Editing Negative temperature

{{Short description|Physical systems hotter than any other}}
{{Use American English|date = February 2019}}
[[File:ColdnessScale.svg|right|thumb|250px|[[International System of Units|SI]] temperature/coldness conversion scale: Temperatures on the Kelvin scale are shown in blue (Celsius scale in green, Fahrenheit scale in red), coldness values in gigabyte per nanojoule are shown in black. Infinite temperature (coldness zero) is shown at the top of the diagram; positive values of coldness/temperature are on the right-hand side, negative values on the left-hand side.]]

Certain [[system (thermodynamics)|systems]] can achieve '''negative thermodynamic temperature'''; that is, their [[Thermodynamic temperature|temperature]] can be expressed as a [[negative number|negative]] quantity on the [[Kelvin]] or [[Rankine scale|Rankine]] scales. This should be distinguished from temperatures expressed as negative numbers on non-[[thermodynamic]] [[Celsius scale|Celsius]] or [[Fahrenheit scale]]s, which are nevertheless higher than [[absolute zero]]. A system with a truly negative temperature on the Kelvin scale is ''hotter'' than any system with a positive temperature.  If a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system.<ref>{{cite journal|date=1956-07-01 |title=Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures |journal=Physical Review |volume=103 |issue=1 |pages=20–28 |doi=10.1103/PhysRev.103.20|bibcode = 1956PhRv..103...20R |author-link1=Norman Foster Ramsey, Jr.|last1=Ramsey|first1=Norman }}</ref><ref>{{cite journal|date=1975-11-18 |title=Comment on: Negative Kelvin temperatures: some anomalies and a speculation |journal=American Journal of Physics |volume=44 |pages=994–995 |url=http://www.physique.usherbrooke.ca/~tremblay/articles/Comment%20on%20%27Negative%20Kelvin%20temperatures,%20Some%20anomalies%20and%20a%20speculation%27.%20Tremblay.pdf|doi=10.1119/1.10248|issue=10|bibcode = 1976AmJPh..44..994T|last1=Tremblay|first1=André-Marie }}</ref> A standard example of such a system is [[population inversion]] in [[laser physics]].

Thermodynamic systems with unbounded [[phase space]] cannot achieve negative temperatures: adding [[heat]] always increases their [[entropy]]. The possibility of a decrease in entropy as energy increases requires the system to "saturate" in entropy. This is only possible if the number of high energy states is limited. For a system of ordinary (quantum or classical) particles such as atoms or dust, the number of high energy states is unlimited (particle momenta can in principle be increased indefinitely). Some systems, however (see the [[#Examples|examples]] below), have a maximum amount of energy that they can hold, and as they approach that maximum energy their entropy actually begins to decrease.<ref>{{cite book|url=https://books.google.com/books?id=-7aP0f5O2CMC|pages= 89–95|title=The Laws of Thermodynamics: A Very Short Introduction|isbn=978-0-19-957219-9|last1=Atkins|first1=Peter W.|author-link1=Peter Atkins|date=2010-03-25|publisher= Oxford University Press|oclc=467748903}}</ref>

== History ==

The possibility of negative temperatures was first predicted by [[Lars Onsager]] in 1949.<ref name = Ons1949>{{cite journal |last=Onsager |first=L. |year=1949 |title= Statistical Hydrodynamics |journal=Il Nuovo Cimento |volume= 6|issue=  2|pages=279–287 |doi=10.1007/BF02780991 |issn=1827-6121 |bibcode=1949NCim....6S.279O |s2cid=186224016 }}</ref>
Onsager was investigating [[Two-dimensional point vortex gas|2D vortices]] confined within a finite area, and realized that since their positions are not independent [[degrees of freedom]] from their momenta, the resulting [[phase space]] must also be bounded by the finite area. Bounded phase space is the essential property that allows for negative temperatures, and can occur in both classical and quantum systems. As shown by Onsager, a system with bounded phase space necessarily has a peak in the entropy as energy is increased. For energies exceeding the value where the peak occurs, the entropy ''decreases'' as energy increases, and high-energy states necessarily have negative Boltzmann temperature.

