The unlikely Carnot efficiency

Work and efficiency fluctuations in a quantum Otto cycle with idle levels

We study the performance of a quantum Otto heat engine with two spins coupled by a Heisenberg interaction, taking into account not only the mean values of work and efficiency but also their fluctuations. We first show that, for this system, the output work and its fluctuations are directly related to the magnetization and magnetic susceptibility of the system at equilibrium with either heat bath. We analyze the regions where the work extraction can be done with low relative fluctuation for a given range of temperatures, while still achieving an efficiency higher than that of a single spin system heat engine. In particular, we find that, due to the presence of "idle" levels, an increase in the interspin coupling can either increase or decrease fluctuations, depending on the other parameters. In all cases, however, we find that the relative fluctuations in work or efficiency remain large, implying that this microscopic engine is not very reliable as a source of work.

Statistics of quantum heat in the Caldeira-Leggett model

Nonequilibrium fluctuation relation lies at the heart of the quantum thermodynamics. Many previous studies have demonstrated that the heat exchange between a quantum system and a thermal bath initially prepared in their own Gibbs states at different temperatures obeys the famous Jarzynski-Wójcik fluctuation theorem. However, this conclusion is obtained under the assumption of Born-Markovian approximation. In this paper, going beyond the Born-Markovian limitation, we investigate the statistics of quantum heat in an exactly non-Markovian relaxation process described by the well-known Caldeira-Leggett model. It is revealed that the Jarzynski-Wójcik fluctuation theorem breaks down in the strongly non-Markovian regime. Moreover, we find the steady-state quantum heat within the non-Markovian framework can be widely tunable by using the quantum reservoir-engineering technique. These results are sharply contrary to the common Born-Markovian predictions. Our results presented in this paper may update the understanding of the quantum thermodynamics in strongly coupled and low-temperature systems. Moreover, the controllable heat may have some potential applications in improving the performance of a quantum heat engine.

Efficiency of a microscopic heat engine subjected to stochastic resetting

We explore the thermodynamics of stochastic heat engines in the presence of stochastic resetting. The setup comprises an engine whose working substance is a Brownian particle undergoing overdamped Langevin dynamics in a harmonic potential with a time-dependent stiffness, with the dynamics interrupted at random times with a resetting to a fixed location. The effect of resetting to the potential minimum is shown to enhance the efficiency of the engine, while the output work is shown to have a nonmonotonic dependence on the rate of resetting. The resetting events are found to drive the system out of the linear response regime, even for small differences in the bath temperatures. Shifting the reset point from the potential minimum is observed to reduce the engine efficiency. The experimental setup for the realization of such an engine is briefly discussed.

Harnessing Viscoelasticity to Suppress Irreversibility Buildup in a Colloidal Stirling Engine

Typically, the rate at which a heat engine can produce useful work is constrained by the buildup of irreversibility with increasing operating speed. Here, using a recently developed reservoir engineering technique, we designed and quantified the performance of a colloidal Stirling engine operating in a viscoelastic bath. While the bath acts like a viscous fluid in the quasistatic limit, and the engine's performance agrees with equilibrium predictions, on reducing the cycle time to the bath's structural relaxation time, the increasingly elastic response of the bath aids suppress the buildup of irreversibility. We show that the elastic energy stored during the isothermal compression step of the Stirling cycle facilitates quick equilibration in the isothermal expansion step. This results in equilibriumlike efficiencies even for cycle times shorter than the equilibration time of the colloidal particle.

Thermodynamic uncertainty relations in mesoscopic devices

We investigate the thermodynamic uncertainty relations (TURs) in mesoscopic devices for all universal symmetry classes of Wigner-Dyson and Dirac (chiral). The observables of interest include the TUR (MS), which is defined in terms of the ratio between the mean noise and mean conductance, as well as a new TUR (R) proposed in this article, which is based on the ensemble mean of the noise-to-conductance ratio. A detailed study is made on the quantum interference corrections associated with the TURs. We also analyze the influence of orbital and sublattice/chiral degrees of freedom for the validity of the observables in these chaotic mesoscopic billiards. Our investigation is based on the concatenation between the Landauer-Büttiker theory, the Mahaux-Wendeinmüller theory, and the TURs. We simulate the universal mesoscopic chaotic quantum dots using the random-matrix theory and compare our numerical results with the pertinent experimental data. The results were obtained for a different number of channels and tunneling rates that vary from the opaque to the ideal regime and, in all cases, demonstrate a clear phenomenological distinction between the TURs. In particular, the opaque regime engenders remarkable differences between the observables, even in the semiclassical regime, which characterizes a clear violation of the central limit theorem. Furthermore, we show that the phenomenology of the quantum interference corrections is strikingly robust, surprisingly exhibiting an order of magnitude greater than the supposedly leading semiclassical term for the TUR (R).

