Original by Scott Chase.
Question: Can you really make a system that has a negative temperature?
Answer: You can, provided you allow some flexibility in the definition of temperature.
To get things started, we need a clear definition of temperature. Actually various kinds of "temperature" appear in the literature of physics (e.g., kinetic temperature, color temperature). The relevant one here is the one from thermodynamics, which is in some sense the most fundamental.
Our intuitive notion is that two systems in thermal contact should exchange no heat, on average, if and only if they are at the same temperature. Let's call the two systems S1 and S2. The combined system, treating S1 and S2 together, is called S3. The important question, consideration of which will lead us to a useful quantitative definition of temperature, is "How will the energy of S3 be distributed between S1 and S2?" I will explain this briefly, but I recommend that you read Kittel and Kroemer (referenced below) for a careful, simple, and thorough explanation of this important and fundamental result.
S3 has many possible internal states (microstates) because the atoms of S3 can share their total energy E in many ways. Let's say there are N of these microstates. Each microstate corresponds to a particular division of the total energy in the two subsystems S1 and S2. Many microstates can correspond to the same division of energy: E1 in S1 and E2 in S2. The only division of the energy that will occur with any significant probability is the one with the overwhelmingly largest number of microstates for the total system S3. That number N(E1, E2) is just the product of the number of states allowed in each subsystem: N(E1,E2) = N1(E1) N2(E2), and since E1 + E2 = E, N(E1, E2) reaches a maximum when N1N2 is stationary with respect to variations of E1 and E2 subject to the total energy constraint.
For convenience, physicists prefer to frame the question in terms of Boltzmann's constant k times the natural logarithm of the number of microstates N, and call this the entropy S (so S = k ln N). The above analysis indicates that two systems are in equilibrium with one another when (∂S/∂E)1 = (∂S/∂E)2 at constant volume and particle number; that is, the rate of increase of entropy S per unit increase in energy E must be the same for both systems (at constant volume and particle number). Otherwise, energy will tend to flow from one subsystem to another as S3 bounces randomly from one microstate to another as the combined system moves towards a state of maximal total entropy. We define the temperature T by T = ∂E/∂S at constant volume and particle number, so that the equilibrium condition becomes the very simple T1 = T2.
This statistical mechanical definition of temperature does in fact correspond to your intuitive notion of temperature for most systems. So long as ∂E/∂S is always positive, T is always positive. For common situations, such as a collection of free particles or particles in a harmonic oscillator potential, adding energy always increases the number of available microstates, increasingly faster with increasing total energy. So temperature increases with increasing energy, from zero, asymptotically approaching positive infinity as the energy increases.
Not all systems have the property that the entropy increases with energy. In some cases, as energy is added to the system, the number of available microstates, or configurations, actually decreases for some range of energies. For example, imagine an ideal "spin-system", a set of N atoms each with spin 1/2 on a one-dimensional wire. The atoms are not free to move from their positions on the wire. The only degree of freedom allowed to them is spin: the spin of a given atom can point in an "upward" or "downward" direction. The total energy of the system, in a magnetic field of strength B pointing down, is (U − D) μB, where μ > 0 is the magnitude of the component of each atom's magnetic moment along the B direction, and U and D are the numbers of atoms with "spin up" and "spin down" respectively. Notice that with this definition, E is zero when half of the spins are up and half are down. E is negative when the majority are down and positive when the majority are up.
The lowest possible energy state (all the spins pointing down) gives the system a total energy of −NμB, and a temperature of absolute zero. There is only one configuration of the system at this energy: all the spins must point down. The entropy is k times the log of the number of microstates, so in this case is k ln 1 = 0. If we now add energy 2μB to the system, one spin is allowed to flip up. There are N ways of having one spin flip up, so the new entropy is k ln N. If we add another quantum of energy, there are a total of N(N−1)/2 allowable configurations with two spins up. The entropy is increasing quickly, and the temperature is rising as well.
But for this system, the entropy does not go on increasing forever. The system has maximal energy NμB when all spins are up. Here there is again only one microstate, and the entropy is again zero. If we remove one quantum of energy from the system, we allow one spin down. At this energy there are N available microstates. The entropy goes on increasing as the energy is lowered. In fact the maximal entropy occurs for total energy zero, i.e., half of the spins up, half down.
So we have created a system where, as we add more and more energy, temperature starts off positive and increases to some large positive number as maximum entropy is approached, with half of all spins up. After that, the temperature becomes a negative number of large absolute value, coming down in magnitude toward zero as the energy increases toward maximum. (You will sometimes find it written that the temperature goes to infinity when half the spins are up, then jumps to minus infinity. This isn't really correct, because these examples of negative temperature always deal with discrete systems; but the above definition of temperature "T = ∂E/∂S at constant volume and particle number" refers to a continuous system—or at least a system that is as good as continuous. So the best we can do for our set of spins is write "T ≈ ΔE/ΔS at constant volume and particle number", so that the calculation of T is now a little nebulous.) If you take two copies of the system, one with positive and one with negative temperature, and put them in thermal contact, heat will flow from the negative-temperature system into the positive-temperature system. It is sometimes said that the negative-temperature system is hotter than the positive-temperature system. On the other hand, the original definition of temperature above was actually rooted in the analysis of everyday systems whose energy and entropy increase together, and simply defining "T = ∂E/∂S at constant volume and particle number" without this assumption is bound to produce strange values of temperature for systems that lie outside this assumption. Negative temperature is just such an example of these strange values, so don't take it too seriously!
Can this system ever by realized in the real world, or is it just a fantastic invention of sinister theoretical condensed matter physicists? Atoms always have other degrees of freedom in addition to spin, usually making the total energy of the system unbounded upward due to the atom's translational degrees of freedom. Thus, only certain degrees of freedom of a particle can have negative temperature. It makes sense to define the "spin temperature" of a collection of atoms as long as one condition is met: the coupling between the atomic spins and the other degrees of freedom is sufficiently weak, and the coupling between atomic spins sufficiently strong, that the timescale for energy to flow from the spins into other degrees of freedom is very large compared to the timescale for thermalisation of the spins among themselves. Then it makes sense to talk about the temperature of the spins separately from the temperature of the atoms as a whole. This condition can easily be met for the case of nuclear spins in a strong external magnetic field.
Nuclear and electron spin systems can be promoted to negative temperatures by suitable radio frequency techniques. Various experiments in the calorimetry of negative temperatures, as well as applications of negative temperature systems as RF amplifiers, can be found in the articles listed below, and the references therein.