More musing about phase transitions

Everyone has seen phase transitions - water freezing and water boiling, for example. These are both examples of "first-order" phase transitions, meaning that there is some kind of "latent heat" associated with the transition. That is, it takes a certain amount of energy to convert 1 g of solid ice into 1 g of liquid water while the temperature remains constant. The heat energy is "latent" because as it goes into the material, it's not raising the temperature - instead it's changing the entropy, by making many more microscopic states available to the atoms than were available before. In our ice-water example, at 0 C there are a certain number of microscopic states available to the water molecules in solid ice, including states where the molecules are slightly displaced from their equilibrium positions in the ice crystal and rattling around. In liquid water at the same temperature, there are many more possible microscopic states available, since the water molecules can, e.g., rotate all over the place, which they could not do in the solid state. (This kind of transition is "first order" because the entropy, which can be thought of as the first derivative of some thermodynamic potential, is discontinuous at the transition.) Because this kind of phase transition requires an input or output of energy to convert material between phases, there really aren't big fluctuations near the transition - you don't see pieces of ice bopping in and out of existence spontaneously inside a glass of icewater.

There are other kinds of phase transitions. A major class of much interest to physicists is that of "second-order" transitions. If one goes to high enough pressure and temperature, the liquid-gas transition becomes second order, right at the critical point where the distinction between liquid and gas vanishes. A second order transition is continuous - that is, while there is a change in the collective properties of the system (e.g., in the ferro- to paramagnetic transition, you can think of the electron spins as many little compass needles; in the ferromagnetic phase the needles all point the same direction, while in the paramagnetic phase they don't), the number of microscopic states available doesn't change across the transition. However, the rate at which microstates become available with changes in energy is different on the two sides of the transition. In second order transitions, you can get big fluctuations in the order of the system near the transition. Understanding these fluctuations ("critical phenomena") was a major achievement of late 20th century theoretical physics.

Here's an analogy to help with the distinction: as you ride a bicycle along a road, the horizontal distance you travel is analogous to increasing the energy available to one of our systems, and the height of the road corresponds to the number of microscopic states available to the system. If you pedal along and come to a vertical cliff, and the road continues on above your head somewhere, that's a bit like the 1st order transition. With a little bit of energy available, you can't easily go back and forth up and down the cliff face. On the other hand, if you are pedaling along and come to a change in the slope of the road, that's a bit like the 2nd order case. Now with a little bit of energy available, you can imagine rolling back and forth over that kink in the road. This analogy is far from perfect, but maybe it'll provide a little help in thinking about these distinctions. One challenge in trying to discuss this stuff with the lay public is that most people only have everyday experience with first-order transitions, and it's hard to explain the subtle distinction between 1st and 2nd order.