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Approach to Equilibrium in Translation-Invariant Quantum Systems: Some Structural Results

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Abstract

We formulate the problem of approach to equilibrium in algebraic quantum statistical mechanics and study some of its structural aspects, focusing on the relation between the zeroth law of thermodynamics (approach to equilibrium) and the second law (increase in entropy). Our main result is that approach to equilibrium is necessarily accompanied by a strict increase in the specific (mean) energy and entropy. In the course of our analysis, we introduce the concept of quantum weak Gibbs state which is of independent interest.

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Notes

  1. For additional information and references about this problem see for example [57, 92]

  2. This terminology is unfortunate; see [14]

  3. The unit will be denoted by \(\mathbbm {1}\).

  4. Our sign and ordering conventions are such that, for density matrices \(\rho _1,\rho _2\), \(S(\rho _1|\rho _2)=\mathop {\textrm{tr}}\nolimits (\rho _1(\log \rho _1-\log \rho _2))\). We refer the reader to [69] for an in-depth discussion of relative entropy.

  5. \(\mathscr {B}(\mathscr {H})\) denotes the \(C^*\)-algebra of all linear operators on the Hilbert space \(\mathscr {H}\).

  6. The interaction N generates the gauge group, \(\vartheta ^\theta =\alpha _N^\theta \).

  7. These sets are disjoint.

  8. \(\Theta \) is an anti-linear \(*\)-automorphism of \(\mathfrak {A}\) satisfying \(\Theta \circ \alpha _\Phi ^t=\alpha _\Phi ^{-t}\circ \Theta \) for all \(t\in \mathbb {R}\) and \(\omega (\Theta (A))=\omega (A^*)\) for all \(A\in \mathfrak {A}\).

  9. The principal of which is that \((\mathfrak {A},\alpha _\Lambda ,\omega _\Lambda )\) has the property of return to equilibrium for all large enough \(\Lambda \).

  10. The converse does not hold.

  11. The error concerns the identity (32) in [66], which implies that the KMS vector representative constructed by Ejima and Ogata in [21] is in the natural cone. Unfortunately, a simple argument shows that this is so only in the commutative case.

  12. See [63, 64].

  13. Any state \(\omega \in \mathscr {S}_\textrm{I}(\mathscr {A})\) restricts to the gauge-invariant state \(\bar{\omega }=\int _0^{2\pi }\omega \circ \vartheta ^\theta \frac{\textrm{d}\theta }{2\pi }\in \mathscr {S}_\textrm{I}(\mathfrak {A})\).

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Acknowledgements

This work was supported by the French Agence Nationale de la Recherche, grant NONSTOPS (ANR-17-CE40-0006-01, ANR-17-CE40-0006-02, ANR-17-CE40-0006-03) and CY Initiative of Excellence, Investissements d’Avenir program (grant ANR-16-IDEX-0008). It was partly developed during VJ’s stay at the CY Advanced Studies, whose support is gratefully acknowledged. VJ also acknowledges the support of NSERC. The authors wish to thank Laurent Bruneau and Aernout van Enter for useful discussions, and Luc Rey-Bellet for pointing them to the reference [66].

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada

    Vojkan Jakšić

  2. Aix Marseille Univ, Universitée de Toulon, CNRS, CPT, Marseille, France

    Claude-Alain Pillet

  3. Institut de Recherche Mathématique Avancée, UMR 7501 Université de Strasbourg et CNRS, 7 rue René-Descartes, 67000, Strasbourg, France

    Clément Tauber

Authors
  1. Vojkan Jakšić
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  2. Claude-Alain Pillet
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  3. Clément Tauber
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Correspondence to Clément Tauber.

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Communicated by Antti Kupiainen.

Dedicated to the memory of Krzysztof Gawȩdzki

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Jakšić, V., Pillet, CA. & Tauber, C. Approach to Equilibrium in Translation-Invariant Quantum Systems: Some Structural Results. Ann. Henri Poincaré 25, 715–749 (2024). https://doi.org/10.1007/s00023-023-01281-6

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