"Temperature bottoms out at 0 K and has no ceiling" ── both halves are wrong. Make β the quantity and the whole range lines up on one axis
Temperature has a floor (absolute zero) and no ceiling ── that is what we are taught. Neither half is quite right. As Episode 2 showed, the real quantity is not \(T\) but \(\beta=1/k_BT\): "gain one unit of energy, and by what factor does the count of ways multiply?" For ordinary matter more energy always means more ways, so \(\beta>0\). But in a system whose energy has a ceiling, the story changes ── near the ceiling, taking more energy makes the count of ways shrink. Then \(\beta<0\), i.e. \(T<0\). And that is not "colder than absolute zero" ── it is hotter than infinite temperature. It was made in nuclear spins in 1951 and in the motional degrees of freedom of cold atoms in 2013.
A gas molecule can move arbitrarily fast. Its energy has no upper bound. So the more energy you add, the more combinations of velocities are possible ── the count of ways \(\Omega\) keeps growing.
Recall the definition \(\beta=d\ln\Omega/dE\) from Episode 2: if \(\Omega\) keeps growing, \(\beta\) is always positive. That is why "ordinary matter always has positive temperature." It is not a law of physics but merely a consequence of energy having no ceiling.
Now consider a system with a ceiling. The simplest is the two-level system from Episode 3: each particle is either down (energy 0) or up (energy \(\Delta E\)). With \(N\) of them the total energy runs from 0 to \(N\Delta E\) ── a genuine ceiling.
The number of states with \(n\) of the \(N\) particles up is \(\Omega=\binom{N}{n}\), with energy \(E=n\Delta E\).
| State | fraction up, p | ways Ω | β = dlnΩ/dE |
|---|---|---|---|
| all down (lowest energy) | 0 | 1 (one way) | \(+\infty\) |
| half and half | 0.5 | maximum | 0 |
| all up (highest energy) | 1 | 1 (one way) | \(-\infty\) |
The count of ways is hill-shaped. The summit is at half and half, and past it adding energy makes the count fall. On the right of the summit \(d\ln\Omega/dE<0\), i.e. \(\beta<0\).
Solving explicitly, the relation to the upper-level occupancy \(p\) is
Raise \(p\) smoothly from 0 to 1 and:
$$\beta:\ +\infty\ \longrightarrow\ 0\ \longrightarrow\ -\infty \qquad\text{(one road, continuous)}$$ $$T:\ 0^{+}\ \longrightarrow\ +\infty\ \ \Big|\ \ -\infty\ \longrightarrow\ 0^{-} \qquad\text{(jumps in the middle)}$$Only \(T\) jumps; \(\beta\) walks straight through without incident. So the eerie point called "infinite temperature" is an apparent singularity created by writing a reciprocal. This is why Episode 2 insisted the real quantity is \(\beta\).
Which of two things is hotter is decided by which way heat flows when you put them in contact. As Episode 2 showed, heat flows toward the larger \(\beta\) (the side whose count of ways explodes on receiving energy is the "cold" one).
So line them up:
| cold ← | β | T | → hot |
|---|---|---|---|
| absolute zero | \(+\infty\) | \(0^+\) | the coldest there is |
| room temperature | 40 /eV | 300 K | |
| the centre of the Sun | small positive | \(10^7\) K | |
| infinite temperature | 0 | \(\pm\infty\) | still only halfway |
| negative temperature | small negative | \(-10^7\) K | hotter still |
| complete inversion (all up) | \(-\infty\) | \(0^-\) | the hottest there is |
They line up on one ordering: the smaller \(\beta\), the hotter. A negative-temperature system has \(\beta<0\), so against any positive-temperature system (\(\beta>0\)) it is always the one that gives energy away ── hence "hot." And \(T=0^-\) (complete inversion) is the maximum temperature.
On the left of the figure is the entropy \(S\) of the two-level system plotted against energy \(E\). It is hill-shaped. The slope of the tangent to this curve is exactly \(1/T=\partial S/\partial E\) ── Episode 2's definition itself.
