Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Consistent thermostatistics forbids negative absolute temperatures

Nature Physics volume 10, pages 67–72 (2014)Cite this article

Subjects

Abstract

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.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Non-negativity of the absolute temperature in quantum systems with a bounded spectrum.

Similar content being viewed by others

References

  1. Callen, H. B. Thermodynamics and an Introduction to Thermostatics (Wiley, 1985).

    MATH  Google Scholar 

  2. Ramsay, N. F. Thermodynamics and statistical mechanics at negative absolute temperatures. Phys. Rev. 103, 20–28 (1956).

    Article  ADS  Google Scholar 

  3. Landsberg, P. T. Heat engines and heat pumps at positive and negative absolute temperature. J. Phys. A 10, 1773–1780 (1977).

    Article  ADS  Google Scholar 

  4. Rapp, A., Mandt, S. & Rosch, A. Equilibration rates and negative absolute temperatures for ultracold atoms in optical lattices. Phys. Rev. Lett. 105, 220405 (2010).

    Article  ADS  Google Scholar 

  5. Purcell, E. M. & Pound, R. V. A nuclear spin system at negative temperature. Phys. Rev. 81, 279–280 (1951).

    Article  ADS  Google Scholar 

  6. Hakonen, P. & Lounasmaa, O. V. Negative absolute temperature—hot spins in spontaneous magnetic order. Science 265, 1821–1825 (1994).

    Article  ADS  Google Scholar 

  7. Braun, S. et al. Negative absolute temperature for motional degrees of freedom. Science 339, 52–55 (2013).

    Article  ADS  Google Scholar 

  8. Peebles, P. J. & Ratra, B. The cosmological constant and dark energy. Rev. Mod. Phys. 75, 559–606 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  9. Loeb, A. Thinking outside the simulation box. Nature Phys. 9, 384–386 (2013).

    Article  ADS  Google Scholar 

  10. Carr, L. D. Negative temperatures? Science 339, 42–43 (2013).

    Article  ADS  Google Scholar 

  11. Sommerfeld, A. Vorlesungen über Theoretische Physik (Band 5): Thermodynamik und Statistik 181–183 (Verlag Harri Deutsch, 2011).

    Google Scholar 

  12. Khinchin, A. I. Mathematical Foundations of Statistical Mechanics (Dover, 1949).

    MATH  Google Scholar 

  13. Huang, K. Statistical Mechanics 2nd edn (Wiley, 1987).

    MATH  Google Scholar 

  14. Gibbs, J. W. Elementary Principles in Statistical Mechanics (Dover, 1960) (Reprint of the 1902 edition).

    MATH  Google Scholar 

  15. Campisi, M., Talkner, P. & Hänggi, P. Fluctuation theorem for arbitrary open quantum systems. Phys. Rev. Lett. 102, 210401 (2009).

    Article  ADS  Google Scholar 

  16. Campisi, M. & Kobe, D. Derivation of the Boltzmann principle. Am. J. Phys. 78, 608–615 (2010).

    Article  ADS  Google Scholar 

  17. Dunkel, J. & Hilbert, S. Phase transitions in small systems: Microcanonical vs. canonical ensembles. Physica A 370, 390–406 (2006).

    Article  ADS  Google Scholar 

  18. Votyakov, E V., Hidmi, H. I., De Martino, A. & Gross, D. H. E. Microcanonical mean-field thermodynamics of self-gravitating and rotating systems. Phys. Rev. Lett. 89, 031101 (2002).

    Article  ADS  Google Scholar 

  19. Becker, R. Theory of Heat (Springer, 1967).

    Book  Google Scholar 

  20. Campisi, M. Thermodynamics with generalized ensembles: The class of dual orthodes. Physica A 385, 501–517 (2007).

    Article  ADS  MathSciNet  Google Scholar 

  21. Hertz, P. Über die mechanischen Grundlagen der Thermodynamik. Ann. Phys. (Leipz.) 33 225–274; 537–552 (1910).

  22. Hoffmann, D. ‘... you can’t say anyone to their face: your paper is rubbish.’ Max Planck as Editor of Annalen der Physik. Ann. Phys. (Berlin) 17, 273–301 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  23. Einstein, A. Bemerkungen zu den P. Hertzschen Arbeiten:‘Über die mechanischen Grundlagen der Thermodynamik’. Ann. Phys. (Leipz.) 34, 175–176 (1911).

    Article  ADS  Google Scholar 

  24. Campisi, M. On the mechanical foundations of thermodynamics: The generalized Helmholtz theorem. Stud. Hist. Philos. Mod. Phys. 36, 275–290 (2005).

    Article  MathSciNet  Google Scholar 

  25. Stanley, R. P. Enumerative Combinatorics 2nd edn, Vol. 1 (Cambridge Studies in Advanced Mathematics, Cambridge Univ. Press, 2000).

    MATH  Google Scholar 

  26. Tremblay, A-M. Comment on ‘Negative Kelvin temperatures: Some anomalies and a speculation’. Am. J. Phys. 44, 994–995 (1975).

    Article  ADS  Google Scholar 

  27. Dunkel, J., Hänggi, P. & Hilbert, S. Nonlocal observables and lightcone averaging in relativistic thermodynamics. Nature Phys. 5, 741–747 (2009).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue E17-412, Cambridge, Massachusetts 02139-4307, USA

    Jörn Dunkel

  2. Max Planck Institute for Astrophysics, Karl-Schwarzschild-Straße 1, Garching 85748, Germany

    Stefan Hilbert

Authors
  1. Jörn Dunkel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Hilbert
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to all aspects of this work.

Corresponding author

Correspondence to Jörn Dunkel.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 581 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dunkel, J., Hilbert, S. Consistent thermostatistics forbids negative absolute temperatures. Nature Phys 10, 67–72 (2014). https://doi.org/10.1038/nphys2815

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys2815

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing