Communication: 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 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 Sigma(E< E) Omega(E') 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. (C) 2014 AIP Publishing LLC.