Irreversibility, the time arrow and a dynamical proof of the second law of thermodynamics

Abstract

We provide a dynamical proof of the second law of thermodynamics, along the lines of an argument of Penrose and Gibbs, making crucial use of the upper semicontinuity of the mean entropy proved by Robinson and Ruelle and Lanford and Robinson. An example is provided by a class of models of quantum spin systems introduced by Emch and Radin. Consequences regarding irreversibility and the time arrow, as well as possible extensions to quantum continuous systems, are discussed.

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Acknowledgements

We should like to thank Lawrence Landau, Heide Narnhofer and Derek W. Robinson for their remarks in a fruitful correspondence. The remarks of professors Oliver Penrose and David Ruelle in correspondence are also gratefully acknowledged. In a previous version we overlooked the fact that the density matrices \((\rho _{t})_{\Lambda }\) and \(\rho _{\Lambda ,t}\), the latter given by (2.10), are not the same, unless there are no interactions. This was pointed out to us by Lawrence Landau, as well as by the reviewer. We are very grateful to the reviewer, who generously provided us with Propositions 2.1 and 2.2, which prove that this difference indeed does not affect the specific entropy. He should be considered a co-author of this paper.

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Correspondence to Walter F. Wreszinski.

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Wreszinski, W.F. Irreversibility, the time arrow and a dynamical proof of the second law of thermodynamics. Quantum Stud.: Math. Found. 7, 125–136 (2020). https://doi.org/10.1007/s40509-019-00203-8

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Keywords

  • Irreversibility
  • Time-arrow
  • Specific entropy
  • Second law