Negative temperature

Let's think about the concept of temperature for a moment. What is a temperature, in terms of how you measure it in real life? When we measure temperature, we put a small system (a thermometer) in contact with a larger system. The thermometer comes to equilibrium with the larger system by exchanging energy until the total entropy of both systems is maximized. What we read off as temperature has to do with how the fluid inside the thermometer expands or contracts.

Systems at negative temperature come to equilibrium by the same means. Recall that, when we put two systems together that are isolated, the total entropy is maximized, S = S1+S2 while the total energy is held constant, E=E1+E2. Thus the sign of dS must be positive. But dS is just related to the temperature, when the energy changes.

A thermometer put in contact with a system at negative temperature will come to equilibrium in the same way it would as if it were put into contact with a system at positive temperature. The two systems will exchange energy until their temperatures are the same. Whatever temperature the thermometer is at in the end will depend on how much energy it exchanged with the system.

Not all systems can have negative temperatures. Only systems that have entropy functions that contain regions of negative slope do. Some systems have entropy functions that increase monotonically with energy for all energies, and thus, these systems cannot have negative slopes. Fluids have the latter kind of entropy function. Magnets can have the former. The basic difference between the two is that fluids can have an infinite energy (because the momenta are unbounded), but magnets can be saturated at a maximum energy.