Intensive entropy?

Entropy in classical thermodynamics [1] is an extensive quantity, which like energy, volume, or particle number, is additive when systems in equivalent thermodynamic states are aggregated. Claude Shannon showed [2] that many of our familiar notions of information correspond to reductions in entropy, emphasizing that entropy is a measure of ignorance, while adding interpretations of information change to physical phenomena involving heat flows [3]. The statistical basis given by Boltzmann and Gibbs for extensivity of entropy in classical thermodynamics is factorability of probability distributions: the microstate of one subsystem is conditionally independent of the microstates of other subsystems, given their macroscopic state variables. Factorability is weakly broken in a global and scale-invariant way in the coarsegrained description of certain chaotic systems, and Tsallis et. al. [4] have argued that for some of these a nonShannon-Boltzmann entropy, which preserves extensivity by avoiding double-counting of the same information, should be used as a foundation for thermodynamics.