Negative-temperature Systems

Richard S. Starling

Measuring temperature is an essential part of science, no matter the discipline. Many reactions have to be conducted at certain temperatures in order for the molecules to react appropriately. Such key reactions include polymerase chain reactions, which are used to copy strands of DNA, and nuclear reactions around the sun and other stars (Markham, 1993). Recently, a popular thermodynamic application is to create negative temperature. Negative temperature doesn’t sound like an unnatural thing; many areas of the world are consistently below 0oC, but this temperature isn’t absolute temperature. Absolute temperature refers to the Kelvin temperature scale, which is 273.15 units higher than the Celsius scale. When dealing with absolute temperatures on the Kelvin scale, it is unlawful by modern thermodynamics for negative temperatures to exist (Schneider, 2013).

A textbook definition of temperature is the measure of the average kinetic energy in a closed system. For ideal gases and gases that behave ideally (real gases under normal or high temperature and normal to low pressure), the temperature is proportional to average kinetic energy, but not necessarily equivalent to it (Shiga, 2010). In supercold gases/atoms, the temperature is not proportional to the average kinetic energy, however, so it has to be defined another way (Shiga, 2010).

Formally, temperature is actually dependent on and determined by the organization of a system, and the disorder (inverse of organization) of a system is referred to as entropy. The equation is simple:

1/T = dS/dE

where T is temperature, S is entropy and dS/dE is the change in entropy with respect to energy. More specifically, temperature is dependent on how added energy affects the entropy of a system. If added energy (+dE) increases the entropy (+dS), which is normally the case, then the temperature increases (Shiga, 2010). To put temperature into a macro-scale, one can think about a classroom full of kindergarten-aged children, with each child representing an atom or particle. If energy is added to the children, say they eat all the leftover candy from Halloween, they will run around the classroom, making it more disorderly. If energy is taken away from these children by putting them to sleep, the classroom will be less disorderly. This example may seem simple and intuitive, but it mirrors what happens in a real gas (and just about everything else) under ideal conditions.

Introduction to Measuring Negative Temperature

Negative-temperature systems deal with a specific set of gases that are composed of supercold or ultracold atoms (Rey, 2009). When calculating the temperature of these systems, it is important to understand the formal definition of temperature (as explained above). The effect of added energy to a negative-temperature system actually makes the system more orderly, thus decreases S. It is difficult to think that when a substance is heated up, the particles slow down and become less disorderly, but in negative temperature systems, there is no lower bound on temperature (Rapp et al., 2010). In normal systems, there is a lower bound on temperature (T=0) because thermodynamics prevents otherwise. In negative-temperature systems, since there isn’t a lower bound, there must be an upper bound, which there is (Rapp et al., 2010). It’s unclear if scientists have concretely found the upper bound to negative-temperature systems, but it does theoretically exist.

Measuring temperature usually isn’t difficult. Traditionally, scientists use a glass thermometer filled with an appropriate substance to measure systems. In extremely cold systems, however, scientists can’t just simply use a thermometer (the glass would shatter). Instead they essentially take high-resolution pictures of a system and use a Gaussian distribution to determine the temperature (Rey, 2009). This technique was used up until 2009 and could accurately measure temperatures as low as a few nano-Kelvin but couldn’t be applied to systems such as optical lattices (a form of negative temperature systems) because the diluteness of the system is too difficult to determine (Rey, 2009). A team at the MIT-Harvard center for ultracold atoms has recently developed a method using spin gradients and magnetic fields to measure temperatures as low as 50 pico-Kelvin. It can also work with negative-temperature systems by switching the magnetic fields of a system (Rey, 2009). In a normal system, most of the particles have about the same kinetic energy with a few random particles having a higher kinetic energy. In a negative-temperature system, most of the atoms settle in a higher energy state with a few having significantly lower kinetic energies. Ulrich Schneider, a physicist at the Ludwig Maximilian University in Munich, describes the situation in macro-world terms. “[the switching of magnetic fields] suddenly shifts the atoms from their most stable, lowest-energy state to the highest possible energy sate, before they can react. It’s like walking through a valley, then instantly finding yourself on the mountain peak” (Merali, 2013).

