1
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Wang Z, Ren J. Thermodynamic Geometry of Nonequilibrium Fluctuations in Cyclically Driven Transport. PHYSICAL REVIEW LETTERS 2024; 132:207101. [PMID: 38829089 DOI: 10.1103/physrevlett.132.207101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/11/2023] [Revised: 09/06/2023] [Accepted: 04/11/2024] [Indexed: 06/05/2024]
Abstract
Nonequilibrium thermal machines under cyclic driving generally outperform steady-state counterparts. However, there is still lack of coherent understanding of versatile transport and fluctuation features under time modulations. Here, we formulate a theoretical framework of thermodynamic geometry in terms of full counting statistics of nonequilibrium driven transports. We find that, besides the conventional dynamic and adiabatic geometric curvature contributions, the generating function is also divided into an additional nonadiabatic contribution, manifested as the metric term of full counting statistics. This nonadiabatic metric generalizes recent results of thermodynamic geometry in near-equilibrium entropy production to far-from-equilibrium fluctuations of general currents. Furthermore, the framework proves geometric thermodynamic uncertainty relations of near-adiabatic thermal devices, constraining fluctuations in terms of statistical metric quantities and thermodynamic length. We exemplify the theory in experimentally accessible driving-induced quantum chiral transport and Brownian heat pump.
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Affiliation(s)
- Zi Wang
- Center for Phononics and Thermal Energy Science, China-EU Joint Lab on Nanophononics, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
| | - Jie Ren
- Center for Phononics and Thermal Energy Science, China-EU Joint Lab on Nanophononics, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
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2
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Gu J. Speed limit, dissipation bound, and dissipation-time trade-off in thermal relaxation processes. Phys Rev E 2023; 108:L052103. [PMID: 38115476 DOI: 10.1103/physreve.108.l052103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2023] [Accepted: 10/15/2023] [Indexed: 12/21/2023]
Abstract
We investigate bounds on speed, nonadiabatic entropy production, and the trade-off relation between them for classical stochastic processes with time-independent transition rates. Our results show that the time required to evolve from an initial to a desired target state is bounded from below by the information-theoretical ∞-Rényi divergence between these states, divided by the total rate. Furthermore, we conjecture and provide extensive numerical evidence for an information-theoretical bound on the nonadiabatic entropy production and a dissipation-time trade-off relation that outperforms previous bounds in some cases..
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Affiliation(s)
- Jie Gu
- Chengdu Academy of Education Sciences, Chengdu 610036, China
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3
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Erdman PA, Noé F. Model-free optimization of power/efficiency tradeoffs in quantum thermal machines using reinforcement learning. PNAS NEXUS 2023; 2:pgad248. [PMID: 37593201 PMCID: PMC10427747 DOI: 10.1093/pnasnexus/pgad248] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/13/2023] [Revised: 07/13/2023] [Accepted: 07/25/2023] [Indexed: 08/19/2023]
Abstract
A quantum thermal machine is an open quantum system that enables the conversion between heat and work at the micro or nano-scale. Optimally controlling such out-of-equilibrium systems is a crucial yet challenging task with applications to quantum technologies and devices. We introduce a general model-free framework based on reinforcement learning to identify out-of-equilibrium thermodynamic cycles that are Pareto optimal tradeoffs between power and efficiency for quantum heat engines and refrigerators. The method does not require any knowledge of the quantum thermal machine, nor of the system model, nor of the quantum state. Instead, it only observes the heat fluxes, so it is both applicable to simulations and experimental devices. We test our method on a model of an experimentally realistic refrigerator based on a superconducting qubit, and on a heat engine based on a quantum harmonic oscillator. In both cases, we identify the Pareto-front representing optimal power-efficiency tradeoffs, and the corresponding cycles. Such solutions outperform previous proposals made in the literature, such as optimized Otto cycles, reducing quantum friction.
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Affiliation(s)
- Paolo A Erdman
- Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
| | - Frank Noé
- Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
- Microsoft Research AI4Science, Karl-Liebknecht Str. 32, 10178 Berlin, Germany
- Department of Physics, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
- Department of Chemistry, Rice University, Houston, TX 77005, USA
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4
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Chen J, Wang Y, Chen J, Su S. Optimal figure of merit of low-dissipation quantum refrigerators. Phys Rev E 2023; 107:044118. [PMID: 37198854 DOI: 10.1103/physreve.107.044118] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/20/2022] [Accepted: 03/21/2023] [Indexed: 05/19/2023]
Abstract
The Drazin inverse of the Liouvillian superoperator provides a solution to determine the dynamics of a time-dependent system governed by the Markovian master equation. Under the condition of slow driving, the perturbation expansion of the density operator of the system in powers of time can be derived. As an application, a finite-time cycle model of the quantum refrigerator driven by a time-dependent external field is established. The method of the Lagrange multiplier is adopted as a strategy to find the optimal cooling performance. The figure of merit given by the product of the coefficient of performance and the cooling rate is taken as a new objective function, and, consequently, the optimally operating state of the refrigerator is revealed. The effects of the frequency exponent determining dissipation characteristics on the optimal performance of the refrigerator are discussed systemically. The results obtained show that the adjacent areas of the state of the maximum figure of merit are the best operation region of low-dissipative quantum refrigerators.
