The other day, someone asked me about negative temperatures. What does it even mean, really? We are often told that negative temperature is "hotter" than infinitely hot. How can that be?

Well, let us look at the definition of temperature. In axiomatic thermodynamics, we may encounter it in the form,

\begin{align}
\frac{1}{T}=\frac{\partial S}{\partial U},\tag{1}
\end{align}

that is, temperature expresses the rate of change in entropy $S$ as a function of the system's internal energy $U$.

Closely related is the definition that arises in the form of the Boltzmann distribution in statistical physics. The probability $P(E)$ of finding a system in a state characterized by energy $E$ is given by

\begin{align}
P(E) \propto e^{-E/kT}.\tag{2}
\end{align}

So let us suppose that there is such a thing as negative temperature. In the case of Eq. (1), this implies that there is an entropy maximum at some finite value of the internal energy $U$, beyond which entropy decreases, so the partial derivative becomes negative. In the case of Eq. (2), negative temperature implies a change in sign in the exponent, which implies probabilities that increase with energy.

How can this happen? Well, it happens of the system's energy is bounded from above. And ultimately, it boils down to simple combinatorics.

Take a system of $k$ particles. Assume that each of these particles can have distinct energy levels between $0$ and $m$. Given a number, $0\le n \le km$, how many different permutations exist of the $k$ particles with total energy $km$?

This is synonymous to asking for the coefficient of $x^n$ in the expression, $(1+x+x^2+\cdots+x^m)^k.$ We get

\begin{align}
N(k,m,n)=\sum_{j=0}^{\left\lfloor \frac{n}{m+1}\right\rfloor}
(-1)^j \binom{k}{j}\binom{n-j(m+1)+k-1}{k-1},\tag{3}
\end{align}

In the limit where $m\to\infty$, this collapses to

\begin{align}
\lim_{m\to\infty} N(k,m,n)=\binom{n+k-1}{k-1}.\tag{4}
\end{align}

Here is what the two cases look like when plotted in terms of the total energy (in arbitrary units):

How to make sense of this plot? The red line shows a conventional system of 6 particles, their energies not bounded from above. As we increase the total energy, the number of ways those energy levels can be expressed in terms of the sums of the energy levels of individual particles continues to increase rapidly.

The green curve shows something different. When the individual particles' energy levels are constrained from above, the curve has a maximum, in this case at half the maximum total energy. Beyond that, the number of ways the total energy can be expressed as a sum of individual energies declines quickly. The maximum energy of the system can only be achieved by one combination: When all individual particles are at maximum energy.

This has profound consequences on entropy and, ultimately, temperature.

The green curve, representing the entropy of the system with its energy bounded from above, has a maximum at half the system's maximum energy. As a result, the temperature reads like this:

This plot tells (almost) the full story: for the bounded system, temperature rises rapidly to infinity and beyond, as the system's total energy continues to increase beyond the halfway point. This result explains why "negative temperature" actually represents a system that is hotter (more energy) than a temperature of plus infinity.

Doesn't make much sense, does it? Well... perhaps it's because for historical or cultural reasons, we've been using the wrong temperature scale all along. How about if we replace the temperature $T$ with its negative reciprocal? Like this:

\begin{align}
\Theta=-\frac{1}{T}.\tag{5}
\end{align}

Now isn't this plot beautiful. This new temperature, $\Theta$, increases monotonically. It's minus infinity at "absolute zero". For systems not bounded by a maximum energy, it never reaches 0; always stays negative. Maximum temperature, corresponding to infinite energy, is zero. For systems bounded in energy, it goes over into positive territory, and the maximum temperature, when the system is at its maximum energy state, is infinite.

Better yet, we can also make this temperature scale... convenient. Just as the Kelvin scale is made convenient by adding 273.15 and define centigrades (Celsius), we can do the same with the $\Theta$ scale. Here it is, rescaled so that the freezing point and melting point of water are at $\Theta'=0$ and $\Theta'=100$, respectively, $\Theta'=373.15(1-273.15/(T+273.15))$ as a function of the conventional temperature value $T$ in centigrades:

The mapping is almost linear. For instance, body temperature, 37 C, is 44.5 degrees on the $\Theta'$ scale.