Temperatura Negativa Fisica

ARTICLESPUBLISHED ONLINE: 8 DECEMBER 2013 | DOI: 10.1038/NPHYS2815

Consistent thermostatistics forbids negativeabsolute temperaturesJrn Dunkel1* and Stefan Hilbert2

Over the past 60 years, a considerable number of theories and experiments have claimed the existence of negative absolutetemperature in spin systems and ultracold quantum gases. This has led to speculation that ultracold gases may be dark-energyanalogues and also suggests the feasibility of heat engines with efficiencies larger than one. Here, we prove that allprevious negative temperature claims and their implications are invalid as they arise from the use of an entropy definitionthat is inconsistent both mathematically and thermodynamically. We show that the underlying conceptual deficiencies canbe overcome if one adopts a microcanonical entropy functional originally derived by Gibbs. The resulting thermodynamicframework is self-consistent and implies that absolute temperature remains positive even for systems with a boundedspectrum. In addition, we propose a minimal quantum thermometer that can be implemented with available experimentaltechniques.

Positivity of absolute temperature T , a key postulate ofthermodynamics1, has repeatedly been challenged boththeoretically24 and experimentally57. If indeed realizable,negative temperature systems promise profound practical andconceptual consequences. They might not only facilitate thecreation of hyper-efficient heat engines24 but could also help7 toresolve the cosmological dark-energy puzzle8,9. Measurements ofnegative absolute temperature were first reported in 1951 by Purcelland Pound5 in seminal work on the population inversion in nuclearspin systems. Five years later, Ramsays comprehensive theoreticalstudy2 clarified hypothetical ramifications of negative temperaturestates, most notably the possibility to achieve Carnot efficiencies > 1 (refs 3,4). Recently, the first experimental realization of anultracold bosonic quantum gas7 with a bounded spectrum hasattracted considerable attention10 as another apparent examplesystem with T < 0, encouraging speculation that cold-atom gasescould serve as laboratory dark-energy analogues.

Here, we show that claims of negative absolute temperaturein spin systems and quantum gases are generally invalid, as theyarise from the use of a popular yet inconsistent microcanonicalentropy definition attributed to Boltzmann11. By means of rigorousderivations12 and exactly solvable examples, we will demonstratethat the Boltzmann entropy, despite being advocated in mostmodern textbooks13, is incompatible with the differential structureof thermostatistics, fails to give sensible predictions for analyticallytractable quantum and classical systems, and violates equipartitionin the classical limit. The general mathematical incompatibilityimplies that it is logically inconsistent to insert negative Boltz-mann temperatures into standard thermodynamic relations, thusexplaining paradoxical (wrong) results for Carnot efficiencies andother observables. The deficiencies of the Boltzmann entropy canbe overcome by adopting a self-consistent entropy concept thatwas derived by Gibbs more than 100 years ago14, but has beenmostly forgotten ever since. Unlike the Boltzmann entropy, Gibbsentropy fulfils the fundamental thermostatistical relations andproduces sensible predictions for heat capacities and other ther-modynamic observables in all exactly computable test cases. The

1Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue E17-412, Cambridge, Massachusetts 02139-4307, USA,2Max Planck Institute for Astrophysics, Karl-Schwarzschild-Strae 1, Garching 85748, Germany. *e-mail: [email protected]

Gibbs formalism yields strictly non-negative absolute temperatureseven for quantum systems with a bounded spectrum, therebyinvalidating all previous negative temperature claims.

Negative absolute temperatures?The seemingly plausible standard argument in favour of negativeabsolute temperatures goes as follows10: assume a suitably designedmany-particle quantum system with a bounded spectrum5,7 canbe driven to a stable state of population inversion, so that mostparticles occupy high-energy one-particle levels. In this case, theone-particle energy distributionwill be an increasing function of theone-particle energy . To fit7,10 such a distributionwith a Boltzmannfactor exp(), must be negative, implying a negative Boltz-mann temperature TB = (kB)1 < 0. Although this reasoningmay indeed seem straightforward, the arguments below clarify thatTB is, in general, not the absolute thermodynamic temperature T ,unless one is willing to abandon the mathematical consistency ofthermostatistics. We shall prove that the parameter TB = (kB)1,as determined by Purcell and Pound5 and more recently also inref. 7 is, in fact, a function of both temperature T and heat capacityC . This function TB(T ,C) can indeed become negative, whereas theactual thermodynamic temperatureT always remains positive.

