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In this work the non-additive entropy is examined. It appears in isolatedparticle systems composed of few components. Therefore, the mixing of isolatedparticle systems S=S1+S2 has been studied. Two cases are considered T1=T2 andT1\leqT2, where T1,T2 are the initial temperatures of the system S1 and S2respectively. The concept of similar systems containing interacting particlesis introduced. These systems are defined by a common temperature and anidentical time evolution process, i.e. the approach to the same thermodynamicequilibrium. The main results are: 1) The properties of the similar particlesystems yield the non-additive entropy and free energy. The Gibbs Paradox isnot a paradox. 2) The relation between the initial temperatures T1 and T2governs the mixing process. 3) In the two cases T1=T2, T1\leqT2 mixing of thesystems S1, S2 results in a uniform union system S=S1+S2. The systems S, S1, S2are similar one to the other. 4) The mixing process is independent of theextensive quantities (volume, particle number, energy) and of the particletype. Only the mean energy plays an important role in the mixing of the systemsS1, 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 anirreversible process. Keywords: Entropy; Similar systems of interactingparticles; Mixing of systems; Thermal equilibrium

Non-additive entropy: Reason and conclusions