Poincaréʹs observation and the origin of Tsallis generalized canonical distributions

In this paper, we present some geometric properties of the maximum entropy Tsallis-distributions under energy constraint. In the case q>1, these distributions are proved to be marginals of uniform distributions on the sphere; in the case q<1, they can be constructed as conditional distributions of a Cauchy law built from the same uniform distribution on the sphere using a gnomonic projection. As such, these distributions reveal the relevance of using Tsallis distributions in the microcanonical setup: an example of such application is given in the case of the ideal gas.