What about systems with more than two energy levels?

Themes > Science > Physics > Cryogenics > Temperature > Is a negative Kelvin temperature possible? > What about systems with more than two energy levels?

Usual systems have more than two excited states and hence more than two energy levels. Can these temperatures be attributed to such systems, too?

Looking at the limits of this two level model may help to elucidate further the concept of temperature. First, each level must be populated enough to avoid random fluctuations of particle number. Keep in mind that the systems are dynamic, particles exchanging energy continually. So the model fails when there are only a few particles on each level.

The above given Boltzmann relation holds true for multi level systems, of course. Now we can clarify what it means when it is said that a system must be in thermal equilibrium:

A system is in thermal equilibrium if particle number on each energy level follows the Boltzmann distribution.

Random fluctuations cause deviations from this distribution, but the larger the fluctuation, the less it is probable, so most of the time the system will be close to that distribution.

For systems with an infinite number of states infinite temperature would also mean infinite energy. Now the relation

E_kin = ³/₂kT

shows that, in contradiction to our conclusions above, infinite temperature means infinite energy. In fact, the number of translational states is very large, near infinite. For a system with a Volume of about 1 cm³ there are more translational states than molecules.

Translational states thus cannot be described by the above model. Free particles that store energy in their translational motion cannot have negative Kelvin temperature.

If you put negative temperature into the Boltzmann formula, the function value goes up exponetially. Each energy level must be populated more than its lower neighbour. Of course it is not possible to keep populating levels like this. It follows that in system that have inversely populated states, these states cannot be in thermal equilibrium with all other states of the same system. Negative absolute temperature is not an equilibrium quantity.