Scale invariance and related properties of q-Gaussian systems

We advance scale-invariance arguments for systems that are governed (or approximated) by a q  -Gaussian distribution, i.e., a power law distribution with exponent Q=1/(1−q)Q=1/(1−q); q∈Rq∈R. The ensuing line of reasoning is then compared with that applying for Gaussian distributions, with emphasis on dimensional considerations. In particular, a Gaussian system may be part of a larger system that is not Gaussian, but, if the larger system is spherically invariant, then it is necessarily Gaussian again. We show that this result extends to q-Gaussian systems via elliptic invariance. The problem of estimating the appropriate value for the Tsallis' parameter q is revisited. A kinetic application is also provided.