System-size scaling of Boltzmann and alternate Gibbs entropies

It has recurrently been proposed that the Boltzmann textbook definition of entropy $S(E)=k\ln Ω(E)$ in terms of the number of microstates $Ω(E)$ with energy $E$ should be replaced by the expression $S_G(E)=k\ln \sum_{E^\prime <E}{Ω(E^\prime )}$ 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.

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