A critique of non-extensive q-entropy for thermal statistics

Recently there have been numerous articles on a new relation between entropy and probability which is non-extensive, and which has an undetermined parameter q that depends on the nature of the thermodynamic system under consideration. For q = 1 this relation corresponds to the Boltzmann-Gibbs entropy, but for other values of q it is claimed that this relation leads to a formalism which is also consistent with the laws of thermodynamics. However, it is shown here that the joint entropy for systems having different values of q is not defined in this formalism, and consequently fundamental thermodynamic concepts like temperature and heat exchange cannot be considered for such systems. Moreover, for q 6= 1 the internal energy is also nonextensive, leading to unphysical consequences that have been ignored in various applications given in the literature. Recently a new relation between entropy and probability has been proposed by C. Tsallis [1] which is non-extensive and depends on a parameter q determined by the thermodynamic system under consideration. For the special case q = 1 this relation reduces to the Boltzmann-Gibbs entropy, but it is claimed that for other values of q this relation gives an alternative formalism that is “entirely consistent” with the laws of thermodynamics. [1]-[6]. It will be shown, however, that this claim is not valid, because the total entropy of