Negative temperatures and the definition of entropy

The concept of negative temperature has recently received renewed interest in the context of debates about the correct definition of the thermodynamic entropy in statistical mechanics. Several researchers have identified the thermodynamic entropy exclusively with the "volume entropy’’ suggested by Gibbs, and have further concluded that by this definition, negative temperatures violate the principles of thermodynamics. We disagree with these conclusions. We demonstrate that volume entropy is inconsistent with the postulates of thermodynamics for systems with non-monotonic energy densities, while a definition of entropy based on the probability distributions of macroscopic variables does satisfy the postulates of thermodynamics. Our results confirm that negative temperature is a valid extension of thermodynamics.