Gibbs volume entropy is incorrect.

@article{Swendsen2015GibbsVE,
  title={Gibbs volume entropy is incorrect.},
  author={Robert H. Swendsen and Jian-Sheng Wang},
  journal={Physical review. E, Statistical, nonlinear, and soft matter physics},
  year={2015},
  volume={92 2},
  pages={
          020103
        }
}
We show that the expression for the equilibrium thermodynamic entropy contains an integral over a surface in phase space, and in so doing, we confirm that negative temperature is a valid thermodynamic concept. This demonstration disproves the claims of several recent papers that the Gibbs entropy, which contains an integral over a volume in phase space, is the correct definition and that thermodynamics cannot be extended to include negative temperatures. We further show that the Gibbs entropy… 
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