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Invertia

Negative Absolute Temperature / Population Inversion Physics

\[\frac{1}{T} = \frac{\partial S}{\partial E} < 0\]
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Invertia is a Pixelated Physics law character, one of our thermodynamics designs. More particles on the top level than the bottom: negative absolute temperature by the Boltzmann definition, still debated. This is a 3" × 3" pixel-art sticker. The physics panel below gives the equation, what each symbol means and the sources.

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3″ × 3″ kiss-cut sticker.

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The physics

Temperature is defined by how entropy changes when you add energy. In a system with a top energy level, like nuclear spins, putting more particles in the upper state than the lower one makes that slope negative: a negative absolute temperature, which is actually 'hotter than infinity'.

Invertia runs a hotel that, unusually, has a top floor.

Most systems have no ceiling: add energy and more particles climb higher, but the lower floors stay the busiest. That is what positive temperature means. The ratio of upper to lower occupants is the Boltzmann factor on this page, less than one whenever T is positive.

Some systems do have a top floor. A nuclear spin in a magnetic field has only a few allowed energies. In 1951 Edward Purcell and Robert Pound at Harvard took a lithium fluoride crystal whose nuclear spins were lined up in a strong field and reversed the field faster than the spins could follow. Afterwards more spins sat in the upper level than in the lower one: Invertia’s penthouse full, the lobby nearly empty. Read through the Boltzmann factor, that population needs T below zero. The inversion then decayed back, through zero, with the crystal’s five-minute relaxation time. Norman Ramsey worked out the thermodynamics in 1956, and in 2013 a Munich team led by Ulrich Schneider and Immanuel Bloch made the same inversion with the motion of ultracold atoms.

In the standard (Boltzmann) definition of entropy, such a system is hotter than any positive temperature, even an infinite one. In contact with any ordinary system, heat flows out of it. That behaviour is observed and undisputed.

The definition is not. Jörn Dunkel and Stefan Hilbert argued in 2014 that the Gibbs entropy is the right one, and with it temperature never goes negative. Daan Frenkel and Patrick Warren, Robert Swendsen and Jian-Sheng Wang, and others defended Boltzmann. That question is still open, though most follow-ups side with Boltzmann.

Either way, by the textbook thermometer Invertia is always the one giving heat to the neighbours.

Top floor first. Heat only leaves me.

Equations

\[\frac{1}{T} = \frac{\partial S}{\partial E} < 0\]
\[\frac{N_{\uparrow}}{N_{\downarrow}} = e^{-\Delta E/k_B T}\]
\[T < 0 \;\Rightarrow\; N_{\uparrow} > N_{\downarrow}\]
SymbolMeaningUnit
\(T\) absolute temperature, in the standard (Boltzmann) definition \(\mathrm{K}\)
\(S\) entropy (Boltzmann form, from the number of states at energy E) \(\mathrm{J/K}\)
\(E\) total energy of the spin system \(\mathrm{J}\)
\(N_{\uparrow}, N_{\downarrow}\) number of particles in the upper and the lower level \(\mathrm{1}\)
\(\Delta E\) energy gap between the two levels \(\mathrm{J}\)
\(k_B\) Boltzmann constant \(\mathrm{J/K}\)

Sources

  1. Purcell & Pound (1951) A Nuclear Spin System at Negative Temperature, Phys. Rev. 81, 279 (reversed state decayed with the 5-min time constant)
  2. Ramsey (1956) Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures, Phys. Rev. 103, 20
  3. Braun et al. (2013) Negative Absolute Temperature for Motional Degrees of Freedom, Science 339, 52
  4. Dunkel & Hilbert (2014) Consistent thermostatistics forbids negative absolute temperatures, Nature Phys. 10, 67
  5. Hilbert, Hänggi & Dunkel (2014) Thermodynamic laws in isolated systems, Phys. Rev. E 90, 062116
  6. Frenkel & Warren (2015) Gibbs, Boltzmann, and negative temperatures, Am. J. Phys. 83, 163
  7. Swendsen & Wang (2016) Negative temperatures and the definition of entropy, Physica A 453, 24
  8. Abraham & Penrose (2017) Physics of negative absolute temperatures, Phys. Rev. E 95, 012125
  9. Pound (2000) Edward Mills Purcell 1912–1997, Biographical Memoirs of the National Academy of Sciences 78 (inverted LiF state decayed with the several-minute spin-lattice time constant)

Source Purcell, E. M.; Pound, R. V. (1951). A Nuclear Spin System at Negative Temperature. Physical Review 81, 279–280.doi.org/10.1103/PhysRev.81.279

Field notes

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