The limited range of states accessible to a system with negative temperature means that negative temperature is associated with emergent ordering of the system at high energies. For example in Onsager's point-vortex analysis negative temperature is associated with the emergence of large-scale clusters of vortices.<ref name=Ons1949 /> This spontaneous ordering in equilibrium statistical mechanics goes against common physical intuition that increased energy leads to increased disorder.

It seems negative temperatures were first found experimentally in 1951, when Purcell and Pound observed evidence for them in the nuclear spins of a [[lithium fluoride]] crystal placed in a magnetic field, and then removed from this field.<ref name = PuPo>{{cite journal |last1 = Purcell |first1 = E. M. |last2 = Pound|first2 = R. V. |year = 1951|title = A nuclear spin system at negative temperature|journal = Physical Review |volume = 81 | issue = 2|pages =279–280 |doi = 10.1103/PhysRev.81.279 |bibcode = 1951PhRv...81..279P }}</ref>  They wrote:

: A system in a negative temperature state is not cold, but very hot, giving up energy to any system at positive temperature put into contact with it. It decays to a normal state through infinite temperature.

== Definition of temperature ==
The absolute [[temperature]] (Kelvin) scale can be loosely interpreted as the average kinetic energy of the system's particles. The existence of negative temperature, let alone negative temperature representing "hotter" systems than positive temperature, would seem paradoxical in this interpretation. The paradox is resolved by considering the more rigorous definition of [[thermodynamic temperature]] in terms of [[Boltzmann's entropy formula]]. This reveals the tradeoff between [[internal energy]] and [[entropy]] contained in the system, with "[[thermodynamic beta|coldness]]", the ''reciprocal'' of temperature, being the more fundamental quantity. Systems with a positive temperature will increase in entropy as one adds energy to the system, while systems with a negative temperature will decrease in entropy as one adds energy to the system.<ref>{{cite book|url=https://books.google.com/books?id=-7aP0f5O2CMC|pages= 10–14|title=The Laws of Thermodynamics: A Very Short Introduction|isbn=978-0-19-957219-9|last1=Atkins|first1=Peter W.|author-link1=Peter Atkins|date=2010-03-25|publisher= Oxford University Press|oclc=467748903}}</ref>

The definition of [[thermodynamic temperature]] {{mvar|T}} is a function of the change in the system's [[entropy]]  {{mvar|S}}  under reversible [[heat transfer]] {{math|''Q''<sub>rev</sub>}}:
: <math>T = \frac{dQ_\mathrm{rev}}{dS}.</math>
Entropy being a [[state function]], the integral of {{mvar|dS}} over any cyclical process is zero. For a system in which the entropy is purely a function of the system's energy {{mvar|E}}, the temperature can be defined as:
: <math> T  = \left( \frac{dS}{dE} \right)^{-1}.</math>

Equivalently, [[thermodynamic beta]], or "coldness", is defined as 
:<math>\beta = \frac{1}{kT} = \frac{1}{k} \frac{dS}{dE},</math>
where {{mvar|k}} is the [[Boltzmann constant]].

Note that in classical thermodynamics, {{mvar|S}} is defined in terms of temperature. This is reversed here, {{mvar|S}} is the [[statistical entropy]], a function of the possible microstates of the system, and temperature conveys information on the distribution of energy levels among the possible microstates. For systems with many degrees of freedom, the statistical and thermodynamic definitions of entropy are generally consistent with each other.

