# Question on negative temperature

1. Apr 11, 2010

### luxiaolei

Hi all, I was thinking I should post this thread in Statistic Physics, however there isnt, so I put it, not sure wether is ok or not.

Here is my question:

1. what exactly is negative temperature? in my mind is , in a bounded system(number of energy levels are constant), when there are more particles in the higher states than in the lowers states. Am I right?

2. In an unbound system, is there a probability than within that system, forms a ''fake'' bounded system in a very short time period, and in that sub bounded system, negative temperature occurs.

2. Apr 11, 2010

### IttyBittyBit

What is temperature? It is the change of thermal energy divided by the change of entropy. If you have a system where it is possible to add thermal energy and have entropy decrease, you have negative temperature. It's that simple. An example of such a system would be the spin states of a finite set of atoms under a magnetic field.

Negative temperature doesn't mean below absolute zero; that is impossible. If you put two similar setups, one with negative temperature and the other with positive temperature in contact, heat will flow from the positive one to the negative one, not the other way around.

To answer (1), yes that is roughly what is happening, but that is just a specific case of a general phenomenon.

To answer (2), you must realize that temperature is defined as an aggregate property of a system. As soon as you start analyzing the 'temperature' of microscopic elements within the system you lose the meaning of temperature.

3. Apr 11, 2010

### luxiaolei

@IttyBittyBit, Great answer! Big Thanks! :)

4. Apr 11, 2010

### xepma

Just to add some clarity to the picture: it's very common to work with the inverse temperature, $$\beta = 1/(k_B T)$$ with $$k_B$$ the Boltzmann constant. Then the zero temperature case corresponds to the (unreachable) limit $$\beta \limit \infty$$. The crossover from positive to negative temperature occurs at $$\beta = 0$$ and is smooth with respect to beta -- the temperature on the other hand has some weird flip when you cross over from positive to negative temperature, although it's a smooth transition (smooth as in smooth with respect to the thermodynamic quantities).

As for question two, I do not really agree with IttyBittyBit on this (unless I misunderstood what he ment).

Temperature can also be introduced in a subsystem of a macroscopic system, as long as this subsystem isn't microscopic. What I mean by that (or,well, what Landau and Lifgarbagez mean by that, since that's my source) is that it's perfectly valid to consider a subsystem of an isolated, but very large system. Although the system as a whole has a fixed energy, the subsystem does not -- energy of the subsystem is constantly hopping with the rest of the system. The subsystem is therefore described by considering it as some system in contact with a heat bath. The rest of the system palys the role of the heat bath, even though the total energy is fixed.

So it is very well possible to talk about the temperature of a subsystem.

As for your question: you have to remember that temperature is introduced after some form of time averaging of the system. The system is constantly fluctuating, and hops from state to state. But the temperature arises as an (implicit) averaging over these fluctuations. So although the systems hops from state to state, it's temperature does not fluctuate.

5. Apr 11, 2010

### SpectraCat

I think you must mean the reverse of this, at least in general. Otherwise there would be no way to return the temperature of a system to positive values, once it had become negative, and you would have a second law violation. Negative temperatures are weird, but not *that* weird. The way I learned to think of it is that systems with negative temperatures have a thermal population inversion (!!) and are thus "hotter than than ought to be allowed". Thus heat will tend to flow out of a system with a negative temperature, until it becomes positive again.

It is important to note in this context that negative temperature systems are always "artificial", in that they require input of a lot of energy under fairly specific conditions to bring about.

6. Apr 11, 2010

### genneth

The other thing which bears remembering is that thermodynamics is a theory of equilibrium states. Temperature, entropy, etc. are undefined when the system has yet to reach equilibrium. In practise, we can relax the requirements somewhat, and talk about quasi-equilibrium, i.e. when the time needed to reach equilibrium is much shorter than other relevant scales. It is possible to go even further, but the concepts stop being quite so universal.

7. Apr 11, 2010

### IttyBittyBit

Ooops! I shouldn't be writing at 2:00 AM!
Yes, I meant that energy should flow from the negative one to the positive one.

8. Apr 15, 2010

### Dickfore

If there is a negative temperature, I can make a Carnot engine with efficiency greater than 1.