Entropies of closed systemsWhen interpreting thermodynamic data of newmany-body states7,one of the first questions to be addressed is the choice of theappropriate thermostatistical ensemble15,16. Equivalence of themicrocanonical and other statistical ensembles cannotin fact,must notbe taken for granted for systems that are characterizedby a non-monotonic2,4,7 density of states (DOS) or that can undergophase-transitions due to attractive interactions17gravity being aprominent example18. Population-inverted systems are generallythermodynamically unstable when coupled to a (non-population-inverted) heat bath and, hence, must be prepared in isolation57.In ultracold quantum gases7 that have been isolated from theenvironment to suppress decoherence, both particle numberand energy are in good approximation conserved. Therefore,

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS2815

barring other physical or topological constraints, any ab initiothermostatistical treatment should start from the microcanonicalensemble. We will first prove that only the Gibbs entropy providesa consistent thermostatistical model for the microcanonical densityoperator. Instructive examples will be discussed subsequently.

We consider a (quantum or classical) system with microscopicvariables governed by the Hamiltonian H = H ( ;V ,A),where V denotes volume and A = (A1, ...) summarizes otherexternal parameters. If the dynamics conserves the energy E , allthermostatistical information about the system is contained in themicrocanonical density operator

( ;E,V ,A)=(EH )

(1)

which is normalized by the DOS

(E,V ,A)=Tr[(EH )]

When considering quantum systems, we assume, as usual,that equation (1) has a well-defined operator interpretation, forexample, as a limit of an operator series. For classical systems, thetrace simply becomes a phase-space integral over . The average ofsome quantity F with respect to is denoted by FTr[F], andwe define the integrated DOS

(E,V ,A)=Tr[(EH )]

which is related to the DOS by differentiation,

=

E

Intuitively, for a quantum system with spectrum {En}, the quantity(En,V ,A) counts the number of eigenstates with energy lessthan or equal to En.

Given the microcanonical density operator from equation (1),one can find two competing definitions for the microcanonicalentropy in the literature1214,17,19,20:

SB(E,V ,A)= kB ln(),

SG(E,V ,A)= kB ln()

where is a constant with dimensions of energy, required tomake the argument of the logarithm dimensionless. The Boltzmannentropy SB is advocated by most modern textbooks13 and usedby most authors nowadays2,4,5,7. The second candidate SG is oftenattributed toHertz21 butwas in fact already derived byGibbs in 1902(ref. 14, Chapter XIV). For this reason, we shall refer to SG as Gibbsentropy.Hertz proved in 1910 that SG is an adiabatic invariant21. Hiswork was highly commended by Planck22 and Einstein, who closeshis comment23 by stating that he would not have written some of hispapers had he been aware of Gibbs comprehensive treatise14.

Thermostatistical consistency conditionsThe entropy S constitutes the fundamental thermodynamicpotential of the microcanonical ensemble. Given S, secondarythermodynamic observables, such as temperature T or pressure p,are obtained by differentiation with respect to the natural controlvariables {E,V ,A}. Denoting partial derivatives with respect toE by a prime, the two formal temperatures associated with SBand SG are given by

TB(E,V ,A)=(SBE

)1=

1kB

=

1kB

(2)

TG(E,V ,A)=(SGE

)1=

1kB

=

1kB

(3)

Note that TB becomes negative if < 0, that is, if the DOS isnon-monotic, whereas TG is always non-negative, because isa monotonic function of E . The question as to whether TB orTG defines the thermodynamic absolute temperature T can bedecided unambiguously by considering the differential structure ofthermodynamics, which is encoded in the fundamental relation

dS =(SE

)dE+

(SV

)dV +

i

(SAi

)dAi

1TdE+

pTdV +

i

aiTdAi (4)

All consistent thermostatistical models, corresponding to pairs(,S) where is a density operator and S an entropy potential, mustsatisfy equation (4). If one abandons this requirement, any relationto thermodynamics is lost.

