Optimal operating protocol to achieve efficiency at maximum power of heat engines

Temperature fluctuations in mesoscopic systems

Temperature is a fundamental concept in thermodynamics. In macroscopic thermodynamics, systems possess their own intrinsic temperature which equals the reservoir temperature when they equilibrate. In stochastic thermodynamics for simple systems at the microscopic level, thermodynamic quantities other than temperature (a deterministic parameter of the reservoir) are stochastic. To bridge the disparity in the perspectives about temperature between the micro- and macroregimes, we assign a generic mesoscopic N-body system an intrinsic fluctuating temperature T in this work. We simplify the complicated dynamics of numerous particles to one stochastic differential equation with respect to T, where the noise term accounts for finite-size effects arising from random energy transfer between the system and the reservoir. Our analysis reveals that these fluctuations make the extensive quantities (in the thermodynamic limit) deviate from being extensive. Moreover, we derive finite-size corrections, characterized by heat capacity of the system, to the Jarzynski equality. A possible violation of the principle of maximum work that scales with N^{-1} is also discussed. Additionally, we examine the impact of temperature fluctuations in a finite-size Carnot engine. We show that irreversible entropy production resulting from the temperature fluctuations of the working substance diminishes the average efficiency of the cycle as η_{C}-〈η〉∼N^{-1}, highlighting the unattainability of the Carnot efficiency η_{C} for mesoscopic heat engines even under the quasistatic limit. Our general framework paves the way for further exploration of nonequilibrium thermodynamics and the corresponding finite-size effects in a mesoscopic regime.

Geodesic path for the optimal nonequilibrium transition: Momentum-independent protocol

Accelerating controlled thermodynamic processes requires an auxiliary Hamiltonian to steer the system into instantaneous equilibrium states. An extra energy cost is inevitably needed in such finite-time operation. We recently developed a geodesic approach to minimize such energy cost for the shortcut to isothermal process. The auxiliary control typically contains momentum-dependent terms, which are hard to be experimentally implemented due to the requirement of constantly monitoring the speed. In this work, we employ a variational auxiliary control without the momentum-dependent force to approximate the exact control. Following the geometric approach, we obtain the optimal control protocol with variational minimum energy cost. We demonstrate the construction of such protocol via an example of Brownian motion with a controllable harmonic potential.

Minimal energy cost to initialize a bit with tolerable error

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.

Microscopic theory of the Curzon-Ahlborn heat engine based on a Brownian particle

The Curzon-Ahlborn (CA) efficiency, as the efficiency at the maximum power (EMP) of the endoreversible Carnot engine, has significant impact on finite-time thermodynamics. However, the CA engine is based on many assumptions. In the past few decades, although a lot of efforts have been made, a microscopic theory of the CA engine is still lacking. By adopting the method of the stochastic differential equation of energy, we formulate a microscopic theory of the CA engine realized with a highly underdamped Brownian particle in a class of nonharmonic potentials. This theory gives microscopic interpretation of all assumptions made by Curzon and Ahlborn. In other words, we find a microscopic counterpart of the CA engine in stochastic thermodynamics. Also, based on this theory, we derive the explicit expression of the protocol associated with the maximum power for any given efficiency, and we obtain analytical results of the power and the efficiency statistics for the Brownian CA engine. Our research brings new perspectives to experimental studies of finite-time microscopic heat engines featured with fluctuations.

Non-Markovian thermal operations boosting the performance of quantum heat engines

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.

Geodesic Path for the Minimal Energy Cost in Shortcuts to Isothermality

Shortcuts to isothermality are driving strategies to steer the system to its equilibrium states within finite time, and enable evaluating the impact of a control promptly. Finding the optimal scheme to minimize the energy cost is of critical importance in applications of this strategy in pharmaceutical drug tests, biological selection, and quantum computation. We prove the equivalence between designing the optimal scheme and finding the geodesic path in the space of control parameters. Such equivalence allows a systematic and universal approach to find the optimal control to reduce the energy cost. We demonstrate the current method with examples of a Brownian particle trapped in controllable harmonic potentials.

Geometric Bound on the Efficiency of Irreversible Thermodynamic Cycles

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.

Optimal finite-time Brownian Carnot engine

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.

Optimizing thermodynamic cycles with two finite-sized reservoirs

We study the nonequilibrium thermodynamics of a heat engine operating between two finite-sized reservoirs with well-defined temperatures. Within the linear response regime, it is found that the uniform temperature of the two reservoirs at final time τ is bounded from below by the entropy production σ_{min}∝1/τ. We discover a general power-efficiency tradeoff depending on the ratio of heat capacities (γ) of the reservoirs for the engine, and a universal efficiency at maximum average power of the engine for arbitrary γ is obtained. For practical purposes, the operation protocol of an ideal gas heat engine to achieve the optimal performance associated with σ_{min} is demonstrated. Our findings can be used to develop a general optimization scenario for thermodynamic cycles with finite-sized reservoirs in real-world circumstances.

Extrapolating the thermodynamic length with finite-time measurements

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.

Work fluctuations due to partial thermalizations in two-level systems

We study work extraction processes mediated by finite-time interactions with an ambient bath-partial thermalizations-as continuous-time Markov processes for two-level systems. Such a stochastic process results in fluctuations in the amount of work that can be extracted and is characterized by the rate at which the system parameters are driven in addition to the rate of thermalization with the bath. We analyze the distribution of work for the case in which the energy gap of a two-level system is driven at a constant rate. We derive analytic expressions for average work and a lower bound for the variance of work showing that such processes cannot be fluctuation-free in general. We also observe that an upper bound for the Monte Carlo estimate of the variance of work can be obtained using Jarzynski's fluctuation-dissipation relation for systems initially in equilibrium. Finally, we analyze work extraction cycles by modifying the Carnot cycle, incorporating processes involving partial thermalizations, and we obtain efficiency at maximum power for such finite-time work extraction cycles under different sets of constraints.