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Affiliation(s)
- Jingyi Chen
- Department of Physics, Xiamen University, Xiamen 361005, People's Republic of China
| | - Youlin Wang
- Department of Physics, Xiamen University, Xiamen 361005, People's Republic of China
| | - Jincan Chen
- Department of Physics, Xiamen University, Xiamen 361005, People's Republic of China
| | - Shanhe Su
- Department of Physics, Xiamen University, Xiamen 361005, People's Republic of China
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5
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Abstract
We apply the adiabatic approximation to slow but finite-time thermodynamic processes and obtain the full counting statistics of work. The average work consists of change in free energy and the dissipated work, and we identify each term as a dynamical- and geometric-phase-like quantity. An expression for the friction tensor, the key quantity in thermodynamic geometry, is explicitly given. The dynamical and geometric phases are proved to be related to each other via the fluctuation-dissipation relation.
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Affiliation(s)
- Jie Gu
- Chengdu Academy of Education Sciences, Chengdu 610036, China
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6
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Scandi M, Barker D, Lehmann S, Dick KA, Maisi VF, Perarnau-Llobet M. Minimally Dissipative Information Erasure in a Quantum Dot via Thermodynamic Length. PHYSICAL REVIEW LETTERS 2022; 129:270601. [PMID: 36638287 DOI: 10.1103/physrevlett.129.270601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/09/2022] [Revised: 11/29/2022] [Accepted: 12/12/2022] [Indexed: 06/17/2023]
Abstract
In this Letter, we explore the use of thermodynamic length to improve the performance of experimental protocols. In particular, we implement Landauer erasure on a driven electron level in a semiconductor quantum dot, and compare the standard protocol in which the energy is increased linearly in time with the one coming from geometric optimization. The latter is obtained by choosing a suitable metric structure, whose geodesics correspond to optimal finite-time thermodynamic protocols in the slow driving regime. We show experimentally that geodesic drivings minimize dissipation for slow protocols, with a bigger improvement as one approaches perfect erasure. Moreover, the geometric approach also leads to smaller dissipation even when the time of the protocol becomes comparable with the equilibration timescale of the system, i.e., away from the slow driving regime. Our results also illustrate, in a single-electron device, a fundamental principle of thermodynamic geometry: optimal finite-time thermodynamic protocols are those with constant dissipation rate along the process.
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Affiliation(s)
- Matteo Scandi
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels, Barcelona 08860, Spain
| | - David Barker
- NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
| | - Sebastian Lehmann
- NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
| | - Kimberly A Dick
- NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
- Centre for Analysis and Synthesis, Lund University, Box 124, 22100 Lund, Sweden
| | - Ville F Maisi
- NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
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7
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Zhao XH, Gong ZN, Tu ZC. Low-dissipation engines: Microscopic construction via shortcuts to adiabaticity and isothermality, the optimal relation between power and efficiency. Phys Rev E 2022; 106:064117. [PMID: 36671114 DOI: 10.1103/physreve.106.064117] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2022] [Accepted: 11/27/2022] [Indexed: 06/17/2023]
Abstract
We construct a microscopic model of low-dissipation engines by driving a Brownian particle in a time-dependent harmonic potential. Shortcuts to adiabaticity and shortcuts to isothermality are introduced to realize the adiabatic and isothermal branches in a thermodynamic cycle, respectively. We derive an analytical formula of the efficiency at maximum power with explicit expressions of dissipation coefficients under the optimized protocols. When the relative temperature difference between the two baths in the cycle is insignificant, this expression satisfies the universal law of efficiency at maximum power up to the quadratic term of the Carnot efficiency. For large relative temperature differences, the efficiency at maximum power tends to be 1/2. Furthermore, we analyze the issue of power at any given efficiency for general low-dissipation engines and then obtain the supremum of the power in three limiting cases, respectively. These expressions of maximum power at given efficiency provide the optimal relations between power and efficiency which are tighter than the results in previous references.
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Affiliation(s)
- Xiu-Hua Zhao
- Department of Physics, Beijing Normal University, Beijing 100875, China
| | | | - Z C Tu
- Department of Physics, Beijing Normal University, Beijing 100875, China
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8
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Ma YH, Chen JF, Sun CP, Dong H. Minimal energy cost to initialize a bit with tolerable error. Phys Rev E 2022; 106:034112. [PMID: 36266886 DOI: 10.1103/physreve.106.034112] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/09/2022] [Accepted: 08/25/2022] [Indexed: 06/16/2023]
Abstract
Landauer's principle imposes a fundamental limit on the energy cost to perfectly initialize a classical bit, which is only reached under the ideal operation with infinitely long time. The question on the cost in the practical operation for a bit has been raised under the constraint by the finiteness of operation time. We discover a raise-up of energy cost by L^{2}(ε)/τ from the Landaeur's limit (k_{B}Tln2) for a finite-time τ initialization of a bit with an error probability ε. The thermodynamic length L(ε) between the states before and after initializing in the parametric space increases monotonously as the error decreases. For example, in the constant dissipation coefficient (γ_{0}) case, the minimal additional cost is 0.997k_{B}T/(γ_{0}τ) for ε=1% and 1.288k_{B}T/(γ_{0}τ) for ε=0.1%. Furthermore, the optimal protocol to reach the bound of minimal energy cost is proposed for the bit initialization realized via a finite-time isothermal process.