Some theorists have proposed using an alternative definition of entropy as a way to resolve perceived inconsistencies between statistical and thermodynamic entropy for small systems and systems where the number of states decreases with energy, and the temperatures derived from these entropies are different.<ref name="DunkelHilbert">{{cite journal|last1 = Dunkel|first1 = Jorn|last2 = Hilbert|first2 = Stefan|title = Consistent thermostatistics forbids negative absolute temperatures|journal = Nature Physics|volume = 10|issue = 1|pages = 67|doi = 10.1038/nphys2815|arxiv = 1304.2066|bibcode = 2014NatPh..10...67D|year = 2013|s2cid = 16757018}}</ref><ref name="HanggiHilbertDunkel">{{cite journal|last1 = Hanggi|first1 = Peter|last2 = Hilbert|first2 = Stefan|last3 = Dunkel|first3 = Jorn|title = Meaning of temperature in different thermostatistical ensembles|journal = Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume = 374|issue = 2064|pages = 20150039|doi = 10.1098/rsta.2015.0039|arxiv = 1507.05713|year = 2016| pmid=26903095 | bibcode=2016RSPTA.37450039H | s2cid=39161351 }}</ref> It has been argued that the new definition would create other inconsistencies;<ref>{{Cite journal|title = Gibbs, Boltzmann, and negative temperatures|journal = American Journal of Physics|date = 2015-02-01|issn = 0002-9505|pages = 163–170|volume = 83|issue = 2|doi = 10.1119/1.4895828|first1 = Daan|last1 = Frenkel|first2 = Patrick B.|last2 = Warren|arxiv = 1403.4299 |bibcode = 2015AmJPh..83..163F |s2cid = 119179342}}</ref> its proponents have argued that this is only apparent.<ref name="HanggiHilbertDunkel"/>

== Heat and molecular energy distribution ==
[[File:NegativeTemperature.webm|300px|thumb|right|When the temperature is negative, higher energy states are more likely to be occupied than low energy ones.]]Negative temperatures can only exist in a system where there are a limited number of energy states (see below). As the temperature is increased on such a system, particles move into higher and higher energy states, so that the number of particles in the lower energy states and in the higher energy states approaches equality.<ref>{{Cite book |last1=Kubo |first1=Ryōgo |title=Statistical mechanics: an advanced course with problems and solutions |last2=Ichimura |first2=Hiroshi |date=1981 |publisher=North-Holland |isbn=978-0-7204-0090-8 |edition=6. print |location=Amsterdam}}</ref>  This is a consequence of the definition of temperature in [[statistical mechanics]] for systems with limited states. By injecting energy into these systems in the right fashion, it is possible to create a system in which there are more particles in the higher energy states than in the lower ones. The system can then be characterized as having a negative temperature.

A substance with a negative temperature is not colder than [[absolute zero]], but rather it is hotter than infinite temperature.<ref name=":0" /> As Kittel and Kroemer put it,<ref>{{cite book |last1=Kittel |first1=C. |author1-link=Charles Kittel |title=Thermal Physics |last2=Kroemer |first2=H. |author2-link=Herbert Kroemer |publisher=[[W. H. Freeman]] |year=1980 |isbn=978-0-7167-1088-2 |edition=2nd |pages=462}}</ref>

{{blockquote|The temperature scale from cold to hot runs +0&thinsp;K,…, +300&thinsp;K,…,+∞&thinsp;K, −∞&thinsp;K,…, −300&thinsp;K,…, −0&thinsp;K. Note that if a system at −300&thinsp;K is brought into thermal contact with an identical system at 300&thinsp;K, the final equilibrium temperature is not 0&thinsp;K, but is ±∞&thinsp;K.
}}

The corresponding inverse temperature scale, for the quantity {{math|1=''β'' = {{sfrac|1|''kT''}}}} (where {{mvar|k}} is the [[Boltzmann constant]]), runs continuously from low energy to high as +∞, …, 0, …, −∞. Because it avoids the abrupt jump from +∞ to −∞, {{mvar|β}} is considered more natural than {{mvar|T}}. Although a system can have multiple negative temperature regions and thus have −∞ to +∞ discontinuities.

In many familiar physical systems, temperature is associated to the [[kinetic energy]] of atoms.<ref>{{Cite journal |last=Nettleton |first=R. E. |date=March 1994 |title=On the relation between thermodynamic temperature and kinetic energy per particle |url=https://cdnsciencepub.com/doi/10.1139/p94-017 |journal=[[Canadian Journal of Physics]] |volume=72 |issue=3–4 |pages=106–112 |doi=10.1139/p94-017 |bibcode=1994CaJPh..72..106N |issn=0008-4204|url-access=subscription }}</ref> Since there is no upper bound on the momentum of an atom, there is no upper bound to the number of energy states available when more energy is added, and therefore no way to get to a negative temperature.<ref name=":0" /> However, in statistical mechanics, temperature can correspond to other degrees of freedom than just kinetic energy (see below).