Simulating Finite-Time Isothermal Processes with Superconducting Quantum Circuits

Finite-time isothermal processes are ubiquitous in quantum-heat-engine cycles, yet complicated due to the coexistence of the changing Hamiltonian and the interaction with the thermal bath. Such complexity prevents classical thermodynamic measurements of a performed work. In this paper, the isothermal process is decomposed into piecewise adiabatic and isochoric processes to measure the performed work as the internal energy change in adiabatic processes. The piecewise control scheme allows the direct simulation of the whole process on a universal quantum computer, which provides a new experimental platform to study quantum thermodynamics. We implement the simulation on ibmqx2 to show the 1/τ scaling of the extra work in finite-time isothermal processes.

Experimental Test of the 1/τ-Scaling Entropy Generation in Finite-Time Thermodynamics

The finite-time dynamics, apart from its fundamental importance in nonequilibrium thermodynamics, is of great significance in designing heat engine cycles. We build an experimental apparatus to test the predicted long-time 1/τ scaling of the irreversible entropy generation in the finite-time (τ) thermodynamic process by compressing dry air in a temperature-controlled water bath. We present the first direct experimental validation of the scaling, utilized in many finite-time thermodynamic models at the long-time regime. The experimental data also demonstrate a clear deviation from the scaling at the short-time regime. We show the optimal control scheme to minimize the irreversible entropy generation in finite-time process. Such optimization shall bring new insight to the practical design of heat engine cycles.

Quantum Finite-Time Thermodynamics: Insight from a Single Qubit Engine

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.

Effect of Finite-Size Heat Source's Heat Capacity on the Efficiency of Heat Engine

Heat engines used to output useful work have important practical significance, which, in general, operate between heat baths of infinite size and constant temperature. In this paper, we study the efficiency of a heat engine operating between two finite-size heat sources with initial temperature difference. The total output work of such heat engine is limited due to the finite heat capacity of the sources. We firstly investigate the effects of different heat capacity characteristics of the sources on the heat engine's efficiency at maximum work (EMW) in the quasi-static limit. Moreover, it is found that the efficiency of the engine operating in finite-time with maximum power of each cycle is achieved follows a simple universality as η=ηC/4+OηC2, where ηC is the Carnot efficiency determined by the initial temperature of the sources. Remarkably, when the heat capacity of the heat source is negative, such as the black holes, we show that the heat engine efficiency during the operation can surpass the Carnot efficiency determined by the initial temperature of the heat sources. It is further argued that the heat engine between two black holes with vanishing initial temperature difference can be driven by the energy fluctuation. The corresponding EMW is proved to be ηMW=2-2.

Optimal Cycles for Low-Dissipation Heat Engines

We consider the optimization of a finite-time Carnot engine characterized by small dissipations. We bound the power with a simple inequality and show that the optimal strategy is to perform small cycles around a given working point, which can be, thus, chosen optimally. Remarkably, this optimal point is independent of the figure of merit combining power and efficiency that is being maximized. Furthermore, for a general class of dissipative dynamics the maximal power output becomes proportional to the heat capacity of the working substance. Since the heat capacity can scale supraextensively with the number of constituents of the engine, this enables us to design optimal many-body Carnot engines reaching maximum efficiency at finite power per constituent in the thermodynamic limit.

Finite-time quantum Otto engine: Surpassing the quasistatic efficiency due to friction

In finite-time quantum heat engines, some work is consumed to drive a working fluid accompanying coherence, which is called "friction." To understand the role of friction in quantum thermodynamics, we present a couple of finite-time quantum Otto cycles with two different baths: Agarwal versus Lindbladian. We solve them exactly and compare the performance of the Agarwal engine with that of the Lindbladian engine. In particular, we find remarkable and counterintuitive results that the performance of the Agarwal engine due to friction can be much higher than that in the quasistatic limit with the Otto efficiency, and the power of the Lindbladian engine can be nonzero in the short-time limit. Based on additional numerical calculations of these outcomes, we discuss possible origins of such differences between two engines and reveal them. Our results imply that, even with an equilibrium bath, a nonequilibrium working fluid brings on the higher performance than what an equilibrium working fluid does.

Boosting the performance of quantum Otto heat engines

To optimize the performance of a heat engine in a finite-time cycle, it is important to understand the finite-time effect of thermodynamic processes. Previously, we have shown that extra work is needed to complete a quantum adiabatic process in finite time, and proved that the extra work follows a C/τ^{2} scaling for long control time τ. There the oscillating part of the extra work is neglected due to the complex energy-level structure of the particular quantum system. However, such oscillation of the extra work cannot be neglected in some quantum systems with simple energy-level structure, e.g., the two-level system or the quantum harmonic oscillator. In this paper, we build the finite-time quantum Otto engine on these simple systems, and find that the oscillating extra work leads to a jagged edge in the constraint relation between the output power and the efficiency. By optimizing the control time of the adiabatic processes, the oscillation in the extra work is utilized to enhance the maximum power and the efficiency. We further design special control schemes with the zero extra work at the specific control time. Compared to the linear control scheme, these special control schemes of the finite-time adiabatic process improve the maximum power and the efficiency of the finite-time Otto engine.