The unlikely Carnot efficiency

The efficiency of an heat engine is traditionally defined as the ratio of its average output work over its average input heat. Its highest possible value was discovered by Carnot in 1824 and is a cornerstone concept in thermodynamics. It led to the discovery of the second law and to the definition of the Kelvin temperature scale. Small-scale engines operate in the presence of highly fluctuating input and output energy fluxes. They are therefore much better characterized by fluctuating efficiencies. In this study, using the fluctuation theorem, we identify universal features of efficiency fluctuations. While the standard thermodynamic efficiency is, as expected, the most likely value, we find that the Carnot efficiency is, surprisingly, the least likely in the long time limit. Furthermore, the probability distribution for the efficiency assumes a universal scaling form when operating close-to-equilibrium. We illustrate our results analytically and numerically on two model systems.

[1]  C. Jarzynski,et al.  Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies , 2005, Nature.

[2]  Clemens Bechinger,et al.  Realization of a micrometre-sized stochastic heat engine , 2011, Nature Physics.

[3]  C. Jarzynski.  Nonequilibrium Equality for Free Energy Differences , 1996, cond-mat/9610209.

[4]  F. Ritort,et al.  The nonequilibrium thermodynamics of small systems , 2005 .

[5]  E. Lutz,et al.  Experimental verification of Landauer’s principle linking information and thermodynamics , 2012, Nature.

[6]  M. Beck,et al.  Irreversibility on the Level of Single-Electron Tunneling , 2011, 1107.4240.

[7]  C. Jarzynski.  Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale , 2011 .

[8]  Massimiliano Esposito,et al.  Ensemble and trajectory thermodynamics: A brief introduction , 2014, 1403.1777.

[9]  H. Linke,et al.  Experimental verification of reciprocity relations in quantum thermoelectric transport , 2013, 1306.3694.

[10]  P. Talkner,et al.  Colloquium: Quantum fluctuation relations: Foundations and applications , 2010, 1012.2268.

[11]  I. Tinoco,et al.  Equilibrium Information from Nonequilibrium Measurements in an Experimental Test of Jarzynski's Equality , 2002, Science.

[12]  Massimiliano Esposito,et al.  Reaching optimal efficiencies using nanosized photoelectric devices , 2009, 0907.4189.

[13]  P. Solinas,et al.  Distribution of entropy production in a single-electron box , 2013, Nature Physics.

[14]  M. Beck,et al.  Test of the fluctuation theorem for single-electron transport , 2013 .

[15]  L. Szilard.  über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen , 1929 .

[16]  J. Maxwell.  Tait's “Thermodynamics” , 1878, Nature.

[17]  N. Sinitsyn,et al.  Fluctuation relation for heat engines , 2011, 1111.7014.

[18]  M. Esposito,et al.  Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems , 2008, 0811.3717.

[19]  U. Seifert.  Stochastic thermodynamics, fluctuation theorems and molecular machines , 2012, Reports on progress in physics. Physical Society.

[20]  Hiroyasu Itoh,et al.  Resolution of distinct rotational substeps by submillisecond kinetic analysis of F1-ATPase , 2001, Nature.

[21]  Carlos Bustamante,et al.  Inter-Subunit Coordination in a Homomeric Ring-ATPase , 2009, Nature.

[22]  H. Touchette.  The large deviation approach to statistical mechanics , 2008, 0804.0327.

[23]  F. Ritort,et al.  Experimental free-energy measurements of kinetic molecular states using fluctuation theorems , 2012, Nature Physics.

[24]  J. Pekola,et al.  Test of the Jarzynski and Crooks fluctuation relations in an electronic system. , 2012, Physical review letters.

[25]  S. Ciliberto,et al.  Heat flux and entropy produced by thermal fluctuations. , 2013, Physical review letters.

[26]  Stephen R. Williams,et al.  Fluctuation theorems. , 2007, Annual review of physical chemistry.

[27]  L. Szilard.  On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings. , 1964, Behavioral science.

[28]  Thermodynamics of a colloidal particle in a time-dependent nonharmonic potential. , 2005, Physical review letters.

[29]  M. Sano,et al.  Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality , 2010 .

[30]  Unifying approach for fluctuation theorems from joint probability distributions. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  G. Crooks.  Path-ensemble averages in systems driven far from equilibrium , 1999, cond-mat/9908420.

[32]  Hanspeter Bieri,et al.  A Brief Introduction to Distributed Cognition© History and Background , 2022 .