Nonreciprocal forces enable cold-to-hot heat transfer between nanoparticles

We study the heat transfer between two nanoparticles held at different temperatures that interact through nonreciprocal forces, by combining molecular dynamics simulations with stochastic thermodynamics. Our simulations reveal that it is possible to construct nano refrigerators that generate a net heat transfer from a cold to a hot reservoir at the expense of power exerted by the nonreciprocal forces. Applying concepts from stochastic thermodynamics to a minimal underdamped Langevin model, we derive exact analytical expressions predictions for the fluctuations of work, heat, and efficiency, which reproduce thermodynamic quantities extracted from the molecular dynamics simulations. The theory only involves a single unknown parameter, namely an effective friction coefficient, which we estimate fitting the results of the molecular dynamics simulation to our theoretical predictions. Using this framework, we also establish design principles which identify the minimal amount of entropy production that is needed to achieve a certain amount of uncertainty in the power fluctuations of our nano refrigerator. Taken together, our results shed light on how the direction and fluctuations of heat flows in natural and artificial nano machines can be accurately quantified and controlled by using nonreciprocal forces.

Generalized continuous Maxwell demons

We introduce a family of generalized continuous Maxwell demons (GCMDs) operating on idealized single-bit equilibrium devices that combine the single-measurement Szilard and the repeated measurements of the continuous Maxwell demon protocols. We derive the cycle distributions for extracted work, information content, and time and compute the power and information-to-work efficiency fluctuations for the different models. We show that the efficiency at maximum power is maximal for an opportunistic protocol of continuous type in the dynamical regime dominated by rare events. We also extend the analysis to finite-time work extracting protocols by mapping them to a three-state GCMD. We show that dynamical finite-time correlations in this model increase the information-to-work conversion efficiency, underlining the role of temporal correlations in optimizing information-to-energy conversion. The effect of finite-time work extraction and demon memory resetting is also analyzed. We conclude that GCMD models are thermodynamically more efficient than the single-measurement Szilard and preferred for describing biological processes in an information-redundant world.

Fluctuations of entropy production of a run-and-tumble particle

Out-of-equilibrium systems continuously generate entropy, with its rate of production being a fingerprint of nonequilibrium conditions. In small-scale dissipative systems subject to thermal noise, fluctuations of entropy production are significant. Hitherto, mean and variance have been abundantly studied, even if higher moments might be important to fully characterize the system of interest. Here, we introduce a graphical method to compute any moment of entropy production for a generic discrete-state system. Then, we focus on a paradigmatic model of active particles, i.e., run-and-tumble dynamics, which resembles the motion observed in several micro-organisms. Employing our framework, we compute the first three cumulants of the entropy production for a discrete version of this model. We also compare our analytical results with numerical simulations. We find that as the number of states increases, the distribution of entropy production deviates from a Gaussian. Finally, we extend our framework to a continuous state-space run-and-tumble model, using an appropriate scaling of the transition rates. The approach presented here might help uncover the features of nonequilibrium fluctuations of any current in biological systems operating out-of-equilibrium.

Thermodynamics and efficiency of sequentially collisional Brownian particles: The role of drivings

Brownian particles placed sequentially in contact with distinct thermal reservoirs and subjected to external driving forces are promising candidates for the construction of reliable engine setups. In this contribution, we address the role of driving forces for enhancing the collisional machine performance. Analytical expressions for thermodynamic quantities such as power output and efficiency are obtained for general driving schemes. A proper choice of these driving schemes substantially increases both power output and efficiency and extends the working regime. Maximizations of power and efficiency, whether with respect to the strength of the force, driving scheme, or both have been considered and exemplified for two kind of drivings: generic power-law and harmonic (sinusoidal) drivings.