== Temperature and disorder ==
{{Unreferenced section|date=July 2024}}
The distribution of energy among the various [[translation (physics)|translational]], [[oscillation|vibrational]], [[rotation]]al, [[Electron configuration|electronic]], and [[atomic nucleus|nuclear]] modes of a system determines the macroscopic temperature. In a "normal" system, thermal energy is constantly being exchanged between the various modes.

However, in some situations, it is possible to isolate one or more of the modes. In practice, the isolated modes still exchange energy with the other modes, but the [[Speed|time scale]] of this exchange is much slower than for the exchanges within the isolated mode. One example is the case of [[atomic nucleus|nuclear]] [[Spin (physics)|spins]] in a strong external [[magnetic field]]. In this case, energy flows fairly rapidly among the spin states of interacting atoms, but energy transfer between the nuclear spins and other modes is relatively slow. Since the energy flow is predominantly within the spin system, it makes sense to think of a spin temperature that is distinct from the temperature associated to other modes.

A definition of [[Temperature#Second law of thermodynamics|temperature]] can be based on the relationship:

:<math>T = \frac{dq_\mathrm{rev}}{dS}</math>

The relationship suggests that a ''positive temperature'' corresponds to the condition where [[entropy]], {{mvar|S}}, increases as thermal energy, {{math|''q''<sub>rev</sub>}}, is added to the system. This is the "normal" condition in the macroscopic world, and is always the case for the translational, vibrational, rotational, and non-spin-related electronic and nuclear modes. The reason for this is that there are an [[Infinity|infinite]] number of these types of modes, and adding more heat to the system increases the number of modes that are energetically accessible, and thus increases the entropy.

== Examples ==

=== Noninteracting two-level particles ===
{{multiple image
   | align     = right
   | direction = vertical
   | footer    = Entropy, thermodynamic beta, and temperature as a function of the energy for a system of {{mvar|N}} noninteracting two-level particles.
   | width1    = 235
   | image1    = Entropy_vs_E_two_state.svg
   | width2    = 235
   | image2    = Beta_vs_E_two_state.svg
   | width3    = 235
   | image3    = Temperature_vs_E_two_state.svg
  }}{{Unreferenced section|date=July 2024}}
The simplest example, albeit a rather nonphysical one, is to consider a system of {{mvar|N}} particles, each of which can take an energy of either {{math|+''ε''}} or {{math|−''ε''}} but are otherwise noninteracting.  This can be understood as a limit of the [[Ising model]] in which the interaction term becomes negligible.  The total energy of the system is

:<math>E = \varepsilon\sum_{i=1}^N \sigma_i = \varepsilon j</math>

where {{mvar|σ<sub>i</sub>}} is the sign of the {{mvar|i}}th particle and {{mvar|j}} is the number of particles with positive energy minus the number of particles with [[negative energy]].  From elementary [[combinatorics]], the total number of [[microstate (statistical mechanics)|microstates]] with this amount of energy is a [[binomial coefficient]]:

:<math>\Omega_E = \binom{N}\frac{N + j}{2} = \frac{N!}{\left(\frac{N + j}{2}\right)! \left(\frac{N - j}{2}\right)!}.</math>

By the [[fundamental assumption of statistical mechanics]], the entropy of this [[microcanonical ensemble]] is

:<math>S = k_\text{B} \ln \Omega_E</math>

We can solve for thermodynamic beta ({{math|1=''β'' = {{sfrac|1|''k''<sub>B</sub>''T''}}}}) by considering it as a [[central difference]] without taking the [[continuum limit]]:

:<math>\begin{align}
  \beta &= \frac{1}{k_\mathrm{B}} \frac{\delta_{2\varepsilon}[S]}{2\varepsilon}\\[3pt]
        &= \frac{1}{2\varepsilon} \left( \ln \Omega_{E+\varepsilon} - \ln \Omega_{E-\varepsilon} \right)\\[3pt]
        &= \frac{1}{2\varepsilon} \ln \left( \frac{\left(\frac{N + j - 1}{2}\right)! \left(\frac{N - j + 1}{2}\right)!}{\left(\frac{N + j + 1}{2}\right)! \left(\frac{N - j - 1}{2}\right)!} \right)\\[3pt]
        &= \frac{1}{2\varepsilon} \ln \left( \frac{N - j + 1}{N + j + 1} \right).
\end{align}</math>

hence the temperature

:<math>T(E) = \frac{2\varepsilon}{k_\text{B}}\left[\ln \left( \frac{(N + 1)\varepsilon - E}{(N + 1)\varepsilon + E} \right)\right]^{-1}.</math>

