A unified thermodynamic formalism describing the efficiency of learning is proposed. First, we derive an inequality, which is more strength than Clausius's inequality, revealing the lower bound of the entropy-production rate of a subsystem. Second, the inequality is transformed to determine the general upper limit for the efficiency of learning. In particular, we exemplify the bound of the efficiency in nonequilibrium quantum-dot systems and networks of living cells. The framework provides a fundamental trade-off relationship between energy and information inheriting in stochastic thermodynamic processes.
We analyze the efficiency of thermal engines (either quantum or classical) working with a single heat reservoir like atmosphere. The engine first gets an energy intake, which can be done in arbitrary non-equilibrium way e.g. combustion of fuel. Then the engine performs the work and returns to the initial state. We distinguish two general classes of engines where the working body first equilibrates within itself and then performs the work (ergodic engine) or when it performs the work before equilibrating (non-ergodic engine). We show that in both cases the second law of thermodynamics limits th…
We present a detailed study of a three-level quantum heat engine operating at maximum efficient power function, a trade-off objective function defined by the product of the efficiency and power output of the engine. First, for near equilibrium conditions, we find general expression for the efficiency and establish universal nature of efficiency at maximum power and maximum efficient power. Then in the high temperature limit, optimizing with respect to one parameter while constraining the other one, we obtain the lower and upper bounds on the efficiency for both strong as well as weak matter-fi…
A long standing open problem whether a heat engine with finite power achieves the Carnot efficiency is investigated. We rigorously prove a general trade-off inequality on thermodynamic efficiency and time interval of a cyclic process with quantum heat engines. In a first step, employing the Lieb-Robinson bound we establish an inequality on the change in a local observable caused by an operation far from support of the local observable. This inequality provides a rigorous characterization of the following intuitive picture that most of the energy released from the engine to the cold bath remain…
We derive an efficiency bound for continuous quantum heat engines absorbing heat from squeezed thermal reservoirs. Our approach relies on a full-counting statistics description of nonequilibrium transport and it is not limited to the framework of irreversible thermodynamics. Our result, a generalized Carnot efficiency bound, is valid beyond the small squeezing and high temperature limit. Our findings are embodied in a prototype three-terminal quantum photoelectric engine where a qubit converts heat absorbed from a squeezed thermal reservoir into electrical power. We demonstrate that in the qua…
The second law of thermodynamics constrains that the efficiency of heat engines, classical or quantum, cannot be greater than the universal Carnot efficiency. We discover another bound for the efficiency of a quantum Otto heat engine consisting of a harmonic oscillator. Dynamics of the engine is governed by the Lindblad equation for the density matrix, which is mapped to the Fokker-Planck equation for the quasi-probability distribution. Applying stochastic thermodynamics to the Fokker-Planck equation system, we obtain the $\hbar$-dependent quantum mechanical bound for the efficiency. It turns …
We derive a unified quantum theory of coherent and incoherent energy transfer between two atoms (donor and acceptor) valid in arbitrary Markovian nanophotonic environments. Our theory predicts a fundamental bound $η_{max} = \frac{γ_a}{γ_d + γ_a}$ for energy transfer efficiency arising from the spontaneous emission rates $γ_{d}$ and $γ_a$ of the donor and acceptor. We propose the control of the acceptor spontaneous emission rate as a new design principle for enhancing energy transfer efficiency. We predict an experiment using mirrors to enhance the efficiency bound by exploiting the dipole orie…
Power and efficiency of heat engines are two conflicting objectives, and a tight efficiency bound is expected to give insights on the fundamental properties of the power-efficiency tradeoff. Here we derive an upper bound on the efficiency of steady-state heat engines, which incorporates higher-order fluctuations of the power. In a prototypical model of nonlinear nanostructured thermoelectrics, we show that the obtained bound is tighter than a well-established efficiency bound based on the thermodynamic uncertainty relation, demonstrating that the higher-order terms have rich information about …
We examine the efficiency of an effective two-terminal thermoelectric device under broken time-reversal symmetry. The setup is derived from a three-terminal thermoelectric device comprising a thermal terminal and two electronic contacts, under a magnetic field. We find that breaking time-reversal symmetry in the presence of the inelastic electron-phonon processes can significantly enhance the figure of merit for delivering electric power by supplying heat from a phonon bath, beyond the one for producing the electric power by investing thermal power from the electronic heat current. The efficie…
We show that for systems with broken time-reversal symmetry the maximum efficiency and the efficiency at maximum power are both determined by two parameters: a "figure of merit" and an asymmetry parameter. In contrast to the time-symmetric case, the figure of merit is bounded from above; nevertheless the Carnot efficiency can be reached at lower and lower values of the figure of merit and far from the so-called strong coupling condition as the asymmetry parameter increases. Moreover, the Curzon-Ahlborn limit for efficiency at maximum power can be overcome within linear response. Finally, alway…
We derive a universal bound on generalized currents in Langevin systems in terms of the mean-square fluctuations of the current and the total entropy production. This bound generalizes a relation previously found by Barato et al. to arbitrary times and transient states. Using the bound, we define a new efficiency for stochastic transport, which measures how close a given system comes to saturating the bound. The existence of such a bounded efficiency implies that stochastic transport is unavoidably accompanied by a fluctuations and dissipation, which cannot be reduced arbitrarily. We apply the…
