Gibbs, Boltzmann, and negative temperatures

In a recent paper, Dunkel and Hilbert [Nat. Phys. 10, 67–72 (2014)] use an entropy definition due to Gibbs to provide a “consistent thermostatistics” that forbids negative absolute temperatures. Here, we argue that the Gibbs entropy fails to satisfy a basic requirement of thermodynamics, namely, that when two bodies are in thermal equilibrium, they should be at the same temperature. The entropy definition due to Boltzmann does meet this test, and moreover, in the thermodynamic limit can be shown to satisfy Dunkel and Hilbert's consistency criterion. Thus, far from being forbidden, negative temperatures are inevitable, in systems with bounded energy spectra.