Could someone provide me with a mathematical proof of why, a system with an absolute negative Kelvin temperature (such that of a spin system) is hotter than any system with a positive temperature (in the sense that if a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system).
Thermodynamics – Proving Negative Absolute Temperatures Are Hotter than Positive Ones
equilibriumspin-chainsstatistical mechanicstemperaturethermodynamics
Related Solutions
Negative temparature makes sense in statistical mechanics only if the associated Hamiltonian is bounded from above; otherwise the trace in the definition of the partition function does not exist.
In particular, a gas cannot have negative temperature since its Hamiltonian contains a kinetic energy term, which is not bounded from above. (Edit: Since the momentum operator is unbounded, the kinetic energy is unbounded, hence a Hamiltonian involving a kinetic term is unbounded. Thus the trace of $e^{-\beta H}$ is undefined for negative $\beta$. Thus the temperature must be positive. This is due to a purely mathematical property of the spectrum, independent of which energies can be realized in practice.)
Thus the question about its pressure at negative temperatures (and the argument with the ideal gas law) is moot.
Negative temperature is observed in certain spin systems living inside a crystal, and in systems of vortices in 2-dimensional plasmas. (This temperature is not the temperature of the crystal or plasma, but of the subsystem.) See, e.g., http://www.physics.umd.edu/courses/Phys404/Anlage_Spring11/Ramsey-1956-Thermodynamics%20and%20S.pdf
In these systems, the concept of pressure doesn't make much sense; at least it seems not to be used in this context. Formally, the pressure is of course defined. In a simple example, see Example 9.2.5 in my book
http://lanl.arxiv.org/abs/0810.1019
it changes sign with the temperature.
So, is there any macroscopic physical meaning to this statement that the negative temperature is "hotter"? Doesn't this mean that the negative temperature required for that laser to exist ought to be so "hot" no mere mortal can handle such a device?
If you go far up enough in the atmosphere, you'll reach regions where the temperature is over 3000 K. Yet rockets aren't exploding in flames when they get there, because the atmosphere is very sparse. It will transfer energy to the rocket, because its temperature is very high, but only incredibly slowly.
In general, human beings cannot detect temperature; we can only detect heat transfer. This is why metal surfaces often feel cold; they're the same temperature as everything else, but they conduct better, so you lose heat faster when touching them. Similarly, if you go up really high into the atmosphere, you won't feel like it's hot. Instead, you'll feel nothing at all.
So our intuitive sensory notions of "hot" or "cold" have very little to do with temperature. So it's indeterminate how "hot" a negative temperature system will feel, because that depends on how quickly it transfers heat to your body. All the temperature means is that it will transfer heat.
Best Answer
Arnold Neumaier's comment about statistical mechanics is correct, but here's how you can prove it using just thermodynamics. Let's imagine two bodies at different temperatures in contact with one another. Let's say that body 1 transfers a small amount of heat $Q$ to body 2. Body 1's entropy changes by $-Q/T_1$, and body 2's entropy changes by $Q/T_2$, so the total entropy change is $$ Q\left(\frac{1}{T_2}-\frac{1}{T_1}\right). $$ This total entropy change must be positive (according to the second law), so if $1/T_1>1/T_2$ then $Q$ has to be negative, meaning that body 2 can transfer heat to body 1 rather than the other way around. It's the sign of $\frac{1}{T_2}-\frac{1}{T_1}$ that determines the direction that heat can flow.
Now let's say that $T_1<0$ and $T_2>0$. Now it's clear that $\frac{1}{T_2}-\frac{1}{T_1}>0$ since both $1/T_2$ and $-1/T_1$ are positive. This means that body 1 (with a negative temperature) can transfer heat to body 2 (with a positive temperature), but not the other way around. In this sense body 1 is "hotter" than body 2.