We investigate maximum efficiency at a given power for low-dissipation heat engines. Close to maximum power, the maximum gain in efficiency scales as a square root of relative loss in power and this scaling is universal for a broad class of systems. For the low-dissipation engines, we calculate the maximum gain in efficiency for an arbitrary fixed power. We show that the engines working close to maximum power can operate at considerably larger efficiency compared to the efficiency at maximum power. Furthermore, we introduce universal bounds on maximum efficiency at a given power for low-dissip…
From thermodynamics point of view, in this era of aiming at energy conservation and sustainability, we need to develop more accurate ways to design thermal power, cooling and heat pump cycles. It has been the general practice in thermodynamic analysis of cycles to use the Carnot efficiency and Carnot coefficient of performance which are the highest upper bound to efficiency and COP of cycles. In the present report through the application of the second law of thermodynamics for irreversible processes, which results in the general inequality relation for the entropy production, we have introduce…
We study a quantum thermal engine model for which the heat transfer law is determined by Einstein's theory of radiation. The working substance of the quantum engine is assumed to be a two-level quantum systems of which the constituent particles obey Maxwell-Boltzmann(M.B.), Fermi-Dirac(F.D.) or Bose-einstein(B.E.) distributions respectively at equilibrium. The thermal efficiency and its bounds at maximum power of these models are derived and discussed in the long and short thermal contact time limits. The similarity and difference between these models are discussed. We also compare the efficie…
We study a thermal engine model for which Newton's cooling law is obeyed during heat transfer processes. The thermal efficiency and its bounds at maximum output power are derived and discussed. This model, though quite simple, can be applied not only to Carnot engines but also to four other types of engines. For the long thermal contact time limit, new bounds, tighter than what were known before, are obtained. In this case,this model can simulate Otto, Joule Brayton, Diesel, and Atkinson engines. While in the short contact time limit, which corresponds to the Carnot cycle, the same efficiency …
We review theoretical approaches to analyzing efficiency of steady state heat to work conversion which is crucial in the timely problem of optimizing efficiency of small-scale heat engines and refrigerators. A rather abstract perspective of non-equilibrium statistical mechanics and dynamical system's theory is taken to view at this very practical problem. Several recently discovered general mechanisms of optimizing the figure of merit of thermoelectric efficiency are discussed, also in connection to breaking time-reversal symmetry of the microscopic equations of motion. Applications of these t…
We study the optimal exergy efficiency and power for thermodynamic systems with Onsager-type "current-force" relationship describing the linear-response to external influences. We derive, in simple analytic forms, the maximum efficiency and optimal efficiency for maximum power for a thermodynamic machine described by a $N\times N$ symmetric Onsager matrix with arbitrary $N$. The figure of merit is expressed in terms of the largest eigenvalue of the "coupling matrix" which is solely determined by the Onsager matrix. Some simple but general relationships between the power and efficiency at the c…
Thermoelectric transport involving an arbitrary number of terminals is discussed in the presence of a magnetic field breaking time-reversal symmetry within the linear response regime using the Landauer-Büttiker formalism. We derive a universal bound on the Onsager coefficients that depends only on the number of terminals. This bound implies bounds on the efficiency and on efficiency at maximum power for heat engines and refrigerators. For isothermal engines pumping particles and for absorption refrigerators these bounds become independent even of the number of terminals. On a technical level, …
The thermodynamic uncertainty relation provides an inequality relating any mean current, the associated dispersion and the entropy production rate for arbitrary non-equilibrium steady states. Applying it here to a general model of a molecular motor running against an external force or torque, we show that the thermodynamic efficiency of such motors is universally bounded by an expression involving only experimentally accessible quantities. For motors pulling cargo through a viscous fluid, a universal bound for the corresponding Stokes efficiency follows as a variant. A similar result holds if …
For thermoelectric transport in the presence of a magnetic field that breaks time-reversal symmetry, a strong bound on the Onsager coefficients is derived within a general set-up using three terminals. Asymmetric Onsager coefficients lead to a maximum efficiency substantially smaller than the Carnot efficiency reaching only η_C/4 in the limit of strong asymmetry. Related bounds are derived for efficiency at maximum power, which can become larger than the Curzon-Ahlborn value η_C/2, and for a cooling device. Our approach reveals that in the presence of reversible currents the standard analysis …
Universal properties of the statistics of stochastic efficiency for mesoscopic time-reversal symmetry broken energy transducers are revealed in the Gaussian approximation. We also discuss how the second law of thermodynamics restricts the statistics of stochastic efficiency. The tight-coupling (reversible) limit becomes unfavorable, characterized by an infinitely broad distribution of efficiency at {\em all times}, when time-reversal symmetry breaking leads to an asymmetric Onsager response matrix. The underlying physics is demonstrated through the integer quantum Hall effect and further elabo…
The efficiency at maximum power (EMP) of irreversible Carnot-like heat engines is investigated based on the weak endoreversible assumption and the phenomenologically irreversible thermodynamics. It is found that the weak endoreversible assumption can reduce to the conventional one for the heat engines working at maximum power. Carnot-like heat engines are classified into three types (linear, superlinear, and sublinear) according to different characteristics of constitutive relations between the heat transfer rate and the thermodynamic force. The EMPs of Carnot-like heat engines are proved to b…
We analyse the process of conversion of near-field thermal radiation into usable work by considering the radiation emitted between two planar sources supporting surface phonon-polaritons. The maximum work flux that can be extracted from the radiation is obtained taking into account that the spectral flux of modes is mainly dominated by these surface modes. The thermodynamic efficiencies are discussed and an upper bound for the first law efficiency is obtained for this process.