One-particle engine with a porous piston

We propose a variation of the classical Szilard engine that uses a porous piston. Such an engine requires neither information about the position of the particle, nor the removal and subsequent insertion of the piston when resetting the engine to continue doing work by lifting a mass against a gravitational field. Though the engine operates in contact with a single thermal reservoir, the reset mechanism acts as a second reservoir, dissipating energy when a mass that has been lifted by the engine is removed to initiate a new operation cycle.

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.

Collective effects on the performance and stability of quantum heat engines

Recent predictions for quantum-mechanical enhancements in the operation of small heat engines have raised renewed interest in their study both from a fundamental perspective and in view of applications. One essential question is whether collective effects may help to carry enhancements over larger scales, when increasing the number of systems composing the working substance of the engine. Such enhancements may consider not only power and efficiency, that is, its performance, but, additionally, its constancy, that is, the stability of the engine with respect to unavoidable environmental fluctuations. We explore this issue by introducing a many-body quantum heat engine model composed by spin pairs working in continuous operation. We study how power, efficiency, and constancy scale with the number of spins composing the engine and introduce a well-defined macroscopic limit where analytical expressions are obtained. Our results predict power enhancements, in both finite-size and macroscopic cases, for a broad range of system parameters and temperatures, without compromising the engine efficiency, accompanied by coherence-enhanced constancy for finite sizes. We discuss these quantities in connection to thermodynamic uncertainty relations.

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.

Efficiency Fluctuations in a Quantum Battery Charged by a Repeated Interaction Process

A repeated interaction process assisted by auxiliary thermal systems charges a quantum battery. The charging energy is supplied by switching on and off the interaction between the battery and the thermal systems. The charged state is an equilibrium state for the repeated interaction process, and the ergotropy characterizes its charge. The working cycle consists in extracting the ergotropy and charging the battery again. We discuss the fluctuating efficiency of the process, among other fluctuating properties. These fluctuations are dominated by the equilibrium distribution and depend weakly on other process properties.

Solution to the Fokker-Planck equation for slowly driven Brownian motion: Emergent geometry and a formula for the corresponding thermodynamic metric

Considerable progress has recently been made with geometrical approaches to understanding and controlling small out-of-equilibrium systems, but a mathematically rigorous foundation for these methods has been lacking. Towards this end, we develop a perturbative solution to the Fokker-Planck equation for one-dimensional driven Brownian motion in the overdamped limit enabled by the spectral properties of the corresponding single-particle Schrödinger operator. The perturbation theory is in powers of the inverse characteristic timescale of variation of the fastest varying control parameter, measured in units of the system timescale, which is set by the smallest eigenvalue of the corresponding Schrödinger operator. It applies to any Brownian system for which the Schrödinger operator has a confining potential. We use the theory to rigorously derive an exact formula for a Riemannian "thermodynamic" metric in the space of control parameters of the system. We show that up to second-order terms in the perturbation theory, optimal dissipation-minimizing driving protocols minimize the length defined by this metric. We also show that a previously proposed metric is calculable from our exact formula with corrections that are exponentially suppressed in a characteristic length scale. We illustrate our formula using the two-dimensional example of a harmonic oscillator with time-dependent spring constant in a time-dependent electric field. Lastly, we demonstrate that the Riemannian geometric structure of the optimal control problem is emergent; it derives from the form of the perturbative expansion for the probability density and persists to all orders of the expansion.

Extracting work from random collisions: A model of a quantum heat engine

We study the statistical distribution of the ergotropy and of the efficiency of a single-qubit battery ad of a single-qubit Otto engine, respectively fueled by random collisions. The single qubit, our working fluid, is assumed to exchange energy with two reservoirs: a nonequilibrium "hot" reservoir and a zero-temperature cold reservoir. The interactions between the qubit and the reservoirs are described in terms of a collision model of open system dynamics. The qubit interacts with the nonequilibrium reservoir (a large ensemble of qudits all prepared in the same pure state) via random unitary collisions and with the cold reservoir (a large ensemble of qubits in their ground state) via a partial swap. Due to the random nature of the interaction with the hot reservoir, fluctuations in ergotropy, heat, and work are present, shrinking with the size of the qudits in the hot reservoir. While the mean, "macroscopic" efficiency of the Otto engine is the same as in the case in which the hot reservoir is a thermal one, the distribution of efficiencies does not support finite moments, so that the mean of efficiencies does not coincide with the macroscopic efficiency.

