Haha, no yeah, that's a very jargony sentence indeed. Phase space is jargon for the set of all allowed configurations of a system. In the simplest case, an ide

Haha, no yeah, that's a very jargony sentence indeed. Phase space is jargon for the set of all allowed configurations of a system. In the simplest case, an ideal gas of n spherical atoms, each atom is specified by 3 independent numbers for position and 3 for momentum, so you need 6 n numbers total to specify the whole configuration... That is the phase space. This phase space is only partly bounded, it is bounded in position (V^n where V is the volume each particle can occupy) but not in momentum (the temperature can always go higher). Because each of these numbers can vary independently from each other, they are called “independent degrees of freedom.” Now, point vortices are an abstract mathematical limit of physical vortices as they get really small. Generally we take these mathematical limits because it's much easier to work with and we are lazy. The article is saying that Onsager chose to explain his theory with point vortices because you can't give them arbitrarily high momentums, in fact their positions have to be tied up with where they are going, so that they are not independent anymore like they were with the gas. So Onsager knew that if he chose this system then the boundary on volume (well, area, the vortices live in 2D) would become a boundary on momentum space and then you couldn't dump infinite energy into the system. And this is important because it means that when you add more energy, the number of allowed configurations with that energy eventually shrinks. This is the negativity of negative temperature, increase something and something else decreases. The thing that is decreasing, a volume in phase space, is also called the entropy. Increase energy, and entropy decreases... most normal systems work the opposite way, entropy increases. As systems share things to maximize total entropy, this overcharged vortex system is going to share energy with any normal system at any temperature, because that normal system will increase its entropy with energy coming into it, and the overcharged vortex system will gain entropy when it loses energy. So it is hotter than any other thing in the world. (I am erasing the jargon word “canonical” from this explanation... It just refers to a particular procedure we use to define momentum coordinates based on your position coordinates. When you are not dealing with a point particle with defined mass in a flat space, you have to make these sorts of choices.)