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I'm reading this article "Thermodynamics and Statistical Mechanics at Negative Absolute Temperature" by Norman F. Ramsey, and I've got problems understanding some part of it:

The essential requirements for a thermodynamical system to be capable of negative temperature are: (1) The elements of the thermodynamical system must be in thermodynamical equilibrium among themselves in order that the system can be described by a temperature at all; (2) there must be an upper limit to the possible energy of the allowed states of the system; and (3) the system must be thermally isolated from all systems which do not satisfy both of the above conditions, i.e., the thermal equilibrium time among the elements of the system must be short compared to the time during which appreciable energy is lost or gained from other systems. The temperature concept is applicable to the system only for time intervals far from either of the above limits.

I understand conditions (1) and (2). I also understand that the thermal equilibrium time among the elements of the system must be short compared to the time during which appreciable energy is lost or gained from other systems, so that there's some time during which we can attribute a temperature to the system. And I also understand that the temperature concept is applicable to the system only for time intervals far from either of the above limits, which means some time interval intermediate between the time the elements of the system reach to thermal equilibrium among themselves, and the time some energy flows to or from the system. But what I don't understand is how this can be related to condition (3). Why being in contact with systems not satisfying conditions (1) and (2) prevents the system from having negative temperature?

edit: The more I think of condition 3, the more it seems trivial to me. Doesn't even worth noting.

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Assume you connect a system with a negative temperature with a system as described by (3). The latter will heat up and have more accessible states when you transfer energy to it. The former will have more accessible states when you transfer energy from it (as it has a negative temperature). Hence, the total number of accessible states (the product of the accessible states of the two parts) must also increase when energy is transferred from the first to the second system.

It might also be said that (A) the second system cannot have a negative temperature, and (B) two systems can only be in equilibrium if they have the same temperature, from which your statement follows. But this is actually circular logic. The circle is that you need to prove (B) first, and part of that proof is the statement in your question.

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  • $\begingroup$ I think I don't get the circular logic. If $S_{1}$ have negative temperature (assuming only con. 1 and 2 are required) and is in contact with $S_{2}$ which doesn't satisfy 1 or 2, then $S_{1}$ can neither be described by a temperature at all, or have a negative temperature; thus any system in contact with it must satisfy both conditions if the two systems are to be described by a negative equilibrium temperature, unless there is a time interval when $S_{1}$ is in equilibrium with itself but no appreciable energy is lost or gained from other systems. $\endgroup$ – Simorq Dec 21 '16 at 8:02

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