This entire proof assumes the microcanonical ensemble with energy fixed and temperature being the emergent property.  In the [[canonical ensemble]], the temperature is fixed and energy is the emergent property.  This leads to ({{mvar|ε}} refers to microstates):
:<math>\begin{align}
  Z(T) &= \sum_{i=1}^N e^{-\varepsilon_i\beta}\\[6pt]
  E(T) &= \frac{1}{Z}\sum_{i=1}^N \varepsilon_i e^{-\varepsilon_i\beta}\\[6pt]
  S(T) &= k_\text{B}\ln(Z) + \frac{E}{T}
\end{align}</math>

Following the previous example, we choose a state with two levels and two particles.  This leads to microstates {{math|1=''ε''<sub>1</sub> = 0}}, {{math|1=''ε''<sub>2</sub> = 1}}, {{math|1=''ε''<sub>3</sub>  = 1}}, and {{math|1=''ε''<sub>4</sub> = 2}}.
:<math>\begin{align}
  Z(T) &= e^{-0\beta} + 2e^{-1\beta} + e^{-2\beta}\\[3pt]
       &= 1 + 2e^{-\beta} + e^{-2\beta}\\[6pt]
  E(T) &= \frac{0e^{-0\beta} + 2 \times 1e^{-1\beta} + 2e^{-2\beta}}{Z}\\[3pt]
       &= \frac{2e^{-\beta} + 2e^{-2\beta}}{Z}\\[3pt]
       &= \frac{2e^{-\beta} + 2e^{-2\beta}}{1 + 2e^{-\beta} + e^{-2\beta}}\\[6pt]
  S(T) &= k_\text{B}\ln\left(1 + 2e^{-\beta} + e^{-2\beta}\right) + \frac{2e^{-\beta} + 2e^{-2\beta}}{\left(1 + 2e^{-\beta} + e^{-2\beta}\right)T}
\end{align}</math>

The resulting values for {{mvar|S}}, {{mvar|E}}, and {{mvar|Z}} all increase with {{mvar|T}} and never need to enter a negative temperature regime.

=== Nuclear spins ===
The previous example is approximately realized by a system of nuclear spins in an external magnetic field.<ref name="PuPo" /><ref>{{cite journal|doi=10.1103/PhysRevE.57.6487|title=Minimax games, spin glasses, and the polynomial-time hierarchy of complexity classes|year=1998|last1=Varga|first1=Peter|journal=Physical Review E|volume=57|issue=6|pages=6487–6492|arxiv=cond-mat/9604030|bibcode = 1998PhRvE..57.6487V |citeseerx=10.1.1.306.470|s2cid=10964509}}</ref> This allows the experiment to be run as a variation of [[nuclear magnetic resonance spectroscopy]]. In the case of electronic and nuclear spin systems, there are only a finite number of modes available, often just two, corresponding to [[Spin (physics)#Properties of spin|spin up and spin down]]. In the absence of a [[magnetic field]], these spin states are ''degenerate'', meaning that they correspond to the same energy. When an external magnetic field is applied, the energy levels are split, since those spin states that are aligned with the magnetic field will have a different energy from those that are anti-parallel to it.

In the absence of a magnetic field, such a two-spin system would have maximum entropy when half the atoms are in the spin-up state and half  are in the spin-down state, and so one would expect to find the system with close to an equal distribution of spins. Upon application of a magnetic field, some of the atoms will tend to align so as to minimize the energy of the system, thus slightly more atoms should be in the lower-energy state (for the purposes of this example we will assume the spin-down state is the lower-energy state).  It is possible to add energy to the spin system using [[radio frequency]] techniques.<ref>{{cite book|last=Ramsey|first=Norman F.|author-link=Norman Ramsey|title=Spectroscopy with coherent radiation: selected papers of Norman F. Ramsey with commentary|year=1998|publisher=World Scientific|location=Singapore; River Edge, N.J.|isbn= 9789810232504 |pages=417|oclc=38753008|series=World Scientific series in 20th century physics, v. 21}}</ref> This causes atoms to ''flip'' from spin-down to spin-up.

