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Clausius vs. Boltzmann-Gibbs entropies

In this work, we demonstrate the inappropriateness of the Boltzmann-Gibbs log-formulation of the physical Clausius entropy $S$ in connecting thermodynamics and phase space statistics. To achieve our goal, we study thermodynamically the simple case of ideal gases embedded in a finite heat bath and compare the derived equations with the phase space statistical ones obtained within the canonical ensemble. We then show that the results of the aforementioned approaches contradict to each other even in the dilute gas limit (infinite heat bath) if we request the identification of the Boltmann-Gibbs formula $\ln(Density\,\, of\,\, States)$ with the thermodynamic entropy $S$.