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arXiv:1710.00245 · The European Physical Journal Special Topics

The smallest absorption refrigerator: the thermodynamics of a system with quantum local detailed balance

Felipe Barra, Cristóbal Lledó

Published Sep 1, 2018• 21 citations
arXivHTMLPDF
DOI
10.1140/epjst/e2018-00084-x

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1 entropy and the laws of thermodynamics 1 1.1 Entropy 2 1.1.1Classical and Statistical Entropy 4 1.1.2Aspects of Entropy 6 1.2 The 0th Law 7 1.3The 1st Law 7 1.4The 2nd Law 8 2 the third law 11 2.1 Consequences of the Third Law 12 2.2 Helium and the Third Law 14 2.3 Non-Equilibrium Degrees of Freedom 15 2.4 The Unattainability of Absolute Zero 19 3 violations of the laws of thermodynamics 23 3.1 The Second Law 23 3.1.1Fluctuations 24 3.2 The Third Law 26 4 the boson driven refrigerator 29 4.1 Introduction of the Refrigerator Model 29 4.2 Discussion 30 4.3 Non-Equilibrium Systems and Thermodynamics 36 4.4 Objective 37 5 non-equilibrium thermodynamics 39 5.1 The Stochastic Master Equation 40 5.2 Local Detailed Balance 41 5.3 The Stochastic Laws of Thermodynamics 42 5.3.1 The First Law 42 5.3.2The Second Law 43 5.4 Tight Coupling and Efficiency 45 6 boson powered refrigeration 49 6.1 Particle Currents 49 6.2 Heat Transport and Effieciency 52 6.3 Cooling power in the Limit of Absolute Zero 56 7 numerically optimized cooling 59 7.1 Conditions for Optimized Cooling 59 7.2 The Very Low Temperature Regime 63 8 finite energy level spacing 67 8.1 Dependency on δ and ∆ 73 8.2 Maximized Cooling Power 75 8.3 Higher Order Transitions 79 VII VIII contents 8.4 Heat Capacity of the Discrete Model 80 8.5 The Unattainability of Absolute Zero 82 9 summary 85 9.1 Summary and Conclusions 85 9.2 Discussion 86 9.3 Recommendations for Further Work 87 a appendix 89 a.1 C++ Source Code for the Continuous Model 89 a.2 C++ Source Code for the Discrete Model 96
Starting from a microscopic system-baths description, we derive the general conditions for a time-local quantum master equation (QME) to satisfy the first and second laws of thermodynamics at the fluctuating level. Using counting statistics, we show that the fluctuating second law can be rephrased as a generalized quantum detailed balance condition (GQDB), i.e., a symmetry of the time-local generators which ensures the validity of the fluctuation theorem. When requiring in addition a strict system-bath energy conservation, the GQDB reduces to the usual notion of detailed balance which characterizes QMEs with Gibbsian steady states. However, if energy conservation is only required on average, QMEs with non-Gibbsian steady states can still maintain a certain level of thermodynamic consistency. Applying our theory to commonly used QMEs, we show that the Redfield equation breaks the GQDB, and that some recently derived approximation schemes based on the Redfield equation (which hold beyond the secular approximation and allow one to derive a QME of Lindblad form) satisfy the GQDB and the average first law. We find that performing the secular approximation is the only way to ensure the first and second laws at the fluctuating level.