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Affiliation(s)
- Yu-Han Ma
- Graduate School of China Academy of Engineering Physics, No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China
| | - Jin-Fu Chen
- Graduate School of China Academy of Engineering Physics, No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China
- Beijing Computational Science Research Center, Beijing 100193, China
- School of Physics, Peking University, Beijing, 100871, China
| | - C P Sun
- Graduate School of China Academy of Engineering Physics, No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China
- Beijing Computational Science Research Center, Beijing 100193, China
| | - Hui Dong
- Graduate School of China Academy of Engineering Physics, No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China
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9
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Ptaszyński K. Non-Markovian thermal operations boosting the performance of quantum heat engines. Phys Rev E 2022; 106:014114. [PMID: 35974499 DOI: 10.1103/physreve.106.014114] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/29/2022] [Accepted: 06/21/2022] [Indexed: 06/15/2023]
Abstract
It is investigated whether non-Markovianity, i.e., the memory effects resulting from the coupling of the system to its environment, can be beneficial for the performance of quantum heat engines. Specifically, two physical models are considered. The first one is a well-known single-qubit Otto engine; the non-Markovian behavior is there implemented by replacing standard thermalization strokes with so-called extremal thermal operations which cannot be realized without the memory effects. The second one is a three-stroke engine in which the cycle consists of two extremal thermal operations and a single qubit rotation. It is shown that the non-Markovian Otto engine can generate more work-per-cycle for a given efficiency than its Markovian counterpart, whereas performance of both setups is superior to the three-stroke engine. Furthermore, both the non-Markovian Otto engine and the three-stroke engine can reduce the work fluctuations in comparison with the Markovian Otto engine, with their relative advantage depending on the performance target. This demonstrates the beneficial influence of non-Markovianity on both the average performance and the stability of operation of quantum heat engines.
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Affiliation(s)
- Krzysztof Ptaszyński
- Institute of Molecular Physics, Polish Academy of Sciences, Mariana Smoluchowskiego 17, 60-179 Poznań, Poland
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10
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Frim AG, DeWeese MR. Geometric Bound on the Efficiency of Irreversible Thermodynamic Cycles. PHYSICAL REVIEW LETTERS 2022; 128:230601. [PMID: 35749204 DOI: 10.1103/physrevlett.128.230601] [Citation(s) in RCA: 9] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/03/2022] [Accepted: 05/17/2022] [Indexed: 06/15/2023]
Abstract
Stochastic thermodynamics has revolutionized our understanding of heat engines operating in finite time. Recently, numerous studies have considered the optimal operation of thermodynamic cycles acting as heat engines with a given profile in thermodynamic space (e.g., P-V space in classical thermodynamics), with a particular focus on the Carnot engine. In this work, we use the lens of thermodynamic geometry to explore the full space of thermodynamic cycles with continuously varying bath temperature in search of optimally shaped cycles acting in the slow-driving regime. We apply classical isoperimetric inequalities to derive a universal geometric bound on the efficiency of any irreversible thermodynamic cycle and explicitly construct efficient heat engines operating in finite time that nearly saturate this bound for a specific model system. Given the bound, these optimal cycles perform more efficiently than all other thermodynamic cycles operating as heat engines in finite time, including notable cycles, such as those of Carnot, Stirling, and Otto. For example, in comparison to recent experiments, this corresponds to orders of magnitude improvement in the efficiency of engines operating in certain time regimes. Our results suggest novel design principles for future mesoscopic heat engines and are ripe for experimental investigation.
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Affiliation(s)
- Adam G Frim
- Department of Physics, University of California, Berkeley, Berkeley, California, 94720
| | - Michael R DeWeese
- Department of Physics, University of California, Berkeley, Berkeley, California, 94720
- Redwood Center For Theoretical Neuroscience, University of California, Berkeley, Berkeley, California, 94720
- Helen Wills Neuroscience Institute, University of California, Berkeley, Berkeley, California, 94720
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11
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Entropy 2022 Best Paper Award. ENTROPY 2022; 24:e24050724. [PMID: 35626607 PMCID: PMC9141619 DOI: 10.3390/e24050724] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/17/2022] [Accepted: 05/18/2022] [Indexed: 02/01/2023]
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12
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Favretti M. Exponential Families with External Parameters. ENTROPY 2022; 24:e24050698. [PMID: 35626581 PMCID: PMC9142008 DOI: 10.3390/e24050698] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 04/25/2022] [Revised: 05/11/2022] [Accepted: 05/12/2022] [Indexed: 02/01/2023]
Abstract
In this paper we introduce a class of statistical models consisting of exponential families depending on additional parameters, called external parameters. The main source for these statistical models resides in the Maximum Entropy framework where we have thermal parameters, corresponding to the natural parameters of an exponential family, and mechanical parameters, here called external parameters. In the first part we we study the geometry of these models introducing a fibration of parameter space over external parameters. In the second part we investigate a class of evolution problems driven by a Fokker-Planck equation whose stationary distribution is an exponential family with external parameters. We discuss applications of these statistical models to thermodynamic length and isentropic evolution of thermodynamic systems and to a problem in the dynamic of quantitative traits in genetics.
