ARTICLES PUBLISHED ONLINE: 8 DECEMBER 2013 | DOI: 10.1038/NPHYS2815 Consistent thermostatistics forbids negative absolute temperatures Jörn Dunkel1* and Stefan Hilbert2 Over the past 60 years, a considerable number of theories and experiments have claimed the existence of negative absolute temperature in spin systems and ultracold quantum gases. This has led to speculation that ultracold gases may be dark-energy analogues and also suggests the feasibility of heat engines with efficiencies larger than one. Here, we prove that all previous negative temperature claims and their implications are invalid as they arise from the use of an entropy definition that is inconsistent both mathematically and thermodynamically. We show that the underlying conceptual deficiencies can be overcome if one adopts a microcanonical entropy functional originally derived by Gibbs. The resulting thermodynamic framework is self-consistent and implies that absolute temperature remains positive even for systems with a bounded spectrum. In addition, we propose a minimal quantum thermometer that can be implemented with available experimental techniques. ositivity of absolute temperature T, a key postulate of Gibbs formalism yields strictly non-negative absolute temperatures thermodynamics1, has repeatedly been challenged both even for quantum systems with a bounded spectrum, thereby Ptheoretically2–4 and experimentally5–7. If indeed realizable, invalidating all previous negative temperature claims. negative temperature systems promise profound practical and conceptual consequences. They might not only facilitate the Negative absolute temperatures? creation of hyper-efficient heat engines2–4 but could also help7 to The seemingly plausible standard argument in favour of negative resolve the cosmological dark-energy puzzle8,9. Measurements of absolute temperatures goes as follows10: assume a suitably designed negative absolute temperature were first reported in 1951 by Purcell many-particle quantum system with a bounded spectrum5,7 can and Pound5 in seminal work on the population inversion in nuclear be driven to a stable state of population inversion, so that most spin systems. Five years later, Ramsay’s comprehensive theoretical particles occupy high-energy one-particle levels. In this case, the study2 clarified hypothetical ramifications of negative temperature one-particle energy distribution will be an increasing function of the states, most notably the possibility to achieve Carnot efficiencies one-particle energy ✏. To fit7,10 such a distribution with a Boltzmann ⌘>1 (refs 3,4). Recently, the first experimental realization of an factor exp( ✏), β must be negative, implying a negative Boltz- 7 / − 1 ultracold bosonic quantum gas with a bounded spectrum has mann ‘temperature’ TB (kBβ)− < 0. Although this reasoning attracted considerable attention10 as another apparent example may indeed seem straightforward,= the arguments below clarify that system with T < 0, encouraging speculation that cold-atom gases TB is, in general, not the absolute thermodynamic temperature T, could serve as laboratory dark-energy analogues. unless one is willing to abandon the mathematical consistency of 1 Here, we show that claims of negative absolute temperature thermostatistics. We shall prove that the parameter TB (kBβ)− , in spin systems and quantum gases are generally invalid, as they as determined by Purcell and Pound5 and more recently= also in arise from the use of a popular yet inconsistent microcanonical ref. 7 is, in fact, a function of both temperature T and heat capacity 11 entropy definition attributed to Boltzmann . By means of rigorous C. This function TB(T,C) can indeed become negative, whereas the derivations12 and exactly solvable examples, we will demonstrate actual thermodynamic temperature T always remains positive. that the Boltzmann entropy, despite being advocated in most modern textbooks13, is incompatible with the differential structure Entropies of closed systems of thermostatistics, fails to give sensible predictions for analytically When interpreting thermodynamic data of new many-body states7, tractable quantum and classical systems, and violates equipartition one of the first questions to be addressed is the choice of the in the classical limit. The general mathematical incompatibility appropriate thermostatistical ensemble15,16. Equivalence of the implies that it is logically inconsistent to insert negative Boltz- microcanonical and other statistical ensembles cannot—in fact, mann ‘temperatures’ into standard thermodynamic relations, thus must not—be taken for granted for systems that are characterized explaining paradoxical (wrong) results for Carnot efficiencies and by a non-monotonic2,4,7 density of states (DOS) or