Regarding negative temperatures…
Here are a couple of ways (neither very intuitive) to think about negative temperatures:
-
Entropy/energy relation. We’ll define some things:
U = internal energy of the system. This is all the energy from particle motion, spin interactions, anywhere energy can hide.
S = entropy of the system. Entropy is no more than a count of how many states are accessible to the system (given its energy, let’s say). That is, if I say the system has energy U, you can look at the various parameters describing the system and count how many ways you can make the system have energy U. Probably a lot of ways. Entropy is the logarithm (base e) of this (often huge) number.
T = temperature. Since S is often thought of as a function of U (i.e., S(U) = logarithm of number of states with energy U), it might be useful to know how S changes as we change U. Temperature, then, is defined as T=1/(dS/dU). It is the inverse of the slope of the S-U curve. (Note: this derivative is a partial derivative, if you care.)
Whew. So, with this non-intuitive (but fundamental) definition of temperature, all we need in order to have negative temperature is to find some point on the S-U curve where increasing the energy lowers the number of possible states (i.e., where S(U) has negative slope). Does this occur? Not usually, but sort of, sometimes. More later.
2) Boltzmann factor. You have a system at temperature T. What state might it be in? The probability of it being in some particular state with energy E given T is given by the Boltzmann factor: P=e[sup]E/kT[/sup]/Z, where Z and k are constants. (The use of E instead of U is unimportant.) It could just so happen that you have a system whose probability distribution of states, in fact, follows the Boltzmann distribution, but only if you say the temperature is negative.
Most normal systems cannot have negative temperature. You can’t have a brick with negative temperature. The concept only comes up in systems that have a highest possible energy. A free particle can’t meet this requirement as you can always stuff more energy into a particle (by making it go faster). But, if you had some isolated contrived system like this:
You have 2 particles. Each can have either energy 0 or E. The possible energies, then, are 0, E, and 2E. That’s it. (Maybe these 2 particles are held still so we can’t add energy in the motion.) There are two ways to make the system have energy E – particle one has energy E and particle two has energy 0 or vice-versa. Thus, there is a peak in the entropy at U=E since S(0)=log(1), S(E)=log(2), and S(2E)=log(1). Thus, the system can have negative temperature (since dS/dU goes negative. (Doing derivatives on discrete values is clearly bad, but the concept holds.))
Physically realizable? Finding these sorts of systems is easy (atomic levels, spins in magnetic fields), but finding isolated versions is not. If the spins (etc.) are coupled to a particle motion, for instance, then you can’t ingnore that motion in your description of the state. All of a sudden, you can’t have negative temperatures anymore. All is not lost, though. You can speak of the temperature of just those degrees of freedom that allow negative temperatures. And, if those degrees of freedom (like spin) don’t interact too much with the “bad” ones (like momentum), then you can actually do some physics with the concept. Particularly, if the time scale for energy transfer between “good” and “bad” degrees of freedom is much longer than the time scale for interaction within just the “good” degrees of freedom, then talking about the temperature of only those degrees can be useful. This approximation aspect, though, doesn’t take away from the reality of negative temperature.
(Relatively) super-short summary of everything above: You can’t take a system and cool it down and get past absolute zero. You can, however, talk about certain parts of certain systems as having negative temperature since the way these sub-systems interact with each other follows familiar statistical mechanics (but only if you let temperature go negative.)