Since we started with over half the atoms in the spin-down state, this initially drives the system towards a 50/50 mixture, so the entropy is increasing, corresponding to a positive temperature. However, at some point, more than half of the spins are in the spin-up position.<ref>{{cite book|last=Levitt|first=Malcolm H.|title=Spin Dynamics: Basics of Nuclear Magnetic Resonance|year=2008|publisher=John Wiley & Sons Ltd|location=West Sussex, England|isbn= 978-0-470-51117-6|pages=273}}</ref> In this case, adding additional energy reduces the entropy, since it moves the system further from a 50/50 mixture. This reduction in entropy with the addition of energy corresponds to a negative temperature.<ref name="kylma">{{cite web|url=http://ltl.tkk.fi/triennial/positive.html |title=Positive and negative picokelvin temperatures}}</ref>

In [[NMR spectroscopy]], such spin flips correspond to pulses with pulse widths over 180° (for a given spin). While relaxation is fast in solids, it can take several seconds in solutions and even longer in gases and in ultracold systems; several hours were reported for silver and rhodium at picokelvin temperatures.<ref name="kylma" /> It is still important to understand that the temperature is negative only with respect to nuclear spins. Other degrees of freedom, such as molecular vibrational, electronic and electron spin levels are at a positive temperature, so the object still has positive sensible heat. Relaxation actually happens by exchange of energy between the nuclear spin states and other states (e.g. through the [[nuclear Overhauser effect]] with other spins).

=== Lasers ===
This phenomenon can also be observed in many [[laser|lasing]] systems, wherein a large fraction of the system's [[atom]]s (for chemical and gas lasers) or [[electron]]s (in [[semiconductor]] lasers) are in excited states. This is referred to as a [[population inversion]].

The [[Hamiltonian (quantum mechanics)|Hamiltonian]] for a single mode of a luminescent radiation field at frequency {{mvar|ν}} is
:<math>H = (h\nu - \mu)a^\dagger a.</math>

The density operator in the [[grand canonical ensemble]] is
:<math>\rho = \frac{e^{-\beta H}}{\operatorname{Tr}\left(e^{-\beta H}\right)}.</math>

For the system to have a ground state, the trace to converge, and the density operator to be generally meaningful, {{math|''βH''}} must be positive semidefinite. So if {{math|''hν'' < ''μ''}}, and {{mvar|H}} is negative semidefinite, then {{mvar|β}} must itself be negative, implying a negative temperature.<ref>{{cite journal
 |last1=Hsu |first1=W.
 |last2=Barakat |first2=R.
 |year=1992
 |title=Statistics and thermodynamics of luminescent radiation
 |journal=[[Physical Review B]]
 |volume=46 |issue=11 |pages=6760–6767
 |bibcode=1992PhRvB..46.6760H
 |doi=10.1103/PhysRevB.46.6760
 |pmid=10002377
}}</ref>

=== Motional degrees of freedom ===
Negative temperatures have also been achieved in motional [[Degrees of freedom (physics and chemistry)|degrees of freedom]]. Using an [[optical lattice]], upper bounds were placed on the kinetic energy, interaction energy and potential energy of cold [[potassium-39]] atoms. This was done by tuning the interactions of the atoms from repulsive to attractive using a [[Feshbach resonance]] and changing the overall harmonic potential from trapping to anti-trapping, thus transforming the [[Bose–Hubbard model|Bose–Hubbard Hamiltonian]] from {{math|''Ĥ'' → −''Ĥ''}}. Performing this transformation adiabatically while keeping the atoms in the [[Mott insulator]] regime, it is possible to go from a low entropy positive temperature state to a low entropy negative temperature state. In the negative temperature state, the atoms macroscopically occupy the maximum momentum state of the lattice. The negative temperature ensembles equilibrated and showed long lifetimes in an anti-trapping harmonic potential.<ref name=":0">{{Cite journal | last1 = Braun | first1 = S. | last2 = Ronzheimer | first2 = J. P. | last3 = Schreiber | first3 = M. | last4 = Hodgman | first4 = S. S. | last5 = Rom | first5 = T. | last6 = Bloch | first6 = I. | last7 = Schneider | first7 = U. | url = https://www.mpg.de/research/negative-absolute-temperature | doi = 10.1126/science.1227831 | title = Negative Absolute Temperature for Motional Degrees of Freedom | journal = Science | volume = 339 | issue = 6115 | pages = 52–55 | year = 2013 | pmid =  23288533|arxiv = 1211.0545 |bibcode = 2013Sci...339...52B | s2cid = 8207974 }}</ref>

