# I How can an atom reach less than absolute zero?

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1. Jun 22, 2017

### Quantum Velocity

2. Jun 22, 2017

### A. Neumaier

They are actually hotter than infinite temperature. The point is that the physical relevant quantity is inverse temperature. Thus negative absolute temperatures are hotter than any positive one. I have given a more detailed explanation at https://www.physicsoverflow.org/28487

3. Jun 22, 2017

### hilbert2

The article uses misleading terms, as is common in popular science. A thermodynamical absolute temperature can't be negative. That would mean a closed system that goes lower in entropy when you pump heat into it in a reversible process, and that has never been observed in any experiment. In the statistical mechanical definition of temperature, an apparent "negative temperature" is possible in a nonequilibrium system where you have a large set of atoms or molecules that are more likely to be in an excited state than in the ground state (this doesn't happen in any system that's in thermal equilibrium, no matter how high temperature it's in).

Talking about the temperature of a single atom doesn't make any sense no matter what definintion of temperature is used, the whole concept requires a very large number of atoms to be useful.

4. Jun 22, 2017

### A. Neumaier

This only holds for systems that can move. See the references given in the link posted in #2.

5. Jun 22, 2017

### hilbert2

Would a system formed by the nuclear spins of small atoms locked in a solid matrix (i.e. interstitial sites of some crystal lattice) count as an immobile system? I saw something like this discussed on an MIT website.

6. Jun 22, 2017

### A. Neumaier

yes. The key question is whether the spectrum is unbounded. If it is, negative $\beta$ and hence negative energy is impossible. For Hamiltonians with a bounded spectrum, a canonical ensemble makes sense even for negative $\beta$, and this has been realized experimentally (not only recently).

7. Jun 22, 2017

### vanhees71

Formally it holds for systems whose energy is unbounded. For temperature in the usual sense to make any sense, the energy must be bounded from below, i.e., there should be a stable ground state. Then using the canonical ensemble you have
$$\hat{\rho}=\frac{1}{Z} \exp(-\beta \hat{H}), \quad \beta=\frac{1}{k_{\text{B}} T},$$
and if the spectrum of $\hat{H}$ is unbound from above, this has only a well-defined trace (normalized to 1 via the partition sum $Z$) for $T>0$.

For systems with a bounded Hamiltonian, $T$ can take any real value, and negative temperature means "population inversion".

8. Jun 22, 2017

### Quantum Velocity

How can they hotter than infinite

9. Jun 22, 2017

### Gigaz

The concept of negative temperature is controversial. Systems may have properties where you could attribute a temperature T<0, but those are not equilibrium states and usually, temperature is something which requires an equilibrium.

10. Jun 22, 2017

### Staff: Mentor

That's answered partway through the LiveScience article in the first post.

We say that A is hotter than B if heat will flow from A to B when they are brought together.

Heat will flow from a system with negative temperature to a system with positive temperature, no matter how great the temperature of the positive-temperature system. In other words, no matter how hot the positive-temperature system, the negative-temperature system is hotter.

11. Jun 22, 2017

### Quantum Velocity

ohhhhhhhhhhhhh

12. Jun 23, 2017

### A. Neumaier

One only needs local equilibrium, and the systems with $T<0$ are in local equilibrium.

13. Jun 23, 2017

### hsdrop

hey guys,
I'm a little more than confused on this subject. Please bear with me as I stumble along with the limited education that I have and the questions that I'm struggling to answer. Also before anyone tells me that this thread is marked for undergrads when I'm asking for a high school level explanation. I just didn't want to start a new thread on the topic when there was one already up.
First off, I thought that you can only heat something so far (to vibrate at planck length) and therefore can not be infinitely heated. I'm also having trouble understanding how to think of negative temperatures. Is it something that can only be shown with math or can someone describe what physically happens when hitting negative temperatures? Also where does the negative scale pick up at on the kelvin scale?

If it would be easier to start a new thread on the topic, let me know and I will.
Thank you to any one ahead of time that will help me understand the concept.

14. Jun 23, 2017

### A. Neumaier

It needs math, but only simple high school math. The physically relevant scale is inverse absolute temperature $\beta=1/T$ (in appropriate units), which is a measure of coldness rather than heat. Because of historical accidents, hotness and not coldness was formalized first. In therms of coldness, the temperature is therefore $T=1/\beta$. Thus $T=0$ corresponds to infinite coldness; it cannot get colder. Zero coldness is already very hot and corresponds to infinite $T$, negative coldness is even less cold, i.e., even hotter. Due to the singularity of the inverse transformation at $\beta=0$, the resulting temperature scale is split into two differently arranged infinite parts, going from 0K (infinitely cold) through 273 K (freezing point of water) through 373 K (boiling point of water) to $\infty$ K = $-\infty$ K (extremely hot) to $-0$ K (the hottest conceivable state).

15. Jun 23, 2017

### hsdrop

I'm sorry, I'm not understanding the math formulas at all or what each letter means in them. Is it possible to break it down even feather for me please. Also what's the difference between measuring hotness compared to measuring coldness? I am under the impression that hot and coldness is a matter of perspective.

16. Jun 23, 2017

### A. Neumaier

You measure one, and you take the inverse to get the other. $T$ is the temperature (what I called hotness) in Kelvin, $\beta$ the corresponding coldness.
Draw the function $\beta=1/T$ in a $(T,\beta)$ diagram, to see that the physically natural coldness order is transformed into a less natural hotness order, since positive and negative coldness transforms differently.

17. Jun 23, 2017

### hsdrop

ok I believe I understood the math this time around. However I'm still having a bit of trouble picturing what it would look like physically (off the graff paper that is)
again thank you for the patient with my eager uneducated mind

18. Jun 23, 2017

### Quantum Velocity

i think heat must flow from the positive to negative for

19. Jun 24, 2017

### A. Neumaier

heat always flows from lower coldness $\beta=1/T$ to higher coldness. This is generally true. In case of positive temperature, it therefore flows from higher temperature to lower. In case of negative $T$ it also flows from small negative temperatures to large negative temperatures. But the pole at T=0 implies that all these negative temperatures are less cold and hence hotter than all the positive temperatures.

Last edited: Jun 24, 2017
20. Jun 24, 2017

### Quantum Velocity

i don't get it