THERMODYNAMIC STABILITY CONDITIONS FOR THE TSALLIS AND RENYI ENTROPIES

Thermodynamic stability conditions (TSCs) are derived for the Tsallis and Renyi entropies. These entropies are monotononically increasing functions of each other, so their TSCs are completely equivalent although different in appearance. The TSC for the Renyi entropy S(q)R is simply that S(q)R be a concave (convex) function of the mean energy E for q > 0 (q < 0). This condition is not in general satisfied. The TSC for the Tsallis entropy S(q)T is then an immediate consequence of the nonlinear functional relation between S(q)T and S(q)R. Due to the nonlinearity, the resulting TSC for S(q)T is not simply related to the concavity (convexity) of S(q)T. The concavity properties of S(q)T are therefore not sufficient to guarantee thermodynamic stability.