Editing Negative temperature (section)

== Examples ==

=== Noninteracting two-level particles ===
{{multiple image
   | align     = right
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   | 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>