### 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.

Original language | English (US) |
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Title of host publication | Stochastic Dynamics Out of Equilibrium - Institut Henri Poincaré, 2017 |

Editors | Giambattista Giacomin, Stefano Olla, Ellen Saada, Herbert Spohn, Gabriel Stoltz, Gabriel Stoltz |

Publisher | Springer New York LLC |

Pages | 581-596 |

Number of pages | 16 |

ISBN (Print) | 9783030150952 |

DOIs | |

State | Published - Jan 1 2019 |

Event | International workshop on Stochastic Dynamics out of Equilibrium, IHPStochDyn 2017 - Paris, France Duration: Jun 12 2017 → Jun 16 2017 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 282 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Conference

Conference | International workshop on Stochastic Dynamics out of Equilibrium, IHPStochDyn 2017 |
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Country | France |

City | Paris |

Period | 6/12/17 → 6/16/17 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- Nonequilibrium thermodynamics
- Statistical mechanics

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## Cite this

*Stochastic Dynamics Out of Equilibrium - Institut Henri Poincaré, 2017*(pp. 581-596). (Springer Proceedings in Mathematics and Statistics; Vol. 282). Springer New York LLC. https://doi.org/10.1007/978-3-030-15096-9_22