So last week, one of my colleagues came up to me and said "What's all this about negative absolute temperature?" She caught me completely off-guard. Even among scientists, it's not a topic that immediately springs to mind, and this is certainly true in my department.
Apparently, there was an article recently published in Science on the topic, and my colleague had been informed of this by an associate of hers, so she came to her local physical chemist for an explanation.
But then she shared this question with students in her advanced class, who are also in my advanced class, and so I had to confront the question again, this time from my students. So, I investigated the Science paper to try to figure out what the deal is.
Details below the orange frosty.
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Understanding what a negative absolute temperature actually means is dependent on understanding what the definition of absolute temperature is in the first place. It turns out that the pertinent definition for understanding negative absolute temperature is not quite the same as the one with which most people are familiar. So we have to make sure when talking about absolute temperature that we're all talking about the same thing.
The origin of the concept of absolute temperature came out of the gas laws, Charles' Law in particular. If you measure the volume of a gas at constant pressure for a number of different temperatures, you observe that the volume increases linearly with the increase in temperature. If you extrapolate from this linear dependence the temperature at which the volume of the gas will be zero, it turns out that, for all gases under ordinary conditions, the temperature is the same: -273.15 degrees Celcius.
This observation served as the basis of a new temperature scale, the Kelvin scale, which assigns the temperature -273.15 degrees Celcius to have a value of zero. It was assumed at that time that zero Kelvin was in fact the absolute zero of temperature, that it was not possible for temperature to go any lower.
In the meantime, during the late 19th and early 20th Centuries, the field of statistical mechanics arose to try to understand the thermal properties of materials in terms of the behavior of the molecules from which those materials are made. An early result was to recognize that absolute temperature was directly proportional to the average kinetic energy (i. e. energy of motion) of the molecules in a substance. From this point of view, absolute zero corresponds to a system where the molecules have no kinetic energy whatsoever. This correspondence bolsters the idea that there can be no temperatures lower than zero Kelvin, since, once all the kinetic energy has been drained from the system, you can't remove any more. Kinetic energy can only have positive values. The idea of negative kinetic energy is absurd; thus, the idea of a negative absolute temperature is also absurd on the basis of this definition.
Despite this powerful and useful picture as a statistical mechanical definition for absolute temperature, it's not the only one, and certainly not the most general one. I will refrain from introducing the thermodynamic definition of temperature for fear of losing readers (though I suspect I've lost more than a few already). Instead, I will approach the more general definition of temperature through the concept of the probability distribution. Such a distribution is a function that relates the value of a statistical variable to the probability that that variable will have that value. For example, if I consider a coin flip, the possible "values" for the statistical variable are "heads" and "tails," while the probabilities associated with each value are 0.5, reflecting the fact that the number of "heads" and "tails" outcomes ought to be equal.
For a system of gas molecules, for example, there is a distribution function for the kinetic energy of each molecule; that is, the function gives the probability that a molecule in the gas will have a particular value for its kinetic energy. Absolute temperature is a parameter in this function; this should not come as a surprise, given that we've seen that absolute temperature is closely linked to kinetic energy. When a system is in equilibrium with its surroundings (i. e. no net heat transfer between the system and its surroundings), what one finds is that there are more molecules (i. e. a high probability) that have low kinetic energies, while fewer (low probability) have high kinetic energies. When the temperature is increased, the number of molecules with higher energy increases, and the number with lower energies decrease, but still, as long as the system is in equilibrium, molecules with lower energy will outnumber molecules with higher energy.
But not all systems are in equilibrium all the time. It is possible to set up a system where more molecules have high energy than low energy. Such a situation is called a population inversion. If you try to apply the distribution function to an inverted population, the only way to make it fit is to reverse the sign of the absolute temperature! It should be clear that in this situation, the negative parameter temperature is not related to the average kinetic energy of the molecules in the system, as it is in the previous definition. After all, there's plenty of energy available in an inverted system. It should also be apparent that a negative temperature does not imply that such a system is colder than a system at absolute zero. In fact, a system with an inverted population and a negative temperature is hotter than one with a positive temperature. If we accept this definition of temperature as a parameter in a distribution, we can come to a better understanding of the meaning of a negative absolute temperature.
It turns out that this concept is not new. I remember reading an article about the concept of negative absolute temperatures in Scientific American more than 30 years ago. However, these inverted states with negative temperatures are not thermodynamically stable because they are not in equilibrium. In common experience, they eventually relax to an equilibrium state with a positive temperature. The novelty in the article by Braun et al published in Science (vol. 339, p. 51, 2013) is in producing a negative temperature state that is thermodynamically stable. How do they do it? Oy. I'll go so far as to say that they use a Bose-Einstein Condensate consisting of potassium-39 atoms. Much of the rest is, frankly, over my head, and this diary is long enough already. Nonetheless, I hope I've managed to expand your understanding on what is meant by the term "temperature."
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