I've read recently about systems with the ability to reach negative temperature: where entropy decreases as energy increases. I have a couple question regarding this. First, how is it possible to reach negative temperature without crossing absolute zero? Is there some discontinuity like a phase transition that allows this to happen? How can negative temperature be related to the kinetic energy of system? Additionally, wouldn't this allow heat to flow from negative to positive temperature, leading to more than 100% efficient engines? Finally, could this have anything to do with negative pressure associated with positive vacuum energy density?