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Affiliation(s)
- Marco Favretti
- Dipartimento di Matematica Tullio Levi-Civita, Università degli Studi di Padova, 35131 Padova, Italy
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13
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Future Perspectives of Finite-Time Thermodynamics. ENTROPY 2022; 24:e24050690. [PMID: 35626573 PMCID: PMC9141533 DOI: 10.3390/e24050690] [Citation(s) in RCA: 15] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Track Full Text] [Download PDF] [Subscribe] [Scholar Register] [Received: 05/06/2022] [Accepted: 05/07/2022] [Indexed: 12/10/2022]
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14
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Eglinton J, Brandner K. Geometric bounds on the power of adiabatic thermal machines. Phys Rev E 2022; 105:L052102. [PMID: 35706185 DOI: 10.1103/physreve.105.l052102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2022] [Accepted: 04/21/2022] [Indexed: 06/15/2023]
Abstract
We analyze the performance of slowly driven meso- and microscale refrigerators and heat engines that operate between two thermal baths with a small temperature difference. Using a general scaling argument, we show that such devices can work arbitrarily close to their Carnot limit only if heat leaks between the baths are fully suppressed. Their power output is then subject to a universal geometric bound that decays quadratically to zero at the Carnot limit. This bound can be asymptotically saturated in the quasistatic limit if the driving protocols are suitably optimized and the temperature difference between the baths goes to zero with the driving frequency. These results hold under generic conditions for any thermodynamically consistent dynamics admitting a well-defined adiabatic-response regime and a generalized Onsager symmetry. For illustration, we work out models of a qubit-refrigerator and a coherent charge pump operating as a cooling device.
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Affiliation(s)
- Joshua Eglinton
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom and Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - Kay Brandner
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom and Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
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15
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Frim AG, DeWeese MR. Optimal finite-time Brownian Carnot engine. Phys Rev E 2022; 105:L052103. [PMID: 35706186 DOI: 10.1103/physreve.105.l052103] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/14/2021] [Accepted: 04/29/2022] [Indexed: 06/15/2023]
Abstract
Recent advances in experimental control of colloidal systems have spurred a revolution in the production of mesoscale thermodynamic devices. Functional "textbook" engines, such as the Stirling and Carnot cycles, have been produced in colloidal systems where they operate far from equilibrium. Simultaneously, significant theoretical advances have been made in the design and analysis of such devices. Here, we use methods from thermodynamic geometry to characterize the optimal finite-time nonequilibrium cyclic operation of the parametric harmonic oscillator in contact with a time-varying heat bath with particular focus on the Brownian Carnot cycle. We derive the optimally parametrized Carnot cycle, along with two other new cycles and compare their dissipated energy, efficiency, and steady-state power production against each other and a previously tested experimental protocol for the Carnot cycle. We demonstrate a 20% improvement in dissipated energy over previous experimentally tested protocols and a ∼50% improvement under other conditions for one of our engines, whereas our final engine is more efficient and powerful than the others we considered. Our results provide the means for experimentally realizing optimal mesoscale heat engines.
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Affiliation(s)
- Adam G Frim
- Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA
| | - Michael R DeWeese
- Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA
- Redwood Center For Theoretical Neuroscience, University of California, Berkeley, Berkeley, California 94720, USA
- Helen Wills Neuroscience Institute, University of California, Berkeley, Berkeley, California 94720, USA
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16
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Van Vu T, Saito K. Thermodynamics of Precision in Markovian Open Quantum Dynamics. PHYSICAL REVIEW LETTERS 2022; 128:140602. [PMID: 35476476 DOI: 10.1103/physrevlett.128.140602] [Citation(s) in RCA: 15] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/12/2021] [Revised: 03/02/2022] [Accepted: 03/18/2022] [Indexed: 06/14/2023]
Abstract
The thermodynamic and kinetic uncertainty relations indicate trade-offs between the relative fluctuation of observables and thermodynamic quantities such as dissipation and dynamical activity. Although these relations have been well studied for classical systems, they remain largely unexplored in the quantum regime. In this Letter, we investigate such trade-off relations for Markovian open quantum systems whose underlying dynamics are quantum jumps, such as thermal processes and quantum measurement processes. Specifically, we derive finite-time lower bounds on the relative fluctuation of both dynamical observables and their first passage times for arbitrary initial states. The bounds imply that the precision of observables is constrained not only by thermodynamic quantities but also by quantum coherence. We find that the product of the relative fluctuation and entropy production or dynamical activity is enhanced by quantum coherence in a generic class of dissipative processes of systems with nondegenerate energy levels. Our findings provide insights into the survival of the classical uncertainty relations in quantum cases.
