# A Thermometer That Could Measure Negative Temperature

Just for fun, I was thinking of trying to design one. How could we design a thermometer that could accurately measure negative temperature of a spin system?

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stewartcs
Just for fun, I was thinking of trying to design one. How could we design a thermometer that could accurately measure negative temperature of a spin system?
What's a spin system?

Also, there is no such thing as negative absolute temperature.

CS

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You can get a negative absolute temperature in certain cases in non-equilibrium thermodynamics. Spins can do this, and presumably that is what the OP is talking abiyt.

What I don't understand about the OP's question is that a thermometer works by being in equilibrium with the thing whose temperature you want to measure. So he is linking together the ideas on equilibrium thermo with non-equilibrium thermo in an odd way.

What's a spin system?

Also, there is no such thing as negative absolute temperature.

CS
Yes there is.

(One over) Temperature is defined as the change of entropy with respect to the change of energy, i.e. roughly

$\frac{1}{T} = \frac{\partial S}{\partial E}$

This means that the temperature is negative whenever the entropy decreases with increasing energy. Usually, the larger the energy of the system is, the larger its entropy will be. For free gases, for instance, the temperature is linear with respect to the mean kinetic energy of the atoms.

However, there are examples where there is a "largest energy state", meaning the system can only contain up to some amount of energy. Usually, this state will be unique and therefore this state doesnt correspond to the case where the system has the largest entropy. It is the lower energy states which correspond to higher entropies.

The system then has the remarkable property, that for the high energy states, it can increase its entropy by decreasing its energy.

As for the question of the topic... I am not sure yet, but my first guess would be: yes.

A thermometer is just some device which is tends to a thermal equilibrium with its surrounding or the "thing it is measuring". By doing so, we can read off in what macrostate the thermometer is and from there we know what the temperature is. So to begin, you would need a device which can have a negative temperature - or else it cannot reach a thermal equilibrium with the thing you're measuring. Next, you need some way of reading out in what macrostate the thermometer is in.

So perhaps you could use a spin system as a thermometer and maybe through some sort of magnet read out its temperature? Just guessing here.

stewartcs
Yes there is.

(One over) Temperature is defined as the change of entropy with respect to the change of energy, i.e. roughly

$\frac{1}{T} = \frac{\partial S}{\partial E}$

This means that the temperature is negative whenever the entropy decreases with increasing energy. Usually, the larger the energy of the system is, the larger its entropy will be. For free gases, for instance, the temperature is linear with respect to the mean kinetic energy of the atoms.
From a thermodynamic standpoint, 0 Kelvin is the lowest absolute temperature - not negative Kelvin. It is selected as the baseline for which thermodynamic equation are used.

However, it seems that for spin systems it can be negative according to this reference:

http://ltl.tkk.fi/triennial/positive.html

Caveat: I know nothing about spin systems nor do I know if that source is reliable.

I would like to see a proper reference for these claims though...so if either of you have one please let me know.

CS

I never claimed that a system can reach 0 Kelvin. On the contrary, in the spin system a negative temperature (or any other for that matter) actually corresponds to a hotter system than a system with a positive temperature. The meaning behind this is that when the systems are in thermal contact there will be a net energy flow from the hotter system to the colder one - just like you would expect. (Please note that most systems are not capable of obtaining a negative temperature)

There is even a point where the temperature is infinite! The interpretation is again subtle. It means that wether we add an infinitesimal amount of energy or subtract an infinitesimal amount the entropy will always decrease. Hence, the system is in a state of maximal entropy and the only way to change this is to perform some amount of work. It is again an example which shows that temperature and energy are really different concepts.

I googled some references, but to be honest, this concept is described in any introductory book on thermodynamics (mine was Schroeder)

* Kittel and Kroemer, Thermal Physics, appendix E.
* N.F. Ramsey, "Thermodynamics and statistical mechanics at negative absolute temperature," Phys. Rev. 103, 20 (1956).
* M.J. Klein,"Negative Absolute Temperature," Phys. Rev. 104, 589 (1956).

stewartcs
I never claimed that a system can reach 0 Kelvin.
Sure you did, in post #4 when you said "[y]es there is". My quote was with reference to absolute temperature scales. Kelvin is an absolute scale, so by saying "yes there is" you are implying that negative Kelvin exists. Now, whether or not you meant that, I do not know. But that's what you wrote. Hence my reply.

On the contrary, in the spin system a negative temperature (or any other for that matter) actually corresponds to a hotter system than a system with a positive temperature. The meaning behind this is that when the systems are in thermal contact there will be a net energy flow from the hotter system to the colder one - just like you would expect. (Please note that most systems are not capable of obtaining a negative temperature)
The entropy of an isolated system during a process always increases, or remains constant if the process is reversible. Since no system is reversible, it always increases, not decreases.

Spin systems are definitely not covered in introductory thermodynamic text books. Nor is the concept of negative absolute temperatures. However, this is beyond the normal macro view of thermodynamics and more in line with condensed matter physics. So, I admit there may be some theoretical description that allows for a "negative temperature"...but that's beyond my area of expertise and definitely beyond the macro view of thermodynamics!

I did find this article that seems to support what you are saying to some degree (however I didn't check the works cited), and appears to be your source for the given references.

http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/neg_temperature.html

CS

Sure you did, in post #4 when you said "[y]es there is". My quote was with reference to absolute temperature scales. Kelvin is an absolute scale, so by saying "yes there is" you are implying that negative Kelvin exists. Now, whether or not you meant that, I do not know. But that's what you wrote. Hence my reply.
To make it absolutely (no pun intended) clear what I meant: yes, negative absolute temperature exists (<0 Kelvin). No, a system cannot reach 0 Kelvin. A point of confusion which mich arise here could be that for a system to go from positive to negative temperature it must "go through" the point of 0 Kelvin. But that's not the case: it reaches a negative temperature by letting the temperature increase!

The entropy of an isolated system during a process always increases, or remains constant if the process is reversible. Since no system is reversible, it always increases, not decreases.
True, but that's why I mentioned that it can only be accomplished by performing some amount of work. The spin system is then no longer isolated, so it entropy can decrease. The complete system (including whatever device you use to perform the work) has ofcourse an increasing entropy.

Spin systems are definitely not covered in introductory thermodynamic text books. Nor is the concept of negative absolute temperatures. However, this is beyond the normal macro view of thermodynamics and more in line with condensed matter physics. So, I admit there may be some theoretical description that allows for a "negative temperature"...but that's beyond my area of expertise and definitely beyond the macro view of thermodynamics!

I did find this article that seems to support what you are saying to some degree (however I didn't check the works cited), and appears to be your source for the given references.

http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/neg_temperature.html

CS
Yea, that's indeed the ref I used. But you can also check wikipedia. It's pretty worthwile to study these concepts, cause it shows the subtleties associated with thermodynamic properties. Also, the exercise of the spin system is pretty funny to work on (I suggest the book by Schroeder: Thermal Physics - his treatment is pretty clear).