Raise the upper-level occupancy \(p\) with the slider. The tangent tips over smoothly: rising (\(T>0\)) → horizontal (\(T=\infty\)) → falling (\(T<0\)). Nowhere is there a discontinuity. The right-hand panel marks the same state on both a \(\beta\) axis and a \(T\) axis ── you can see the \(\beta\) side pass through quietly while only the \(T\) side leaps from one end to the other.
| Year | System | Method |
|---|---|---|
| 1951 | nuclear spins (LiF crystal) Purcell and Pound | Align the spins in a magnetic field, then reverse the field suddenly. The spins cannot follow, and find themselves biased toward the high-energy side ── a population inversion. The spin system alone holds \(T<0\) for several minutes |
| 2013 | motional degrees of freedom of cold atoms Braun et al. | An optical lattice puts a ceiling on the kinetic energy of potassium atoms (band structure), and the sign of the interaction is flipped. What was new is that negative temperature was achieved in motional degrees of freedom themselves |
| routinely | laser population inversion | More population upstairs than downstairs = formally \(T<0\). But it is not in thermal equilibrium, so whether to call it a "temperature" is a matter of taste (see the honest line below) |
The answer goes back to section 01 ── everyday systems have no ceiling on kinetic energy. Molecules can always go faster, so \(\Omega\) keeps growing and \(\beta\) stays positive. To make a negative temperature you need to be able to treat a ceilinged degree of freedom on its own, cut off from the rest.
The nuclear-spin experiment worked because the spin system took minutes to exchange heat with the lattice (= the degree of freedom with no ceiling). For those minutes, the spins alone could carry a negative temperature as an independent "system." Put the other way round, negative temperature is always a quasi-equilibrium state ── wait long enough and heat always leaks into the unbounded degrees of freedom and the temperature returns to positive.
Established: that with \(1/T=\partial S/\partial E\) as the definition, \(T<0\) is definable in a system whose energy has a ceiling and whose entropy is hill-shaped in \(E\); that such a system always releases energy on contact with a positive-temperature system (= hotter than any positive temperature); the realisations in nuclear spins (Purcell–Pound 1951) and in the motional degrees of freedom of cold atoms (Braun et al. 2013); and that using \(\beta=1/k_BT\) makes the entire range of temperature a single continuous axis ── all standard physics.
What is debated: (1) Negative temperature holds only in quasi-equilibrium. Wait for relaxation with unbounded degrees of freedom (lattice vibrations, say) and it always returns to positive; the premise is that there is a time window in which the subsystem alone can be treated as equilibrated. (2) The answer depends on which entropy you adopt. The body uses the usual (Boltzmann) entropy \(S=k_B\ln\Omega\); adopt the Gibbs volume entropy instead and the temperature is never negative ── a dispute broke out around 2014 over whether negative temperature exists at all, and it has not fully settled. The majority position is the one taken here (negative temperature is physically meaningful), but do note that the claim is definition-dependent. (3) A laser's population inversion can formally be written as \(T<0\), but being far from thermal equilibrium, some are cautious about calling it a temperature. (4) The figure idealises \(N\) independent two-level systems and ignores interactions.
In a system whose energy has a ceiling, the count of ways \(\Omega\) is hill-shaped as a function of \(E\). Past the summit (half and half) \(d\ln\Omega/dE<0\), i.e. \(\beta<0\) and \(T<0\). For a two-level system \(\beta=\frac{1}{\Delta E}\ln\frac{1-p}{p}\), so you get there just by raising the occupancy \(p\) from 0 to 1.
And a negative temperature is not "cold" but hotter than any positive temperature ── heat flows toward the larger \(\beta\), so a \(\beta<0\) system is always the giver. Ordered from cold: \(0^+\to+\infty\ |\ -\infty\to0^-\), and only \(T\) jumps; \(\beta\) walks quietly through. Episode 2's insistence that "the real quantity is \(\beta\)" pays off here.
Realised in nuclear spins (1951) and cold atoms (2013). But it requires being able to isolate the ceilinged degrees of freedom, and waiting always returns it to positive temperature ── negative temperature is always a quasi-equilibrium phenomenon.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen, raising the upper-level occupancy tips the tangent from rising to horizontal to falling, and only T leaps across ±∞. "See the answer" opens each solution.