=== Two-dimensional vortex motion ===
The two-dimensional systems of vortices confined to a finite area can form thermal equilibrium states at negative temperature,<ref>{{cite journal
 |last1=Montgomery |first1=D. C.
 |year=1972
 |title=Two-dimensional vortex motion and "negative temperatures"
 |journal=[[Physics Letters A]]
 |volume=39 |issue=1 |pages=7–8
 |bibcode=1972PhLA...39....7M
 |doi=10.1016/0375-9601(72)90302-7
}}</ref><ref>{{cite journal
 |last1=Edwards |first1=S. F. |author-link1=Sam Edwards (physicist)
 |last2=Taylor |first2=J. B. |author-link2=J. Bryan Taylor
 |year=1974
 |title=Negative Temperature States of Two-Dimensional Plasmas and Vortex Fluids
 |journal=[[Proceedings of the Royal Society of London A]]
 |volume=336 |issue=1606 |pages=257–271
 |bibcode=1974RSPSA.336..257E
 |doi=10.1098/rspa.1974.0018
 |jstor=78450
|s2cid=120771020 }}</ref> and indeed negative temperature states were first predicted by Onsager in his analysis of classical point vortices.<ref>{{Cite journal| doi = 10.1007/BF02780991| issn = 1827-6121| volume = 6| issue = 2| pages = 279–287| last = Onsager| first = L.| title = Statistical hydrodynamics| journal = Il Nuovo Cimento (1943-1954)| accessdate = 2019-11-17| date = 1949-03-01| bibcode = 1949NCim....6S.279O| s2cid = 186224016| url = https://doi.org/10.1007/BF02780991| url-access = subscription}}</ref> Onsager's prediction was confirmed experimentally for a system of [[Quantum vortex|quantum vortices]] in a Bose–Einstein condensate in 2019.<ref name=Gau19>{{cite journal |last1=Gauthier |first1=G. |last2=Reeves |first2=M. T. |last3=Yu |first3=X. |last4=Bradley |first4=A. S.|last5=Baker|first5=M. A.|last6=Bell|first6=T. A.|last7=Rubinsztein-Dunlop|first7=H.|last8=Davis|first8=M. J.|last9=Neely|first9=T. W. |year=2019 |title=Giant vortex clusters in a two-dimensional quantum fluid |journal=Science |volume=364 |issue= 6447|pages=1264–1267 |doi=10.1126/science.aat5718 |pmid=31249054 |arxiv= 1801.06951|bibcode= 2019Sci...364.1264G|s2cid=195750381 }}</ref><ref>{{cite journal |last1=Johnstone |first1=S. P. |last2=Groszek|first2=A. J.|last3=Starkey|first3=P. T.|last4=Billinton|first4=C. J.|last5=Simula|first5=T. P.|last6=Helmerson|first6=K.|year=2019 |title= Evolution of large-scale flow from turbulence in a two-dimensional superfluid|volume=365 |issue=6447|pages=1267–1271 |doi=10.1126/science.aat5793 |pmid=31249055 |journal=Science |arxiv=1801.06952 |bibcode=2019Sci...364.1267J |s2cid=4948239 }}</ref>

== See also ==
* [[Negative resistance]]