Survival and extreme statistics of work, heat, and entropy production in steady-state heat engines

We derive universal bounds for the finite-time survival probability of the stochastic work extracted in steady-state heat engines and the stochastic heat dissipated to the environment. We also find estimates for the time-dependent thresholds that these quantities do not surpass with a prescribed probability. At long times, the tightest thresholds are proportional to the large deviation functions of stochastic entropy production. Our results entail an extension of martingale theory for entropy production, for which we derive universal inequalities involving its maximum and minimum statistics that are valid for generic Markovian dynamics in nonequilibrium stationary states. We test our main results with numerical simulations of a stochastic photoelectric device.

Obtaining efficient thermal engines from interacting Brownian particles under time-periodic drivings

We introduce an alternative route for obtaining reliable cyclic engines, based on two interacting Brownian particles under time-periodic drivings which can be used as a work-to-work converter or a heat engine. Exact expressions for the thermodynamic fluxes, such as power and heat, are obtained using the framework of stochastic thermodynamic. We then use these exact expression to optimize the driving protocols with respect to output forces, their phase difference. For the work-to-work engine, they are solely expressed in terms of Onsager coefficients and their derivatives, whereas nonlinear effects start to play a role since the particles are at different temperatures. Our results suggest that stronger coupling generally leads to better performance, but careful design is needed to optimize the external forces.

Modeling of biomolecular machines in non-equilibrium steady states

Numerical computations have become a pillar of all modern quantitative sciences. Any computation involves modeling-even if often this step is not made explicit-and any model has to neglect details while still being physically accurate. Equilibrium statistical mechanics guides both the development of models and numerical methods for dynamics obeying detailed balance. For systems driven away from thermal equilibrium, such a universal theoretical framework is missing. For a restricted class of driven systems governed by Markov dynamics and local detailed balance, stochastic thermodynamics has evolved to fill this gap and to provide fundamental constraints and guiding principles. The next step is to advance stochastic thermodynamics from simple model systems to complex systems with tens of thousands or even millions of degrees of freedom. Biomolecules operating in the presence of chemical gradients and mechanical forces are a prime example for this challenge. In this Perspective, we give an introduction to isothermal stochastic thermodynamics geared toward the systematic multiscale modeling of the conformational dynamics of biomolecular and synthetic machines, and we outline some of the open challenges.

Current fluctuations in nonequilibrium discontinuous phase transitions

Discontinuous phase transitions out of equilibrium can be characterized by the behavior of macroscopic stochastic currents. But while much is known about the average current, the situation is much less understood for higher statistics. In this paper, we address the consequences of the diverging metastability lifetime-a hallmark of discontinuous transitions-in the fluctuations of arbitrary thermodynamic currents, including the entropy production. In particular, we center our discussion on the conditional statistics, given which phase the system is in. We highlight the interplay between integration window and metastability lifetime, which is not manifested in the average current, but strongly influences the fluctuations. We introduce conditional currents and find, among other predictions, their connection to average and scaled variance through a finite-time version of large deviation theory and a minimal model. Our results are then further verified in two paradigmatic models of discontinuous transitions: Schlögl's model of chemical reactions, and a 12-state Potts model subject to two baths at different temperatures.

Universal Bounds on Fluctuations in Continuous Thermal Machines

We study bounds on ratios of fluctuations in steady-state time-reversal energy conversion devices. In the linear response regime, we prove that the relative fluctuations (precision) of the output current (power) is always lower bounded by the relative fluctuations of the input current (heat current absorbed from the hot bath). As a consequence, the ratio between the fluctuations of the output and input currents are bounded both from above and below, where the lower (upper) bound is determined by the square of the averaged efficiency (square of the Carnot efficiency) of the engine. The saturation of the lower bound is achieved in the tight-coupling limit when the determinant of the Onsager response matrix vanishes. Our analysis can be applied to different operational regimes, including engines, refrigerators, and heat pumps. We illustrate our findings in two types of continuous engines: two-terminal coherent thermoelectric junctions and three-terminal quantum absorption refrigerators. Numerical simulations in the far-from-equilibrium regime suggest that these bounds apply more broadly, beyond linear response.

