Special Scale-Invariant Occupancy of Phase Space Makes the Entropy Sq Additive

Phase space can be constructed for N equal subsystems that could be (probabilistically) either independent or correlated. If they are independent, Boltzmann-Gibbs entropy S_{BG} equivalent to - k \sum_i p_i ln p_i is strictly additive in the sense that S_{BG}(N)=N S_{BG}(1). If they have (collectively) special scale-invariant correlations, the entropy S_q equivalent to k[1- \sum_i p_i^q]/(q-1) (with S_1=S_{BG}) satisfies, for some value of q \neq 1, S_q(N)=NS_q(1), and is therefore additive, hence extensive. We exhibit two paradigmatic systems (one discrete and one continuous) for which the entropy S_q is additive, whereas S_{BG} is neither strictly nor asymptotically so. We conjecture that this mechanism is deeply related to the nearly ubiquitous emergence, in natural and artificial complex systems, of scale-free structures.