== References ==
{{Reflist}}

== Further reading ==
* {{cite book
 |last1=Kittel |first1=C. |author1-link=Charles Kittel
 |last2=Kroemer |first2=H. |author2-link=Herbert Kroemer
 |year=1980
 |title=Thermal Physics
 |edition=2nd
 |publisher=[[W. H. Freeman]]
 |isbn=978-0-7167-1088-2
}}
* {{cite book
 |last1=Castle |first1=J.
 |last2=Emmerich |first2=W.
 |last3=Heikes |first3=R.
 |last4=Miller |first4=R.
 |last5=Rayne |first5=J.
 |year=1965
 |title=Science by Degrees: Temperature from Zero to Zero
 |publisher=[[Walker and Company]]
 |lccn=64023985
}}
* {{cite journal|doi=10.1126/science.1227831|title=Negative Absolute Temperature for Motional Degrees of Freedom|year=2013|last1=Braun|first1=S.|last2=Ronzheimer|first2=J. P.|last3=Schreiber|first3=M.|last4=Hodgman|first4=S. S.|last5=Rom|first5=T.|last6=Bloch|first6=I.|last7=Schneider|first7=U.|journal=Science|volume=339|issue=6115|pages=52–5|url=https://www.mpg.de/research/negative-absolute-temperature|pmid=23288533 |arxiv = 1211.0545 |bibcode = 2013Sci...339...52B |s2cid=8207974}}
* {{cite journal|title= Thermal Time Scales in a Color Glass Condensate|doi= 10.1103/PhysRevC.73.017901|year= 2006|journal= Physical Review C|volume= 73|issue= 17901|pages= 017901|arxiv=hep-ph/0505199|bibcode = 2006PhRvC..73a7901P|last1= Parihar|first1= V.|last2= Widom|first2= A.|last3= Srivastava|first3= Y. |s2cid= 119090586}}
* {{cite journal |title= Atomic Gases at Negative Kinetic Temperature  |doi= 10.1103/PhysRevLett.95.040403 |pmid= 16090784 |year= 2005 |journal= Physical Review Letters |volume= 95 |issue= 4|pages= 040403 |arxiv=cond-mat/0501344|bibcode = 2005PhRvL..95d0403M |last1= Mosk |first1= A. |s2cid= 1156732 }}
* {{cite journal |title= Control of Local Relaxation Behavior in Closed Bipartite Quantum Systems |doi= 10.1103/PhysRevE.72.016117 |pmid= 16090046 |year= 2005 |journal= Physical Review E |volume= 72 |issue= 7|pages= 016117 |arxiv=quant-ph/0502181|bibcode = 2005PhRvE..72a6117S |last1= Schmidt |first1= Harry |last2= Mahler |first2= Günter |s2cid= 17987338 }}
* {{cite journal |title= Anti-shielding Effect and Negative Temperature in Instantaneously Reversed Electric Fields and Left-Handed Media |doi= 10.1238/Physica.Regular.068a00087 |year= 2003 |journal= Physica Scripta |volume= 68 |issue= 1 |pages= 87–97|arxiv=cond-mat/0302351|bibcode = 2003PhyS...68...87S |last1= Shen |first1= Jian-Qi |s2cid= 118894011 }}
* {{cite conference|url=https://video.ethz.ch/speakers/zurich_physics_colloquium/8ebe0f93-e6f3-4ce1-b824-1dc587042cc1.html|format=movie|title=Towards Quantum Magnetism with Ultracold Atoms|last=Ketterle|first=Wolfgang|date=22 Sep 2010|conference=The Zurich Physics Colloquium|conference-url=http://www.multimedia.ethz.ch/speakers/zurich_physics_colloquium|location = ETH Zurich, ITS-MMS; Switzerland|access-date=1 Jan 2016|quote=Negative temperature, at about 48min. 53sec.}}
* {{cite journal|date=2013-01-04 |title=Negative Temperatures? |journal=Science |volume=339 |pages=42–43 |doi=10.1126/science.1232558|last1=Carr|first1=Lincoln D.|issue=6115|pmid=23288530 |bibcode = 2013Sci...339...42C |s2cid=124095369 }}

== External links ==
* {{cite web|author=Moriarty, Philip|title=−K: Negative Temperatures|url=http://www.sixtysymbols.com/videos/negative_tempertaures.htm|website=Sixty Symbols|publisher= [[Brady Haran]] for the [[University of Nottingham]]}}

[[Category:Temperature]]
[[Category:Entropy]]
[[Category:Magnetism]]
[[Category:Laser science]]