Heat fluctuations in a harmonic chain of active particles

One of the major challenges in stochastic thermodynamics is to compute the distributions of stochastic observables for small-scale systems for which fluctuations play a significant role. Hitherto much theoretical and experimental research has focused on systems composed of passive Brownian particles. In this paper, we study the heat fluctuations in a system of interacting active particles. Specifically we consider a one-dimensional harmonic chain of N active Ornstein-Uhlenbeck particles, with the chain ends connected to heat baths of different temperatures. We compute the moment-generating function for the heat flow in the steady state. We employ our general framework to explicitly compute the moment-generating function for two example single-particle systems. Further, we analytically obtain the scaled cumulants for the heat flow for the chain. Numerical Langevin simulations confirm the long-time analytical expressions for first and second cumulants for the heat flow for a two-particle chain.

Violating the thermodynamic uncertainty relation in the three-level maser

Nanoscale heat engines are subject to large fluctuations which affect their precision. The thermodynamic uncertainty relation (TUR) provides a trade-off between output power, fluctuations, and entropic cost. This trade-off may be overcome by systems exhibiting quantum coherence. This Letter provides a study of the TUR in a prototypical quantum heat engine, the Scovil-Schulz-DuBois maser. Comparison with a classical reference system allows us to determine the effect of quantum coherence on the performance of the heat engine. We identify analytically regions where coherence suppresses fluctuations, implying a quantum advantage, as well as regions where fluctuations are enhanced by coherence. This quantum effect cannot be anticipated from the off-diagonal elements of the density matrix. Because the fluctuations are not encoded in the steady state alone, TUR violations are a consequence of coherence that goes beyond steady-state coherence. While the system violates the conventional TUR, it adheres to a recent formulation of a quantum TUR. We further show that parameters where the engine operates close to the conventional limit are prevalent and TUR violations in the quantum model are not uncommon.

Tuning the performance of a micrometer-sized Stirling engine through reservoir engineering

Colloidal heat engines are paradigmatic models to understand the conversion of heat into work in a noisy environment - a domain where biological and synthetic nano/micro machines function. While the operation of these engines across thermal baths is well-understood, how they function across baths with noise statistics that is non-Gaussian and also lacks memory, the simplest departure from the thermal case, remains unclear. Here we quantified the performance of a colloidal Stirling engine operating between an engineered memoryless non-Gaussian bath and a Gaussian one. In the quasistatic limit, the non-Gaussian engine functioned like a thermal one as predicted by theory. On increasing the operating speed, due to the nature of noise statistics, the onset of irreversibility for the non-Gaussian engine preceded its thermal counterpart and thus shifted the operating speed at which power is maximum. The performance of nano/micro machines can be tuned by altering only the nature of reservoir noise statistics.

Coherences and the thermodynamic uncertainty relation: Insights from quantum absorption refrigerators

The thermodynamic uncertainty relation, originally derived for classical Markov-jump processes, provides a tradeoff relation between precision and dissipation, deepening our understanding of the performance of quantum thermal machines. Here, we examine the interplay of quantum system coherences and heat current fluctuations on the validity of the thermodynamics uncertainty relation in the quantum regime. To achieve the current statistics, we perform a full counting statistics simulation of the Redfield quantum master equation. We focus on steady-state quantum absorption refrigerators where nonzero coherence between eigenstates can either suppress or enhance the cooling power, compared with the incoherent limit. In either scenario, we find enhanced relative noise of the cooling power (standard deviation of the power over the mean) in the presence of system coherence, thereby corroborating the thermodynamic uncertainty relation. Our results indicate that fluctuations necessitate consideration when assessing the performance of quantum coherent thermal machines.

Heat Driven Transport in Serial Double Quantum Dot Devices

Studies of thermally induced transport in nanostructures provide access to an exciting regime where fluctuations are relevant, enabling the investigation of fundamental thermodynamic concepts and the realization of thermal energy harvesters. We study a serial double quantum dot formed in an InAs/InP nanowire coupled to two electron reservoirs. By means of a specially designed local metallic joule-heater, the temperature of the phonon bath in the vicinity of the double quantum dot can be enhanced. This results in phonon-assisted transport, enabling the conversion of local heat into electrical power in a nanosized heat engine. Simultaneously, the electron temperatures of the reservoirs are affected, resulting in conventional thermoelectric transport. By detailed modeling and experimentally tuning the interdot coupling, we disentangle both effects. Furthermore, we show that phonon-assisted transport is sensitive to excited states. Our findings demonstrate the versatility of our design to study fluctuations and fundamental nanothermodynamics.