Equation (4) imposes stringent constraints on possible entropycandidates. For example, for an adiabatic (that is, isentropic)volume change with dS= 0 and other parameters fixed (dAi = 0),one finds the consistency condition

p=T(SV

)=

(EV

)=

HV

(5)

More generally, for any adiabatic variation of some parameterA {V ,Ai} of the Hamiltonian H , one must have (Supple-mentary Information)

T(SA

)E

=

(EA

)S

=

HA

(6)

where T (S/E)1 and subscripts S and E indicate quantitieskept constant, respectively. The first equality in equation (6) followsdirectly from equation (4). The second equality demands correctidentification of thermodynamic quantities with statistical expec-tation values, guaranteeing for example that mechanically mea-sured gas pressure agrees with abstract thermodynamic pressure.The conditions in equation (6) not only ensure that the thermo-dynamic potential S fulfils the fundamental differential relation(equation (4)). For a given density operator , they can be used toseparate consistent entropy definitions from inconsistent ones.

Using only the properties of themicrocanonical density operatoras defined in equation (1), one finds24

TG

(SGA

)=

1

ATr[(EH )

]=

1Tr[

A(EH )

]= Tr

[(HA

)(EH )

]=

HA

(7)

This proves that the pair (,SG) fulfils equation (6) and, hence,constitutes a consistent thermostatistical model for the mi-crocanonical density operator . Moreover, because generallyTB(SB/A) 6=TG(SG/A), it is a trivial corollary that the Boltz-mann entropy SB violates equation (6) and hence cannot be athermodynamic entropy, implying that it is inconsistent to insertthe Boltzmann temperature TB into equations of state or efficiencyformulae that assume validity of the fundamental thermodynamicrelations (equation (4)).

Similarly to equation (7), it is straightforward to show that, forstandard classical Hamiltonian systems with confined trajectoriesand a finite ground-state energy, only the Gibbs temperature TGsatisfies themathematically rigorous equipartition theorem12

iHj

Tr

[(iHj

)

]= kBTG ij (8)

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NATURE PHYSICS DOI: 10.1038/NPHYS2815 ARTICLESfor all canonical coordinates = (1,...). The key steps of the proofare identical to those in equation (7); that is, one merely exploitsthe chain rule relation (EH )/=(H/)(EH ), whichholds for any variable in the Hamiltonian H . Equation (8) isessentially a phase-space version of Stokes theorem12, relatinga surface (flux) integral on the energy shell to the enclosedphase-space volume.

Small systemsDifferences between SB and SG are negligible for most macroscopicsystems with monotonic DOS , but can be significant for smallsystems12. This can already be seen for a classical ideal gas in d-spacedimensions, where17

(E,V )=EdN/2V N , =(2m)dN/2

N !hd0(dN/2+1)

for N identical particles of mass m and Planck constant h.From this, one finds that only the Gibbs temperature yieldsexact equipartition

E =(dN21)kBTB, (9)

E =dN2

kBTG (10)

Clearly, equation (9) yields paradoxical results for dN = 1, where itpredicts negative temperature TB< 0 and heat capacity CB< 0, andalso for dN = 2, where the temperature TB must be infinite. This isa manifestation of the fact that SB is not an exact thermodynamicentropy. In contrast, the Gibbs entropy SG produces the reasonableequation (10), which is a special case of the more generalequipartition theorem (equation (8)).