that can undergo other observables. The deficiencies of the Boltzmann entropy can phase-transitions due to attractive interactions17—gravity being a be overcome by adopting a self-consistent entropy concept that prominent example18. Population-inverted systems are generally was derived by Gibbs more than 100 years ago14, but has been thermodynamically unstable when coupled to a (non-population- mostly forgotten ever since. Unlike the Boltzmann entropy, Gibbs’ inverted) heat bath and, hence, must be prepared in isolation5–7. entropy fulfils the fundamental thermostatistical relations and In ultracold quantum gases7 that have been isolated from the produces sensible predictions for heat capacities and other ther- environment to suppress decoherence, both particle number modynamic observables in all exactly computable test cases. The and energy are in good approximation conserved. Therefore, 1Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue E17-412, Cambridge, Massachusetts 02139-4307, USA, 2Max Planck Institute for Astrophysics, Karl-Schwarzschild-Straße 1, Garching 85748, Germany. *e-mail: [email protected] NATURE PHYSICS VOL 10 JANUARY 2014 www.nature.com/naturephysics 67 | | | ǟɥƐƎƏƓɥ !,(++-ɥ4 +(2'#12ɥ (,(3#"ƥɥ++ɥ1(%'32ɥ1#2#15#"ƥɥ ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS2815 barring other physical or topological constraints, any ab initio Note that TB becomes negative if !0 < 0, that is, if the DOS is thermostatistical treatment should start from the microcanonical non-monotic, whereas TG is always non-negative, because ⌦ is ensemble. We will first prove that only the Gibbs entropy provides a monotonic function of E. The question as to whether TB or a consistent thermostatistical model for the microcanonical density TG defines the thermodynamic absolute temperature T can be operator. Instructive examples will be discussed subsequently. decided unambiguously by considering the differential structure of We consider a (quantum or classical) system with microscopic thermodynamics, which is encoded in the fundamental relation variables ⇠ governed by the Hamiltonian H H(⇠ V , A), = ; where V denotes volume and A (A1,...) summarizes other @S @S @S = dS dE dV dAi external parameters. If the dynamics conserves the energy E, all = @E + @V + @A ✓ ◆ ✓ ◆ i ✓ i ◆ thermostatistical information about the system is contained in the 1 p ai X microcanonical density operator dE dV dAi (4) ⌘ T + T + i T δ(E H) X ⇢(⇠ E,V ,A) − (1) All consistent thermostatistical models, corresponding to pairs ; = ! (⇢,S) where ⇢ is a density operator and S an entropy potential, must which is normalized by the DOS satisfy equation (4). If one abandons this requirement, any relation to thermodynamics is lost. !(E,V ,A) Tr δ(E H) Equation (4) imposes stringent constraints on possible entropy = [ − ] candidates. For example, for an adiabatic (that is, isentropic) When considering quantum systems, we assume, as usual, volume change with dS 0 and other parameters fixed (dAi 0), that equation (1) has a well-defined operator interpretation, for one finds the consistency= condition = example, as a limit of an operator series. For classical systems, the trace simply becomes a phase-space integral over ⇠. The average of @S @E @H p T (5) some quantity F with respect to ⇢ is denoted by F Tr F⇢ , and = @V =− @V =− @V we define the integrated DOS h i⌘ [ ] ✓ ◆ ✓ ◆ ⌧ More generally, for any adiabatic variation of some parameter ⌦(E,V ,A) Tr ⇥(E H) Aµ V ,Ai of the Hamiltonian H, one must have (Supple- = [ − ] mentary2{ Information)} which is related to the DOS ! by differentiation, @S @E @H @⌦ T (6) ! ⌦ 0 @Aµ E =− @Aµ S =− @Aµ = @E ⌘ ✓ ◆ ✓ ◆ ⌧ 1 Intuitively, for a quantum system with spectrum E , the quantity where T (@S/@E)− and subscripts S and E indicate quantities { n} ⌘ ⌦(En,V ,A) counts the number of eigenstates with energy less kept constant, respectively. The first equality in equation (6) follows than or equal to En. directly from equation (4). The second equality demands correct Given the microcanonical density operator from equation (1), identification of thermodynamic quantities with statistical expec- one can find two competing definitions for the microcanonical tation values, guaranteeing for example that mechanically mea- entropy in the literature12–14,17,19,20: sured gas pressure agrees with abstract thermodynamic pressure. The conditions in equation (6) not only ensure that the thermo- S (E,V ,A) k ln(✏!), B = B dynamic potential S fulfils the fundamental differential relation (equation (4)). For a given density operator ⇢, they can be used to SG(E,V ,A) kB ln(⌦) = separate consistent entropy definitions from inconsistent ones. where ✏ is a constant with dimensions of energy, required to Using only the properties of