Efficiency Fluctuations of Stochastic Machines Undergoing a Phase Transition

We study the efficiency fluctuations of a stochastic heat engine made of N interacting unicyclic machines and undergoing a phase transition in the macroscopic limit. Depending on N and on the observation time, the machine can explore its whole phase space or not. This affects the engine efficiency that either strongly fluctuates on a large interval of equiprobable efficiencies (ergodic case) or fluctuates close to several most likely values (nonergodic case). We also provide a proof that despite the phase transition, the decay rate of the efficiency distribution at the reversible efficiency remains largest one although other efficiencies can now decay equally fast.

Stochastic Floquet quantum heat engines and stochastic efficiencies

Based on the notion of quantum trajectory, we present a stochastic theoretical framework for Floquet quantum heat engines. As an application, the large deviation functions of two types of stochastic efficiencies for a two-level Floquet quantum heat engine are investigated. We find that the statistics of one efficiency agree well with the predictions of the universal theory of efficiency fluctuations developed by Verley et al. [Phys. Rev. E 90, 052145 (2014)10.1103/PhysRevE.90.052145], whereas the statistics of the other efficiency do not. The reason for this discrepancy is attributed to the lack of fluctuation theorems for the latter type of efficiency.

Inferring Entropy Production from Short Experiments

We provide a strategy for the exact inference of the average as well as the fluctuations of the entropy production in nonequilibrium systems in the steady state, from the measurements of arbitrary current fluctuations. Our results are built upon the finite-time generalization of the thermodynamic uncertainty relation, and require only very short time series data from experiments. We illustrate our results with exact and numerical solutions for two colloidal heat engines.

Nonlinear Thermoelectricity with Electron-Hole Symmetric Systems

In the linear regime, thermoelectric effects between two conductors are possible only in the presence of an explicit breaking of the electron-hole symmetry. We consider a tunnel junction between two electrodes and show that this condition is no longer required outside the linear regime. In particular, we demonstrate that a thermally biased junction can display an absolute negative conductance, and hence thermoelectric power, at a small but finite voltage bias, provided that the density of states of one of the electrodes is gapped and the other is monotonically decreasing. We consider a prototype system that fulfills these requirements, namely, a tunnel junction between two different superconductors where the Josephson contribution is suppressed. We discuss this nonlinear thermoelectric effect based on the spontaneous breaking of electron-hole symmetry in the system, characterize its main figures of merit, and discuss some possible applications.

Work distribution in thermal processes

We find the moment generating function (mgf) of the nonequilibrium work for open systems undergoing a thermal process, i.e., when the stochastic dynamics maps thermal states into time-dependent thermal states. The mgf is given in terms of a temperaturelike scalar satisfying a first-order ordinary differential equation. We apply the result to some paradigmatic situations: a levitated nanoparticle in a breathing optical trap, a Brownian particle in a box with a moving piston, and a two-state system driven by an external field, where the work mgfs are obtained for different timescales and compared with Monte Carlo simulations.

Efficiency fluctuations and noise induced refrigerator-to-heater transition in information engines

Understanding noisy information engines is a fundamental problem of non-equilibrium physics, particularly in biomolecular systems agitated by thermal and active fluctuations in the cell. By the generalized second law of thermodynamics, the efficiency of these engines is bounded by the mutual information passing through their noisy feedback loop. Yet, direct measurement of the interplay between mutual information and energy has so far been elusive. To allow such examination, we explore here the entire phase-space of a noisy colloidal information engine, and study efficiency fluctuations due to the stochasticity of the mutual information and extracted work. We find that the average efficiency is maximal for non-zero noise level, at which the distribution of efficiency switches from bimodal to unimodal, and the stochastic efficiency often exceeds unity. We identify a line of anomalous, noise-driven equilibrium states that defines a refrigerator-to-heater transition, and test the generalized integral fluctuation theorem for continuous engines.

Energy dissipation bounds for autonomous thermodynamic cycles

How much free energy is irreversibly lost during a thermodynamic process? For deterministic protocols, lower bounds on energy dissipation arise from the thermodynamic friction associated with pushing a system out of equilibrium in finite time. Recent work has also bounded the cost of precisely moving a single degree of freedom. Using stochastic thermodynamics, we compute the total energy cost of an autonomously controlled system by considering both thermodynamic friction and the entropic cost of precisely directing a single control parameter. Our result suggests a challenge to the usual understanding of the adiabatic limit: Here, even infinitely slow protocols are energetically irreversible.

