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Entropy and the second law for driven, or quenched, thermally isolated systems

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  • Seifert, Udo

Abstract

The entropy of a thermally isolated system should not decrease after a quench or external driving. For a classical system following Hamiltonian dynamics, we show how this statement emerges for a large system in the sense that the extensive part of the entropy change does not become negative. However, for any finite system and small driving, the mean entropy change can well be negative. We derive these results using as micro-canonical entropy a variant recently introduced by Swendsen and co-workers called ”canonical”. This canonical entropy is the one of a canonical ensemble with the corresponding mean energy. As we show by refining the micro-canonical Crooks relation, the same results hold true for the two more conventional choices of micro-canonical entropy given either by the area of a constant energy shell, the Boltzmann entropy, or the volume underneath it, the Gibbs volume entropy. These results are exemplified with quenched N-dimensional harmonic oscillators.

Suggested Citation

  • Seifert, Udo, 2020. "Entropy and the second law for driven, or quenched, thermally isolated systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 552(C).
    • Handle: RePEc:eee:phsmap:v:552:y:2020:i:c:s0378437119310696
    DOI: 10.1016/j.physa.2019.121822
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    References listed on IDEAS

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    1. Swendsen, Robert H., 2017. "The definition of the thermodynamic entropy in statistical mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 467(C), pages 67-73.
    2. Allahverdyan, A.E. & Nieuwenhuizen, Th.M., 2002. "A mathematical theorem as the basis for the second law: Thomson's formulation applied to equilibrium," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 305(3), pages 542-552.
    3. Matty, Michael & Lancaster, Lachlan & Griffin, William & Swendsen, Robert H., 2017. "Comparison of canonical and microcanonical definitions of entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 467(C), pages 474-489.
    4. Franzosi, Roberto, 2018. "Microcanonical entropy for classical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 494(C), pages 302-307.
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