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 *S _{1}*
and

*S _{3}* has many possible internal states (microstates) because the atoms
of

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 *S _{3}* bounces randomly from one
microstate to another as the combined system moves towards a state of maximal total entropy. We define the
temperature

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.