Quantum Optics Group (LMU) - Quantum Many Body Systems Division (MPQ)

Negative Absolute Temperature

Artistic illustration of five different thermal Boltzmann distributions. The first container on the left shows a gas at a very small positive temperature, close to absolute zero. Most atoms are close to the lowest energy state which is given by the lower energy bound, indicated by the cover at the bottom. The second container also shows a gas at positive temperature, but at a much higher temperature. Some atoms also occupy high energy states. In the center container, the gas is at positive or negative infinite temperature, which are physically the same. All energies are equally likely. In this case, both a lower and an upper energy bound is required. In the fourth container, the gas is at negative temperature, at a large negative value. For negative temperatures, an upper energy bound is required, but not necessarily a lower. The gas in the container on the right is at negative temperature, at a very small negative value. Most atoms are close to the maximum energy.

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Artists impression of thermal distributions: For positive temperatures (blue balls) most atoms occupy low energy states and only a few atoms have high energies. For negative temperatures (red balls), the distribution is inverted and most atoms occupy high energy states. The vertical axis represents energy.

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Same as above, but with both distribution next to each other.

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The same distribution as above in inverse order, here the negative temperature situation is sketched below the positive temperature one - even though it consists of higher energies.

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The thermal (Boltzmann) distribution can be illustrated with balls that are distributed on a hilly landscape, which provides both a lower and upper bound for the potential energy of the balls. At positive temperatures (left figure), as they are common in everyday life, most balls lie in the valley around minimum potential energy. They barely move and therefore also possess minimum kinetic energy. States with small total energy are therefore more likely than those with large total energy – the usual Boltzmann distribution. At infinite temperature (central figure) the balls spread evenly over low and high energies in an identical landscape. Here, all energy states are equally probable. At negative temperatures (right figure), however, most balls wander on top of the hill, at the upper limit of potential energy. Also their kinetic energy is maximal. Energy states with large total energy are occupied more than those with small total energy – the Boltzmann distribution is inverted.

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