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Communication: System-size scaling of Boltzmann and alternate Gibbs entropies
2. J. W. Gibbs, Elementary Principles in Statistical Mechanics (Yale University Press, New Haven, 1902).
12. T. L. Hill, An Introduction to Statistical Thermodynamics (Dover Publications, New York, 1986).
13. M. Kardar, Statistical Physics of Particles (Cambridge University Press, Cambridge, 2007).
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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 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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