Mathematically Speaking

Working through and understanding the very basic mathematics of negative-temperature systems is fairly easy because most of it is conceptualizing the idea of a negative magnetic field, throwing off negative signs here and there. If the formal definition of temperature is used,

1/T = dS/dE


(dS/dE) T = 1

As mentioned above, added energy (+dE) usually increases the entropy (+dS) and energy taken away (-dE) usually decreases the entropy (-dS). It’s easy to see why a normal system can’t have negative temperatures because if  dE is positive, then  dS is positive, and if dE  is negative, then dS is negative, and when they are divided, they always give a positive result. Using negative-temperature systems, however, added energy decreases the entropy, so there is a positive divided by a negative, making negative temperatures mathematically very possible.


In a normal system, there is a lower bound of energy (E=0), but in negative-temperature systems, since there isn’t a lower bound, Braun et al (2013) has found that there must be an upper bound to the system. The Boltzman factor is used to determine kinetic energy in a system, and in the case of negative-temperature systems, the Boltzman factor is negative. The distribution of temperatures needs to be normalizable (because a Gaussian distribution has to be performed on it), and since there is no lower bound, there must be an upper bound (Braun et al., 2013), which reinforces Rapp et al’s assertion of an upper bound. The real-life applications of these negative-temperature systems are unclear, but some theoretical uses do come out of the nature of the system. A Carnot engine is the basic, hypothetical engine that is used to determine efficiency. It is modelled by:

n = 1 – (Tc/Th)

where n is the efficiency, Tc is the temperature of the cool air and Th is the temperature of the hot air. In a normal system, since temperature cannot be negative, the efficiency of the engine is bound (n<1). One unintuitive application of negative-temperature systems is making a Carnot engine with efficiency greater than 1 (n>1) (Braun et al, 2013). If it were the case, scientists could essentially create energy. It is unclear (and untested) whether or not this application is even possible, but if it were, it would certainly be ground-breaking.


According to Dunkel & Hilbert (2014), all the claims about negative-temperature systems are invalid. Thinking about negative absolute temperatures is hard to conceptualize, and because scientists have been using the Boltzman equation to determine the temperatures (which according to Dunkel & Hilbert cannot be used for negative kinetic energy), the claims are invalid. In order to prove the impossibility of negative temperatures, the Gibbs entropy equation, which essentially forbids negative temperature, has to be taken into consideration (Dunkel & Hilbert, 2014). The rumors about Carnot engines with efficiencies greater than 1 is impossible because temperatures can’t be negative, which makes sense because an engine with more than 100% efficiency would produce more energy than absorbed, defying the first law of thermodynamics (Dunkel & Hilbert, 2014). The claims by Dunkel & Hilbert seem a little bit biased, but they do make sense.

Overall, negative temperature systems are completely hypothetical and don’t have any true real-life applications right now. It seems that under certain conditions, they may be able to be used, but until then, it is probably best to remain skeptical about the subject. Like many physical anomalies, such as the higgs boson, the intuition and math supports it, but the actual experimental evidence is still lacking. This lack of evidence puts a grey cloud on the matter; that is, scientists don’t know what the future of negative-temperature systems looks like right now, but the ambiguity leads to many differing opinions and theories, which is how very important topics in physics get resolved.


Works Cited


Braun, S., Ronzheiimer, P. J., Schreiber, M., Hodgman, S. S., Rom, T., Bloch, I., & Schneider, U. (2013). Negative absolute temperature for motional degrees of freedom. Science, doi: 10.1126/science.1227831.

Dunkel, J. & Hilbert, S. (2014). Consistent thermodynamics forbids negative absolute temperatures. Nature Physics, doi: 10.1038/nphys2815.

Markham, A. F. (1993). The polymerase chain reaction: a tool for molecular medicine. British Medical Journal, 306, 441-446.

Merali, Z. (2013). Quantum gas goes below absolute zero. Nature, doi: 10.1038/nature.2013.12146.

Rapp, A., Mandt, S., & Rosch, A. (2010). Equilibrium rates and negative absolute temperatures for ultracold atoms in optical lattices. Physical Review Letters, doi: 10.1103/PhysRevLett.105.220405.

Rey, M. A. (2009). Viewpoint: the super cool atom thermometer. Physics, doi: 10.1103/Physics.2.103.

Schneider, U. (2013). A temperature below absolute zero. Max-Planck-Gesellschaft.

Shiga, D. (2010). How to create temperatures below absolute zero. New Scientist, 2789, 15.