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Affiliation(s)
- Tan Van Vu
- Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
| | - Keiji Saito
- Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
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17
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Zhen YZ, Egloff D, Modi K, Dahlsten O. Inverse linear versus exponential scaling of work penalty in finite-time bit reset. Phys Rev E 2022; 105:044147. [PMID: 35590656 DOI: 10.1103/physreve.105.044147] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2021] [Accepted: 04/04/2022] [Indexed: 06/15/2023]
Abstract
A bit reset is a basic operation in irreversible computing. This costs work and dissipates energy in the computer, creating a limit on speeds and energy efficiency of future irreversible computers. It was recently shown by Zhen et al. [Phys. Rev. Lett. 127, 190602 (2021)0031-900710.1103/PhysRevLett.127.190602] that for a finite-time reset protocol, the additional work on top of the quasistatic protocol can always be minimized by considering a two-level system, and then be lower bounded through a thermodynamical speed limit. An important question is to understand under what protocol parameters, including a bit reset error and maximum energy shift, this penalty decreases exponentially vs inverse linearly in the protocol time. Here we provide several analytical results to address this question, as well as numerical simulations of specific examples of protocols.
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Affiliation(s)
- Yi-Zheng Zhen
- Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei 230026, China
| | - Dario Egloff
- Institute of Theoretical Physics, Technical University Dresden, D-01062 Dresden, Germany
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
| | - Kavan Modi
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - Oscar Dahlsten
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
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18
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Ye Z, Holubec V. Maximum efficiency of low-dissipation heat pumps at given heating load. Phys Rev E 2022; 105:024139. [PMID: 35291093 DOI: 10.1103/physreve.105.024139] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2021] [Accepted: 02/09/2022] [Indexed: 06/14/2023]
Abstract
We derive an analytical expression for maximum efficiency at fixed power of heat pumps operating along a finite-time reverse Carnot cycle under the low-dissipation assumption. The result is cumbersome, but it implies simple formulas for tight upper and lower bounds on the maximum efficiency and various analytically tractable approximations. In general, our results qualitatively agree with those obtained earlier for endoreversible heat pumps. In fact, we identify a special parameter regime when the performance of the low-dissipation and endoreversible devices is the same. At maximum power, heat pumps operate as work to heat converters with efficiency 1. Expressions for maximum efficiency at given power can be helpful in the identification of more practical operation regimes.
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Affiliation(s)
- Zhuolin Ye
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany
| | - Viktor Holubec
- Department of Macromolecular Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, CZ-180 00 Praha, Czech Republic
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19
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Barker D, Scandi M, Lehmann S, Thelander C, Dick KA, Perarnau-Llobet M, Maisi VF. Experimental Verification of the Work Fluctuation-Dissipation Relation for Information-to-Work Conversion. PHYSICAL REVIEW LETTERS 2022; 128:040602. [PMID: 35148140 DOI: 10.1103/physrevlett.128.040602] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2021] [Revised: 11/23/2021] [Accepted: 12/22/2021] [Indexed: 06/14/2023]
Abstract
We study experimentally work fluctuations in a Szilard engine that extracts work from information encoded as the occupancy of an electron level in a semiconductor quantum dot. We show that as the average work extracted per bit of information increases toward the Landauer limit k_{B}Tln2, the work fluctuations decrease in accordance with the work fluctuation-dissipation relation. We compare the results to a protocol without measurement and feedback and show that when no information is used, the work output and fluctuations vanish simultaneously, contrasting the information-to-energy conversion case where increasing amount of work is produced with decreasing fluctuations. Our study highlights the importance of fluctuations in the design of information-to-work conversion processes.
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Affiliation(s)
- David Barker
- NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
| | - Matteo Scandi
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona) 08860, Spain
| | - Sebastian Lehmann
- NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
| | - Claes Thelander
- NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
| | - Kimberly A Dick
- NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
- Centre for Analysis and Synthesis, Lund University, Box 124, 22100 Lund, Sweden
| | | | - Ville F Maisi
- NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
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20
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Van Vu T, Hasegawa Y. Lower Bound on Irreversibility in Thermal Relaxation of Open Quantum Systems. PHYSICAL REVIEW LETTERS 2021; 127:190601. [PMID: 34797124 DOI: 10.1103/physrevlett.127.190601] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/16/2021] [Accepted: 10/04/2021] [Indexed: 06/13/2023]
Abstract
We consider the thermal relaxation process of a quantum system attached to single or multiple reservoirs. Quantifying the degree of irreversibility by entropy production, we prove that the irreversibility of the thermal relaxation is lower bounded by a relative entropy between the unitarily evolved state and the final state. The bound characterizes the state discrepancy induced by the nonunitary dynamics, and thus reflects the dissipative nature of irreversibility. Intriguingly, the bound can be evaluated solely in terms of the initial and final states and the system Hamiltonian, thereby providing a feasible way to estimate entropy production without prior knowledge of the underlying coupling structure. This finding refines the second law of thermodynamics and reveals a universal feature of thermal relaxation processes.
