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On the Nonequilibrium Entropy of Large and Small Systems

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Stochastic Dynamics Out of Equilibrium (IHPStochDyn 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 282))

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Abstract

Thermodynamics makes definite predictions about the thermal behavior of macroscopic systems in and out of equilibrium. Statistical mechanics aims to derive this behavior from the dynamics and statistics of the atoms and molecules making up these systems. A key element in this derivation is the large number of microscopic degrees of freedom of macroscopic systems. Therefore, the extension of thermodynamic concepts, such as entropy, to small (nano) systems raises many questions. Here we shall reexamine various definitions of entropy for nonequilibrium systems, large and small. These include thermodynamic (hydrodynamic), Boltzmann, and Gibbs-Shannon entropies. We shall argue that, despite its common use, the last is not an appropriate physical entropy for such systems, either isolated or in contact with thermal reservoirs: physical entropies should depend on the microstate of the system, not on a subjective probability distribution. To square this point of view with experimental results of Bechhoefer we shall argue that the Gibbs-Shannon entropy of a nano particle in a thermal fluid should be interpreted as the Boltzmann entropy of a dilute gas of Brownian particles in the fluid.

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Notes

  1. 1.

    We are taking for granted here an assumed underlying (approximately) equal a priori probability of different microstates for a specified macrostate.

  2. 2.

    The derivation of Eq. (27), due to Boltzmann, is straightforward. Divide the \(\gamma -\)space into regions \(\varDelta _\alpha \), with \(\alpha =1,\ldots ,M\), and let \(N_\alpha \) be the number of particles in \(\varDelta _\alpha \). Then, one has that \(|\varGamma _f|\sim \prod \frac{|\varDelta _\alpha |^{N_\alpha }}{N_\alpha !}\). Using Stirling’s formula, one obtains Eq. (27), see [4] for details.

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Acknowledgements

We thank John Bechhoefer, Rafaël Chetrite, Stanislas Leibler, Eugene Speer and Bingkan Xue for fruitful discussions. The work of JLL was supported by an AFOSR grant FA9550-16-1-0037. The work of PS has been partly supported by grants from the Simons Foundation to Stanislas Leibler through The Rockefeller University (Grant 345430) and the Institute for Advanced Study (Grant 345801). DAH, JLL, and PS thank the Institute for Advanced Study for its hospitality during the elaboration of this work.

Author information

Authors and Affiliations

  1. Department of Mathematics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Sheldon Goldstein

  2. Department of Physics, Princeton University, Jadwin Hall, Washington Road, Princeton, NJ, 08544-0708, USA

    David A. Huse

  3. Departments of Mathematics and Physics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Joel L. Lebowitz

  4. The Center for Studies in Physics and Biology and Laboratory of Living Matter, Rockefeller University, New York, NY, 10065, USA

    Pablo Sartori

Authors
  1. Sheldon Goldstein
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  2. David A. Huse
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Corresponding author

Correspondence to Joel L. Lebowitz .

Editor information

Editors and Affiliations

  1. Laboratoire de Probabilités, Statistiques et Modélisation (UMR 8001), Université Paris Diderot, Paris, France

    Giambattista Giacomin

  2. CEREMADE, CNRS UMR 7534, Université Paris Dauphine - PSL, Paris, France

    Stefano Olla

  3. CNRS UMR 8145, MAP5, Université Paris Descartes, Paris, France

    Ellen Saada

  4. Zentrum Mathematik, Technische Universität München, Garching, Bayern, Germany

    Herbert Spohn

  5. CERMICS, Ecole des Ponts, Marne-La-Vallée, France

    Gabriel Stoltz

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Goldstein, S., Huse, D.A., Lebowitz, J.L., Sartori, P. (2019). On the Nonequilibrium Entropy of Large and Small Systems. In: Giacomin, G., Olla, S., Saada, E., Spohn, H., Stoltz, G. (eds) Stochastic Dynamics Out of Equilibrium. IHPStochDyn 2017. Springer Proceedings in Mathematics & Statistics, vol 282. Springer, Cham. https://doi.org/10.1007/978-3-030-15096-9_22

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