On the thermodynamic stability conditions of Tsallis' entropy

The thermodynamic stability condition (TSC) of Tsallis' entropy is revisited. As Ramshaw [Phys. Lett. A {\bf 198} (1995) 119] has already pointed out, the concavity of Tsallis' entropy with respect to the internal energy is not sufficient to guarantee thermodynamic stability for all values of $q$ due to the non-additivity of Tsallis' entropy. Taking account of the non-additivity the differential form of the TSC for Tsallis entropy is explicitly derived. It is shown that the resultant TSC for Tsallis' entropy is equivalent to the positivity of the standard specific heat. These results are consistent with the relation between Tsallis and R\'enyi entropies.