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Affiliation(s)
- Tan Van Vu
- Department of Information and Communication Engineering, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
| | - Yoshihiko Hasegawa
- Department of Information and Communication Engineering, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
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21
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Zhen YZ, Egloff D, Modi K, Dahlsten O. Universal Bound on Energy Cost of Bit Reset in Finite Time. PHYSICAL REVIEW LETTERS 2021; 127:190602. [PMID: 34797137 DOI: 10.1103/physrevlett.127.190602] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2021] [Accepted: 09/28/2021] [Indexed: 06/13/2023]
Abstract
We consider how the energy cost of bit reset scales with the time duration of the protocol. Bit reset necessarily takes place in finite time, where there is an extra penalty on top of the quasistatic work cost derived by Landauer. This extra energy is dissipated as heat in the computer, inducing a fundamental limit on the speed of irreversible computers. We formulate a hardware-independent expression for this limit in the framework of stochastic processes. We derive a closed-form lower bound on the work penalty as a function of the time taken for the protocol and bit reset error. It holds for discrete as well as continuous systems, assuming only that the master equation respects detailed balance.
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Affiliation(s)
- Yi-Zheng Zhen
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
| | - Dario Egloff
- Institute of Theoretical Physics, Technische Universität Dresden, D-01062 Dresden, Germany
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
| | - Kavan Modi
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - Oscar Dahlsten
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
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22
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Chen JF, Sun CP, Dong H. Extrapolating the thermodynamic length with finite-time measurements. Phys Rev E 2021; 104:034117. [PMID: 34654162 DOI: 10.1103/physreve.104.034117] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2021] [Accepted: 08/27/2021] [Indexed: 11/07/2022]
Abstract
The thermodynamic length, though providing a lower bound for the excess work required in a finite-time thermodynamic process, is determined by the properties of the equilibrium states reached by the quasistatic process and is thus beyond the direct experimental measurement. We propose an experimental strategy to measure the thermodynamic length of an open classical or quantum system by extrapolating finite-time measurements. The current proposal enables the measurement of the thermodynamic length for a single control parameter without requiring extra effort to find the optimal control scheme, and is illustrated with examples of the quantum harmonic oscillator with tuning frequency and the classical ideal gas with changing volume. Such a strategy shall shed light on the experimental design of the lacking platforms to measure the thermodynamic length.
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Affiliation(s)
- Jin-Fu Chen
- Beijing Computational Science Research Center, Beijing 100193, China.,Graduate School of China Academy of Engineering Physics, No. 10 Xibeiwang East Road, Haidian District, Beijing 100193, China
| | - C P Sun
- Beijing Computational Science Research Center, Beijing 100193, China.,Graduate School of China Academy of Engineering Physics, No. 10 Xibeiwang East Road, Haidian District, Beijing 100193, China
| | - Hui Dong
- Graduate School of China Academy of Engineering Physics, No. 10 Xibeiwang East Road, Haidian District, Beijing 100193, China
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23
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Johal RS, Mehta V. Quantum Heat Engines with Complex Working Media, Complete Otto Cycles and Heuristics. ENTROPY (BASEL, SWITZERLAND) 2021; 23:1149. [PMID: 34573774 PMCID: PMC8468726 DOI: 10.3390/e23091149] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/20/2021] [Revised: 08/26/2021] [Accepted: 08/29/2021] [Indexed: 11/17/2022]
Abstract
Quantum thermal machines make use of non-classical thermodynamic resources, one of which include interactions between elements of the quantum working medium. In this paper, we examine the performance of a quasi-static quantum Otto engine based on two spins of arbitrary magnitudes subject to an external magnetic field and coupled via an isotropic Heisenberg exchange interaction. It has been shown earlier that the said interaction provides an enhancement of cycle efficiency, with an upper bound that is tighter than the Carnot efficiency. However, the necessary conditions governing engine performance and the relevant upper bound for efficiency are unknown for the general case of arbitrary spin magnitudes. By analyzing extreme case scenarios, we formulate heuristics to infer the necessary conditions for an engine with uncoupled as well as coupled spin model. These conditions lead us to a connection between performance of quantum heat engines and the notion of majorization. Furthermore, the study of complete Otto cycles inherent in the average cycle also yields interesting insights into the average performance.
