System-size scaling of Boltzmann and alternate Gibbs entropies

Abstract: It has recurrently been proposed that the Boltzmann textbook definition of entropy $S(E)=k\ln \Omega (E)$ in terms of the number of microstates $\Omega (E)$ with energy $E$ should be replaced by the expression $S_G(E)=k\ln \sum_{E^\prime <E}{\Omega (E^\prime )}$ examined by Gibbs. Here, we show that $S_G$ either is equivalent to $S$ in the macroscopic limit or becomes independent of the energy exponentially fast as the system size increases. The resulting exponential scaling makes the realistic use of $S_G$ unfeasible and leads in general to temperatures that are inconsistent with the notions of hot and cold.