Entropy ambiguity in a system in equilibrium with a finite heat bath

Abstract

Plastino and Plastino [Phys. Lett. A 193 (1994) 140] have recently shown that the probability distribution for a system in equilibrium with a finite heat bath having a power-law density of states is given by the generalized canonical distribution of Tsallis [J. Stat. Phys. 52 (1988) 479]. Here we show that the entropy S of such a system is however ambiguous, and is not uniquely given by the Tsallis entropy (or its additive equivalent the Rényi entropy). The ambiguity is due to the fact that the extension of the Boltzmann formula S = ln W to nonuniform probability distributions requires an additional postulate, which in the present context may be taken to be the definition of the entropy of the finite heat bath. Two reasonable alternative definitions D1 and D2 lead to the conventional Boltzmann-Gibbs-Shannon entropy and the Rényi entropy, respectively. Since the Tsallis distribution maximizes the Rényi entropy, definition D2 preserves the entropy maximum principle, whereas D1 does not. The distinction between D1 and D2 vanishes for large heat baths, while small baths exhibit thermodynamic peculiarities such as unequal system and bath temperatures.