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Affiliation(s)
- Ramandeep S. Johal
- Department of Physical Sciences, Indian Institute of Science Education and Research Mohali, Sector 81, S.A.S. Nagar, Manauli PO 140306, Punjab, India;
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24
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Miller HJD, Mohammady MH, Perarnau-Llobet M, Guarnieri G. Joint statistics of work and entropy production along quantum trajectories. Phys Rev E 2021; 103:052138. [PMID: 34134351 DOI: 10.1103/physreve.103.052138] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2020] [Accepted: 04/30/2021] [Indexed: 11/07/2022]
Abstract
In thermodynamics, entropy production and work quantify irreversibility and the consumption of useful energy, respectively, when a system is driven out of equilibrium. For quantum systems, these quantities can be identified at the stochastic level by unravelling the system's evolution in terms of quantum jump trajectories. We here derive a general formula for computing the joint statistics of work and entropy production in Markovian driven quantum systems, whose instantaneous steady states are of Gibbs form. If the driven system remains close to the instantaneous Gibbs state at all times, then we show that the corresponding two-variable cumulant generating function implies a joint detailed fluctuation theorem so long as detailed balance is satisfied. As a corollary, we derive a modified fluctuation-dissipation relation (FDR) for the entropy production alone, applicable to transitions between arbitrary steady states, and for systems that violate detailed balance. This FDR contains a term arising from genuinely quantum fluctuations, and extends an analogous relation from classical thermodynamics to the quantum regime.
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Affiliation(s)
- Harry J D Miller
- Department of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
| | - M Hamed Mohammady
- RCQI, Institute of Physics, Slovak Academy of Sciences, Bratislava 84511, Slovakia
| | | | - Giacomo Guarnieri
- School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland.,Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
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25
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Miller HJD, Mohammady MH, Perarnau-Llobet M, Guarnieri G. Thermodynamic Uncertainty Relation in Slowly Driven Quantum Heat Engines. PHYSICAL REVIEW LETTERS 2021; 126:210603. [PMID: 34114847 DOI: 10.1103/physrevlett.126.210603] [Citation(s) in RCA: 26] [Impact Index Per Article: 8.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/14/2020] [Accepted: 03/18/2021] [Indexed: 06/12/2023]
Abstract
Thermodynamic uncertainty relations express a trade-off between precision, defined as the noise-to-signal ratio of a generic current, and the amount of associated entropy production. These results have deep consequences for autonomous heat engines operating at steady state, imposing an upper bound for their efficiency in terms of the power yield and its fluctuations. In the present Letter we analyze a different class of heat engines, namely, those which are operating in the periodic slow-driving regime. We show that an alternative TUR is satisfied, which is less restrictive than that of steady-state engines: it allows for engines that produce finite power, with small power fluctuations, to operate close to reversibility. The bound further incorporates the effect of quantum fluctuations, which reduces engine efficiency relative to the average power and reliability. We finally illustrate our findings in the experimentally relevant model of a single-ion heat engine.
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Affiliation(s)
- Harry J D Miller
- Department of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
| | - M Hamed Mohammady
- RCQI, Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, Bratislava 84511, Slovakia
| | | | - Giacomo Guarnieri
- School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
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26
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Vallejo A, Romanelli A, Donangelo R. Qubit thermodynamics far from equilibrium: Two perspectives about the nature of heat and work in the quantum regime. Phys Rev E 2021; 103:042105. [PMID: 34005920 DOI: 10.1103/physreve.103.042105] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/04/2020] [Accepted: 03/15/2021] [Indexed: 11/07/2022]
Abstract
Considering an entropy-based division of energy transferred into heat and work, we develop an alternative theoretical framework for the thermodynamic analysis of two-level systems. When comparing these results with those obtained using the standard definitions of these quantities, we observe the appearance of a different term of work, which represents the energy cost of rotating the Bloch vector in the presence of the external field that defines the local Hamiltonian. Additionally, we obtain explicit expressions for the temperature, the heat capacity, and the internal entropy production of the system in both paradigms. In order to illustrate our findings we study, from both perspectives, matter-radiation interaction processes for two different systems.
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Affiliation(s)
- Andrés Vallejo
- Facultad de Ingeniería, Universidad de la República, 11300 Montevideo, Uruguay
| | - Alejandro Romanelli
- Facultad de Ingeniería, Universidad de la República, 11300 Montevideo, Uruguay
| | - Raúl Donangelo
- Facultad de Ingeniería, Universidad de la República, 11300 Montevideo, Uruguay
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27
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Izumida Y. Hierarchical Onsager symmetries in adiabatically driven linear irreversible heat engines. Phys Rev E 2021; 103:L050101. [PMID: 34134349 DOI: 10.1103/physreve.103.l050101] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2021] [Accepted: 04/19/2021] [Indexed: 11/07/2022]
Abstract
In existing linear response theories for adiabatically driven cyclic heat engines, Onsager symmetry is identified only phenomenologically, and a relation between global and local Onsager coefficients, defined over one cycle and at any instant of a cycle, respectively, is not derived. To address this limitation, we develop a linear response theory for the speed of adiabatically changing parameters and temperature differences in generic Gaussian heat engines obeying Fokker-Planck dynamics. We establish a hierarchical relationship between the global linear response relations, defined over one cycle of the heat engines, and the local ones, defined at any instant of the cycle. This yields a detailed expression for the global Onsager coefficients in terms of the local Onsager coefficients. Moreover, we derive an efficiency bound, which is tighter than the Carnot bound, for adiabatically driven linear irreversible heat engines based on the detailed global Onsager coefficients. Finally, we demonstrate the application of the theory using the simplest stochastic Brownian heat engine model.
