TY - GEN

T1 - On the Nonequilibrium Entropy of Large and Small Systems

AU - Goldstein, Sheldon

AU - Huse, David A.

AU - Lebowitz, Joel L.

AU - Sartori, Pablo

N1 - Funding Information:
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.

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - Nonequilibrium thermodynamics

KW - Statistical mechanics

UR - http://www.scopus.com/inward/record.url?scp=85069531104&partnerID=8YFLogxK

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U2 - 10.1007/978-3-030-15096-9_22

DO - 10.1007/978-3-030-15096-9_22

M3 - Conference contribution

AN - SCOPUS:85069531104

SN - 9783030150952

T3 - Springer Proceedings in Mathematics and Statistics

SP - 581

EP - 596

BT - Stochastic Dynamics Out of Equilibrium - Institut Henri Poincaré, 2017

A2 - Giacomin, Giambattista

A2 - Olla, Stefano

A2 - Saada, Ellen

A2 - Spohn, Herbert

A2 - Stoltz, Gabriel

A2 - Stoltz, Gabriel

PB - Springer New York LLC

T2 - International workshop on Stochastic Dynamics out of Equilibrium, IHPStochDyn 2017

Y2 - 12 June 2017 through 16 June 2017

ER -