of systems; Thermal equilibrium