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Affiliation(s)
- Yuki Izumida
- Department of Complexity Science and Engineering, Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa 277-8561, Japan
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28
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Ye Z, Holubec V. Maximum efficiency of absorption refrigerators at arbitrary cooling power. Phys Rev E 2021; 103:052125. [PMID: 34134287 DOI: 10.1103/physreve.103.052125] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2020] [Accepted: 04/30/2021] [Indexed: 06/12/2023]
Abstract
We consider absorption refrigerators consisting of simultaneously operating Carnot-type heat engine and refrigerator. Their maximum efficiency at given power (MEGP) is given by the product of MEGPs for the internal engine and refrigerator. The only subtlety of the derivation lies in the fact that the maximum cooling power of the absorption refrigerator is not limited just by the maximum power of the internal refrigerator, but, due to the first law, also by that of the internal engine. As a specific example, we consider the simultaneous absorption refrigerators composed of low-dissipation (LD) heat engines and refrigerators, for which the expressions for MEGPs are known. The derived expression for maximum efficiency implies bounds on the MEGP of LD absorption refrigerators. It also implies that a slight decrease in power of the absorption refrigerator from its maximum value results in a large nonlinear increase in efficiency, observed in heat engines, whenever the ratio of maximum powers of the internal engine and the refrigerator does not diverge. Otherwise, the increase in efficiency is linear as observed in LD refrigerators. Thus, in all practical situations, the efficiency of LD absorption refrigerators significantly increases when their cooling power is slightly decreased from its maximum.
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Affiliation(s)
- Zhuolin Ye
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany
| | - Viktor Holubec
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany
- Charles University, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách 2, CZ-180 00 Praha, Czech Republic
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29
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Costea M, Petrescu S, Feidt M, Dobre C, Borcila B. Optimization Modeling of Irreversible Carnot Engine from the Perspective of Combining Finite Speed and Finite Time Analysis. ENTROPY 2021; 23:e23050504. [PMID: 33922290 PMCID: PMC8146341 DOI: 10.3390/e23050504] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/19/2021] [Revised: 04/17/2021] [Accepted: 04/18/2021] [Indexed: 11/16/2022]
Abstract
An irreversible Carnot cycle engine operating as a closed system is modeled using the Direct Method and the First Law of Thermodynamics for processes with Finite Speed. Several models considering the effect on the engine performance of external and internal irreversibilities expressed as a function of the piston speed are presented. External irreversibilities are due to heat transfer at temperature gradient between the cycle and heat reservoirs, while internal ones are represented by pressure losses due to the finite speed of the piston and friction. Moreover, a method for optimizing the temperature of the cycle fluid with respect to the temperature of source and sink and the piston speed is provided. The optimization results predict distinct maximums for the thermal efficiency and power output, as well as different behavior of the entropy generation per cycle and per time. The results obtained in this optimization, which is based on piston speed, and the Curzon–Ahlborn optimization, which is based on time duration, are compared and are found to differ significantly. Correction have been proposed in order to include internal irreversibility in the externally irreversible Carnot cycle from Curzon–Ahlborn optimization, which would be equivalent to a unification attempt of the two optimization analyses.
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Affiliation(s)
- Monica Costea
- Department of Engineering Thermodynamics, University POLITEHNICA of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania; (S.P.); (C.D.); (B.B.)
- Correspondence: ; Tel.: +40-021-402-9339
| | - Stoian Petrescu
- Department of Engineering Thermodynamics, University POLITEHNICA of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania; (S.P.); (C.D.); (B.B.)
| | - Michel Feidt
- Laboratory of Energetics, Theoretical and Applied Mechanics (LEMTA), URA CNRS 7563, University of Lorraine, 54518 Vandoeuvre-lès-Nancy, France;
| | - Catalina Dobre
- Department of Engineering Thermodynamics, University POLITEHNICA of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania; (S.P.); (C.D.); (B.B.)
| | - Bogdan Borcila
- Department of Engineering Thermodynamics, University POLITEHNICA of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania; (S.P.); (C.D.); (B.B.)
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30
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Dann R, Kosloff R, Salamon P. Quantum Finite-Time Thermodynamics: Insight from a Single Qubit Engine. ENTROPY (BASEL, SWITZERLAND) 2020; 22:E1255. [PMID: 33287023 PMCID: PMC7712823 DOI: 10.3390/e22111255] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/06/2020] [Revised: 10/29/2020] [Accepted: 11/01/2020] [Indexed: 02/01/2023]
Abstract
Incorporating time into thermodynamics allows for addressing the tradeoff between efficiency and power. A qubit engine serves as a toy model in order to study this tradeoff from first principles, based on the quantum theory of open systems. We study the quantum origin of irreversibility, originating from heat transport, quantum friction, and thermalization in the presence of external driving. We construct various finite-time engine cycles that are based on the Otto and Carnot templates. Our analysis highlights the role of coherence and the quantum origin of entropy production.
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Affiliation(s)
- Roie Dann
- The Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel;
| | - Ronnie Kosloff
- The Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel;
| | - Peter Salamon
- Department of Mathematics and Statistics, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182-7720, USA;
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