  In this paper we discuss about the validity of the Shannon entropy functional
in connection with the correct Gibbs-Hertz probability distribution function.
We show that there is no contradiction in using the Shannon-Gibbs functional
and restate the validity of information theory applied to equilibrium
statistical mechanics. We show that under these assumptions, entropy is always
a monotone function of energy, irrespective to the shape of the density of
states, leading always to positive temperatures even in the case of inverted
population systems. In the second part we assume the validity of the Shannon
entropy and thermodynamic temperature, T=dE/dS, extended to systems under
non-equilibrium steady state. Contrary to equilibrium, we discuss the
possibility and meaning of a negative temperature in this case. Finally we
discuss on Carnot cycles operating with a non-equilibrium bath possessing a
negative temperature and leading to apparent efficiencies larger than one, due
to a wrong accounting af all the energy and entropy fluxes present in the
system, including the external driving forces.
