Over the past 60 years, several theories and experiments have claimed the existence of negative absolute temperatures in certain quantum systems. This fuelled speculations about hyper-efficient heat engines that could act as perpetual motion machines, and also about ultracold atom gases as models for the mysterious Dark Energy in our Universe. Such claims led Jörn Dunkel at MIT (USA) and Stefan Hilbert at MPA (Germany) to re-evaluate the underlying thermodynamical formalism.
Their analysis shows that the widely adopted Boltzmann formalism is inconsistent, whereas the approach proposed 100 years ago by Gibbs remains consistent. In the Gibbs framework, none of the previous experiments provide evidence for negative absolute temperatures, which also implies that hyper-efficient engines remain out of reach, and that cold atom gases are less likely to mimic Dark Energy.
Temperature quantifies our perception of 'hot' and 'cold'. Temperature is also fundamental for predicting the efficiencies of machines that convert heat into usable work. For decades, physics students have learned that temperature is always positive when measured on the Kelvin scale. An important consequence of this assumption is that the efficiency of a heat engine is always smaller than one, i.e. only a fraction of the energy put in as heat, e.g. by burning fuel in the engine of a car, can be harnessed to perform useful work, like propelling a car.
However, there have been both theoretical and experimental claims over the past 60 years that there are certain systems with a negative absolute temperature. Even though these are very special systems -- nuclear spin systems or ultracold atom gases -- this would have profound conceptual and practical consequences. Such systems might not only facilitate the construction of hyper-efficient heat engines. They might also serve as a laboratory model for the mysterious Dark Energy, which was postulated by astrophysicists to explain the accelerated expansion of the Universe. "We actually have no idea what Dark Energy is on a very fundamental level," says Stefan Hilbert from MPA. "So we wanted to find out if these results would indeed shed light on Dark Energy." This, however, meant going back to the basics of thermodynamics.
Most textbooks advocate the formalism introduced by Ludwig Boltzmann to relate the thermodynamic temperature of a system to its internal structure. For many systems, this formalism works just fine. However, "when we examined Boltzmann's definitions in detail, we found serious inconsistencies that lead to nonsense results for many systems," says Stefan Hilbert. Dunkel and Hilbert found that these inconsistencies can be avoided by using a slightly different formalism that was derived by Gibbs already more than 100 years ago, but has mostly been forgotten since then.
A feature of Gibbs' formalism is that temperature never becomes negative. As Dunkel and Hilbert show, the number determined in recent experiments claiming negative temperatures in ultracold atom gases is not the actual thermodynamic temperature, but rather a complex function of temperature and another quantity, known as heat capacity. The thermodynamic temperature in fact remained positive in these experiments, which makes it less likely that these systems behave like Dark Energy.
"In most cases, the difference between the Boltzmann temperature and the Gibbs temperature is negligible," explains Stefan Hilbert. "But in extreme physical conditions, as is the case for these systems with allegedly negative temperature, only Gibbs provides the correct description." To directly test this, Dunkel and Hilbert propose a straight-forward experiment: If there is a single atom in a trap that allows the atom to move only in one direction, then the pressure should be negative at both ends if the Boltzmann description is correct, while the pressure should be positive for the Gibbs case.
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