That SG also is themore appropriate choice for isolated quantumsystems, as relevant to the interpretation of the experimentsby Purcell and Pound5 and Braun et al.7, can be readilyillustrated by two other basic examples: for a simple harmonicoscillator with spectrum

En= h(n+

12

), n= 0,1,...,

we find by inversion and analytic interpolation = 1+ n = 1/2+ E/(h) and, hence, from the Gibbs entropySG= kB ln the caloric equation of state

kBTG=h2+E

which, when combined with the quantum virial theorem, yields anequipartition-type statement for this particular example (equipar-tition is not a generic feature of quantum systems). Furthermore,T =TG gives a sensible prediction for the heat capacity,

C =(TE

)1= kB

accounting for the fact that even a single oscillator can serve asminimal quantum heat reservoir. More precisely, the energy of aquantum oscillator can be changed by performing work througha variation of its frequency , or by injecting or removing energyquanta, corresponding to heat transfer in the thermodynamicpicture. The Gibbs entropy SG quantifies these processes in asensible manner. In contrast, the Boltzmann entropy SB= kB ln()with = (h)1 assigns the same constant entropy to all energystates, yielding the nonsensical result TB = for all energyeigenvalues En and making it impossible to compute the heat

capacity of the oscillator. The failure of the Boltzmann entropy SBfor this basic example should raise doubts about its applicability tomore complex quantum systems7.

That SB violates fundamental thermodynamic relations not onlyfor classical but also for quantum systems can be further illustratedby considering a quantum particle in a one-dimensional infinitesquare-well of length L, for which the spectral formula

En= an2/L2, a= h2 2/(2m), n= 1,2,..., (11)

implies = n = LE/a. In this case, the Gibbs entropy

SG = kB ln gives

kBTG= 2E, pGTG

(SGL

)=

2EL

as well as the heat capacity C = kB/2, in agreement withphysical intuition. In particular, the pressure equation is consistentwith condition (equation (5)), as can be seen by differentiatingequation (11) with respect to the volume L,

pEL=

2EL= pG

That is, pG coincides with the mechanical pressure as obtainedfrom kinetic theory19.

In contrast, we find from SB= kB ln() with = L/(2Ea) for

the Boltzmann temperature

kBTB=2E < 0

Although this result in itself seems questionable, unless one believesthat a quantum particle in a one-dimensional box is a dark-energycandidate, it also implies a violation of equation (5), because

pBTB

(SBL

)=

2EL6= p

This contradiction corroborates that SB cannot be the correctentropy for quantum systems.

We still mention that one sometimes encounters the adhoc convention that, because the spectrum in equation (11) isnon-degenerate, the thermodynamic entropy should be zerofor all states. However, such a postulate entails several otherinconsistencies (Supplementary Information). Focusing on theexample at hand, the convention S = 0 would again imply thenonsensical result T =, misrepresenting the physical fact thatalso a single degree of freedom in a box-like confinement can storeheat in finite amounts.

Measuring TB instead of TFor classical systems, the equipartition theorem (equation (8))implies that an isolated classical gas thermometer shows, strictlyspeaking, the Gibbs temperature T = TG, not TB. When broughtinto (weak) thermal contact with an otherwise isolated system, agas thermometer indicates the absolute temperature T of the com-pound system. In the quantumcase, theGibbs temperatureT can bedetermined with the help of a bosonic oscillator that is prepared inthe ground state and then weakly coupled to the quantum system ofinterest, because (kBT )1 is proportional to the probability that theoscillator has remained in the ground state after some equilibrationperiod (Methods). Thus, the Gibbs entropy provides not only theconsistent thermostatistical description of isolated systems but alsoa sound practical basis for classical and quantum thermometers.

