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Entanglement and the foundations of statistical mechanics

Abstract

Statistical mechanics is one of the most successful areas of physics. Yet, almost 150 years since its inception, its foundations and basic postulates are still the subject of debate. Here we suggest that the main postulate of statistical mechanics, the equal a priori probability postulate, should be abandoned as misleading and unnecessary. We argue that it should be replaced by a general canonical principle, whose physical content is fundamentally different from the postulate it replaces: it refers to individual states, rather than to ensemble or time averages. Furthermore, whereas the original postulate is an unprovable assumption, the principle we propose is mathematically proven. The key element in this proof is the quantum entanglement between the system and its environment. Our approach separates the issue of finding the canonical state from finding out how close a system is to it, allowing us to go even beyond the usual boltzmannian situation.

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Figure 1: The equiprobable state of the universe corresponding to the restriction R.
Figure 2: Bounding deviations from the average using Levy’s lemma.
Figure 3: Example: A system of spins.

References

  1. Mandl, F. Statistical Physics 2nd edn, Ch. 2, 40–41 (Wiley, New York, 1988).

    Google Scholar 

  2. Khinchin, A. I. Mathematical Foundations of Statistical Mechanics Ch. 3, 44–47 (Dover, New York, 1949).

    MATH  Google Scholar 

  3. Bocchieri, P. & Loinger, A. Ergodic foundation of quantum statistical mechanics. Phys. Rev. 114, 948–951 (1959).

    Article  ADS  MathSciNet  Google Scholar 

  4. Lloyd, S. Black Holes, Demons, and the Loss of Coherence. Ch. 3, 79–93, Thesis, Rockefeller Univ. (1988).

  5. Gemmer, J., Michel, M. & Mahler, G. Quantum Thermodynamics Vol. 657 (Lecture Notes in Physics, Springer, Berlin, 2004).

    Book  Google Scholar 

  6. Goldstein, S., Lebowitz, J. L., Tumulka, R. & Zanghì, N. Canonical typicality. Phys. Rev. Lett. 96, 050403 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  7. Landau, L. D. & Lifshitz, E. M. Statistical Physics Ch. 3, 78–80 (Pergamon, London, 1958).

    Google Scholar 

  8. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information Ch. 9, 403–409 (Cambridge Univ. Press, Cambridge, 2000).

    MATH  Google Scholar 

  9. Lubkin, E. Entropy of an n-system from its correlation with a k-reservoir. J. Math. Phys. 19, 1028–1031 (1978).

    Article  ADS  Google Scholar 

  10. Lloyd, S. & Pagels, H. Complexity as thermodynamic depth. Ann. Phys. 188, 186–213 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  11. Page, D. N. Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291–1294 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  12. Sen, S. Average entropy of a quantum subsystem. Phys. Rev. Lett. 77, 1–3 (1996).

    Article  ADS  Google Scholar 

  13. Sommers, H.-J. & Zyczkowski, K. Statistical properties of random density matrices. J. Phys. A: Math. Gen. 37, 8457–8466 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  14. Milman, V. D. & Schechtman, G. Asymptotic Theory of Finite-Dimensional Normed Spaces Ch. 2, 5–6 (Lecture Notes in Mathematics, Vol. 1,200, Springer, Berlin, 1986) and 140–141 Appendix V.

    MATH  Google Scholar 

  15. Hayden, P., Leung, D. W. & Winter, A. Aspects of generic entanglement. Commun. Math. Phys. 265, 95–117 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  16. Lloyd, S. Capacity of the noisy quantum channel. Phys. Rev. A 55, 1613–1622 (1997).

    Article  ADS  MathSciNet  Google Scholar 

  17. Horodecki, M., Oppenheim, J. & Winter, A. Partial quantum information. Nature 436, 673–676 (2005).

    Article  ADS  Google Scholar 

  18. Popescu, S., Short, A. J. & Winter, A. Entanglement and the foundations of statistical mechanics: Individual states vs averages. Preprint at <http://arxiv.org/abs/quant-ph/0511225> (2005).

  19. Tasaki, H. From quantum dynamics to the canonical distribution: General picture and a rigorous example. Phys. Rev. Lett. 80, 1373–1376 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  20. Michel, M., Mahler, G. & Gemmer, J. Fourier’s law from Schrödinger dynamics. Phys. Rev. Lett. 95, 180602 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  21. Breuer, H. P., Gemmer, J. & Michel, M. Non-Markovian quantum dynamics: Correlated projection superoperators and Hilbert space averaging. Phys. Rev. E 73, 016139 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  22. Gemmer, J. & Michel, M. Thermalization of quantum systems by finite baths. Europhys. Lett. 73, 1–7 (2006).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Y. Aharonov and N. Linden for discussions. S.P., A.J.S. and A.W. acknowledge support through the UK EPSRC project ‘QIP IRC’. In addition, S.P. also acknowledges support through EPSRC ‘Engineering-Physics’ grant GR/527405/01 and A.W. acknowledges a University of Bristol Research Fellowship.

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Correspondence to Anthony J. Short.

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Popescu, S., Short, A. & Winter, A. Entanglement and the foundations of statistical mechanics. Nature Phys 2, 754–758 (2006). https://doi.org/10.1038/nphys444

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