The widely debated feasibility of thermodynamic machines achieving Carnot efficiency at finite power has been convincingly dismissed. Yet, the common wisdom that efficiency can only be optimal in the limit of infinitely-slow processes overlooks the dual scenario of infinitely-fast processes. We corroborate that efficient engines at divergent power output are not theoretically impossible, framing our claims within the theory of Stochastic Thermodynamics. We inspect the case of an electronic quantum dot coupled to three particle reservoirs to illustrate the physical rationale.
We discuss the efficiency of a heat engine operating in a nonequilibrium steady state maintained by two heat reservoirs. Within the general framework of linear irreversible thermodynamics we derive a universal upper bound on the efficiency of the engine operating at arbitrary fixed power. Furthermore, we show that a slight decrease of the power below its maximal value can lead to a significant gain in efficiency. The presented analysis yields the exact expression for this gain and the corresponding upper bound.
We derive upper and lower bounds for the efficiency of an isothermal molecular machine operating at maximum power. The upper bound is reached when the activated state is close to the fueling or reactant state (Eyring-like), while the lower bound is reached when the activated state is close to the product state (Kramers-like).
The Carnot-like heat engines are classified into three types (normal-, sub- and super-dissipative) according to relations between the minimum irreversible entropy production in the "isothermal" processes and the time for completing those processes. The efficiencies at maximum power of normal-, sub- and super-dissipative Carnot-like heat engines are proved to be bounded between $η_C/2$ and $η_C/(2-η_C)$, $η_C /2$ and $η_C$, 0 and $η_C/(2-η_C)$, respectively. These bounds are also shared by linear, sub- and super-linear irreversible Carnot-like engines [Tu and Wang, Europhys. Lett. 98, 40001 (20…
We show the validity of some results of finite-time thermodynamics, also within the quasi-static framework of classical thermodynamics. First, we consider the efficiency at maximum work (EMW) from finite source and sink modelled as identical thermodynamic systems. The near-equilibrium regime is characterized by expanding the internal energy upto second order (i.e. upto linear response) in the difference of initial entropies of the source and the sink. It is shown that the efficiency is given by a universal expression $2 η_C / (4-η_C)$, where $η_C$ is the Carnot efficiency. Then, different size…
The problem of inference is applied to the process of work extraction from two constant heat capacity reservoirs, when the thermodynamic coordinates of the process are not fully specified. The information that is lacking, includes both the specific value of a temperature as well as the label of the reservoir to which it is assigned. The estimates for thermal efficiency reveal that uncertainty regarding the exact labels, reduces the maximal efficiency below the Carnot value, its minimum value being the well known Curzon-Ahlborn value. We also make an average estimate of the efficiency {\it befo…
Aiming to explore physical limits of wind turbines, we develop a model for determining the work extractable from a compressible fluid flow. The model employs conservation of mass, energy and entropy and leads to a universal bound for the efficiency of the work extractable from kinetic energy. The bound is reached for a sufficiently slow, weakly-forced quasi-one-dimensional, dissipationless flow. In several respects the bound is similar to the Carnot limit for the efficiency of heat-engines. More generally, we show that the maximum work-extraction demands a contribution from the enthalpy, and i…
We propose a generalized model of a heat engine and calculate the minimum and maximum bounds on the efficiency at maximum power. We obtain a universal form of generalized extreme bounds on the efficiency at maximum power. Our model unifies the bounds on the efficiency and the universality features are observed for various heat engine models. Even though our model is a direct generalization of low-dissipation heat engines, the bounds on the efficiency obtained at a single target function capture those observed in the actual power plants working at different dissipation levels.