It remains to clarify why previous experiments5,7 measuredTB and not the absolute temperature T . The authors of ref. 7,




for example, estimate temperature by fitting a quasi-exponentialBoseEinstein function to their experimentally obtained one-particle energy distributions10. Their system contains N 1particles with Hamiltonian HN and DOS N . The formally exactmicrocanonical one-particle density operator reads

1=TrN1[N ] =TrN1[(EHN )]

N(12)

To obtain an exponential (canonical) fitting formula, as usedin the experiments, one first has to rewrite 1 in the equiva-lent form 1= exp[ln1]. Applying a standard steepest descentapproximation13,19 to the logarithm and assuming discrete one-particle levels E`, one finds for the relative occupancy p` of one-particle level E` the canonical form

p`'eE`/(kBTB)

Z, Z =

`

eE`/(kBTB) (13)

The key observation here is that the exponential approxima-tion (equation (13)) features TB and not the absolute ther-modynamic Gibbs temperature T = TG. This becomes obvi-ous by writing equation (12) for a given one-particle energyE` as p` = N1(E E`)/N (E) = exp[lnN1(E E`)]/N (E)and expanding lnN1(E E`) for small E`, which gives p` exp[E`/(kBTB,N1)], where kBTB,N1 N1(E)/N1(E), inagreement with equation (2). That is, TB in equation (13) is actuallythe Boltzmann temperature of the (N1)-particle system.

Hence, by fitting the one-particle distribution, one determinesthe Boltzmann temperature TB, which can be negative, whereas thethermodynamic Gibbs temperature T =TG is always non-negative.The formal definitions ofTG andTB imply the exact general relation(Supplementary Information)

TB=TG

1kB/C(14)

where C = (TG/E)1 is the total thermodynamic heat capacityassociated with T =TG. As evident from equation (14), differencesbetween TG and TB become relevant only if |C | is close to or smallerthan kB; in particular, TB is negative if 0

NATURE PHYSICS DOI: 10.1038/NPHYS2815 ARTICLEStary Information). Such constraints lead to a strong effectivecoupling between the spin degrees of freedom, thereby invalidatingbasic assumptions in the derivation of canonical distributions,such as equation (13).

In summary, for systems with a bounded spectrum, the effectiveBoltzmann temperature TB differs not only quantitatively butalso qualitatively from the actual thermodynamic temperatureT =TG> 0. Unfortunately, the measurement conventions adoptedby Braun et al.7, and similarly those by Purcell and Pound5, aredesigned to measure TB instead of TG.

Carnot efficiencies > 1 ?The above arguments show that the Boltzmann entropy SB isnot a consistent thermodynamic entropy, neither for classical norfor quantum systems, whereas the Gibbs entropy SG providesa consistent thermodynamic formalism in the low-energy limit(small quantum systems), in the high-energy limit (classicalsystems) and in between. Regrettably, SB has become so widelyaccepted nowadays that, even when its application to exoticstates of matter7 leads to dubious claims, these are rarelyquestioned. One example are speculations2,4,7 that population-inverted systems can drive Carnot machines with efficiency>1. Toevaluate such statements, recall that a Carnot cycle, by definition,consists of four successive steps: isothermal expansion; isentropicexpansion; isothermal compression; isentropic compression. Thetwo isothermal steps require a hot and cold bath with temperaturesTH and TC, respectively, and the two isentropic steps canbe thought of as place-holders for other work-like parametervariations (changes of external magnetic fields, and so on). Theassociated Carnot efficiency

= 1TC

TH(15)

owes its popularity to the fact that it presents an upper bound forother heat engines19. To realize values > 1, one requires either TCor TH to be negative. At least formally, this seems to be achievableby considering systems as in Fig. 1 and naively inserting positive andnegative Boltzmann temperature valuesTB0 into equation (15).

Speculations2,4,7 of this type are unsubstantiated for severalreasons. First, TB is not a consistent thermodynamic temperature,and, if at all, one should use the absolute temperature T =TG>0 inequation (15), which immediately forbids > 1. Second, to changeback and forth between population-inverted states with TB < 0and non-inverted states with TB > 0, work must be performednon-adiabatically26, for example, by rapidly switching a magneticfield. As the thermodynamic entropy is not conserved duringsuch switching processes, the resulting cycle is not of the Carnottype anymore and requires careful energy balance calculations3.In particular, such an analysis has to account for the peculiar factthat, when the heat engine is capable of undergoing populationinversion, both a hot and cold bath may inject heat into the system.Properly defined efficiencies of thermodynamic cycles that involvesystemswith lower and upper energy bounds are, in general, not justsimple functions ofTG orTB. Naive application of equation (15) canbe severely misleading in those cases.

