Negative Temperature Without Statistical Mechanics

Question

In plenty of the qualitative thermodynamics that predates the statistical description, temperature is assumed to always be positive; many key inequalities related to the second law often involve multiplying or dividing by temperature without changing the sign which would only be valid were temperature always positive (the description of 'Availability' on p168 of Blundell & Blundell is what has brought this question up).

I am aware that the second law is more 'properly' viewed as a result of statistical mechanics, and the classical assumption of infinitely many energy states is what leads to strictly positive temperature. Is there any way to see this without reference to statistics?

I know there are plenty of questions here about negative temperature, but I don't think this is a duplicate (sorry if it is).

Answer 1

Here is an argument which forbids 'physical bodies' from having a negative temperature (adapted from Landau and Lifshitz - Statisitcal Physics I).

Consider a body divided into a large number of small, but macroscopic parts. Let E$_{\alpha}$, M$_{\alpha}$, P$_{\alpha}$ and S$_{\alpha}$ denote the energy, mass, momentum and entropy of the parts. (The momentum P$_{\alpha}$ corresponds only to the macroscopic motion of the entire ${\alpha}^{th}$ body i.e., all the molecules move in the same direction as opposed to thermal motion)

The entropy S$_{\alpha}$ of each part is a function of its internal energy which is the total energy E$_{\alpha}$ minus the energy due to macroscopic motion $\frac{P_{\alpha}^2}{2M_{\alpha}}$.

$$S = \sum S_{\alpha}(E_{\alpha} - \frac{P_{\alpha}^2}{2M_{\alpha}} )$$

Also, we know: $$ \frac{1}{T} = \frac{\partial S }{\partial U} $$

Let us consider a 'body' forming a closed system and as a whole at rest. If the temperature was negative, to increase entropy we would have to decrease the internal energy (argument of entropy).

This is possible here only by increasing the energy due to macroscopic motion (since total energy E$_{\alpha}$ is constant), which means that the momentum of individual parts would keep on increasing . For momentum conservation ($\sum P_{\alpha} = 0$) to hold, the parts of the body must fly off into various directions. Concisely put, the body would seek to break up into dispersing parts, spontaneously.

So, a physical body cannot be under equilibrium conditions with $T<0$.