On a final note, groundbreaking experiments such as those byPurcell and Pound5 and Braun et al.7 are essential for verifying theconceptual foundations of thermodynamics and thermostatistics.Such studies disclose previously unexplored regimes, therebyenabling us to test and, where necessary, expand theoreticalconcepts that will allow us to make predictions and are essentialfor the development of new technologies. However, the correctinterpretation of data and the consistent formulation of heatand work exchange15 under extreme physical conditions (forexample, at ultracold or ultrahot27 temperatures, or on atomic orastronomical scales) require special care when it comes to applying

the definitions and conventions that constitute a specific theoreticalframework. When interpreted within a consistent thermostatisticaltheory, as developed by Gibbs14 more than a century ago, neitherthe work of Purcell and Pound5 nor recent experiments7 provideevidence for negative absolute temperatures. Unfortunately, thisalsomeans that cold atomgases are less likely tomimic dark energy.

MethodsMinimal quantum thermometer. A simple quantum thermometer for measuringthe thermodynamic Gibbs temperature T =TG can be realized with a heavy atomin a one-dimensional harmonic trap. The measurement protocol is as follows:before coupling thermometer and system, one must prepare the isolated system ina state with well-defined energy E = ES and the thermometer oscillator with smallangular frequency in the ground state ET= h/2. After coupling the thermometerto the system, the total energy remains conserved, but redistribution of energymay take place. A measurement of the thermometer energy after a sufficiently longequilibration period will produce an oscillator eigenvalue E T= h(n

+1/2), where

n {0,...,b(EE0)/(h)c}, with E0 denoting the systems ground state and bxc theinteger part. If the total energy remains conserved and the thermometer oscillatoris non-degenerate, the probability P[E T|E] of measuring a specific oscillator energyE T is equal to the microcanonical probability of finding the system in a stateE S = E (E

TET) E :

P[E T|E] =g (E+ETE T)

(E)

where g (E S) is the degeneracy of the level E

S of the system, and

(E)=E SE

g (E S)

Assuming that the energy levels lie sufficiently dense ( 0) we can approximatethe discrete probabilities P[E T|E]'p(E

T|E)dE

T by the probability density

p(E T|E)=(E+ETE T)

(E)

This distribution can be obtained by repeating the experiment many times, and asimple estimator for the (inverse) absolute temperatureT >0 is (equation (3))

1kBT=(E)(E)

= p(ET|E) (16)

In practice, one would measure p(E T|E) for E

T > ET = h/2 and extrapolate toE T= ET. The thermometer equation (16) is applicable to systems with and withoutpopulation inversion. The precision of this minimal thermometer is set by theoscillator frequency and the number of measurements.

Received 10 July 2013; accepted 18 October 2013; published online8 December 2013

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AcknowledgementsWe thank I. Bloch, W. Hofstetter and U. Schneider for constructive discussions. We aregrateful to M. Campisi for pointing out equation (14), and to P. Kopietz, P. Talkner,R. E. Goldstein and, in particular, P. Hnggi for helpful comments.

Author contributionsAll authors contributed to all aspects of this work.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints. Correspondenceand requests for materials should be addressed to J.D.

Competing financial interestsThe authors declare no competing financial interests.

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Consistent thermostatistics forbids negative absolute temperaturesNegative absolute temperatures?Entropies of closed systemsThermostatistical consistency conditionsSmall systems Measuring instead of TQuantum systems with a bounded spectrumCarnot efficiencies >1 ?MethodsMinimal quantum thermometer.

Figure 1 Non-negativity of the absolute temperature in quantum systems with a bounded spectrum.ReferencesAcknowledgementsAuthor contributionsAdditional informationCompeting financial interests

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Temperatura Negativa Fisica