Information-Theoretical Bound of the Irreversibility in Thermal Relaxation Processes

We establish that entropy production, which is crucial to the characterization of thermodynamic irreversibility, is obtained through a variational principle involving the Kulback-Leibler divergence. A simple application of this representation leads to an information-theoretical bound on entropy production in thermal relaxation processes; this is a stronger inequality than the conventional second law of thermodynamics. This bound is also interpreted as a constraint on the possible path of a thermal relaxation process in terms of information geometry. Our results reveal a hidden universal law inherent to general thermal relaxation processes.

Efficiency of a cyclic quantum heat engine with finite-size baths

In this paper we investigate the relationship between the efficiency of a cyclic quantum heat engine with the Hilbert space dimension of the thermal baths. By means of a general inequality, we show that the Carnot efficiency can be obtained only when both the hot and cold baths are infinitely large. By further introducing a specific model where the baths are constituted of ensembles of finite-dimensional particles, we further demonstrate the relationship between the engine's power and efficiency, with the dimension of the working substance and the bath particles.

Out-of-Equilibrium Clock Model at the Verge of Criticality

We consider an out-of-equilibrium lattice model consisting of 2D discrete rotators, in contact with heat reservoirs at different temperatures. The equilibrium counterpart of such a model, the clock model, exhibits three phases: a low-temperature ordered phase, a quasiliquid phase, and a high-temperature disordered phase, with two corresponding phase transitions. In the out-of-equilibrium model the simultaneous breaking of spatial symmetry and thermal equilibrium gives rise to directed rotation of the spin variables. In this regime the system behaves as a thermal machine converting heat currents into motion. In order to quantify the susceptibility of the machine to the thermodynamic force driving it out of equilibrium, we introduce and study a dynamical response function. We show that the optimal operational regime for such a thermal machine occurs when the out-of-equilibrium disturbance is applied around the critical temperature at the boundary between the first two phases, namely, where the system is mostly susceptible to external thermodynamic forces and exhibits a sharper transition. We thus argue that critical fluctuations in a system of interacting motors can be exploited to enhance the machine overall dynamic and thermodynamic performances.

Operational approach to quantum stochastic thermodynamics

We set up a framework for quantum stochastic thermodynamics based solely on experimentally controllable but otherwise arbitrary interventions at discrete times. Using standard assumptions about the system-bath dynamics and insights from the repeated interaction framework, we define internal energy, heat, work, and entropy at the trajectory level. The validity of the first law (at the trajectory level) and the second law (on average) is established. The theory naturally allows one to treat incomplete information and it is able to smoothly interpolate between a trajectory-based and an ensemble level description. We use our theory to compute the thermodynamic efficiency of recent experiments reporting on the stabilization of photon number states using real-time quantum feedback control. Special attention is paid to limiting cases of our general theory, where we recover or contrast it with previous results. We point out various interesting problems, which the theory is able to address rigorously, such as the detection of quantum effects in thermodynamics.

Operationally Accessible Bounds on Fluctuations and Entropy Production in Periodically Driven Systems

For periodically driven systems, we derive a family of inequalities that relate entropy production with experimentally accessible data for the mean, its dependence on driving frequency, and the variance of a large class of observables. With one of these relations, overall entropy production can be bounded by just observing the time spent in a set of states. Among further consequences, the thermodynamic efficiency both of isothermal cyclic engines like molecular motors under a periodic load and of cyclic heat engines can be bounded using experimental data without requiring knowledge of the specific interactions within the system. We illustrate these results for a driven three-level system and for a colloidal Stirling engine.

Efficiency Fluctuations in Microscopic Machines

Nanoscale machines are strongly influenced by thermal fluctuations, contrary to their macroscopic counterparts. As a consequence, even the efficiency of such microscopic machines becomes a fluctuating random variable. Using geometric properties and the fluctuation theorem for the total entropy production, a "universal theory of efficiency fluctuations" at long times, for machines with a finite state space, was developed by Verley et al. [Nat. Commun. 5, 4721 (2014)NCAOBW2041-172310.1038/ncomms5721; Phys. Rev. E 90, 052145 (2014)PRESCM1539-375510.1103/PhysRevE.90.052145]. We extend this theory to machines with an arbitrary state space. Thereby, we work out more detailed prerequisites for the "universal features" and explain under which circumstances deviations can occur. We also illustrate our findings with exact results for two nontrivial models of colloidal engines.

Physically consistent numerical solver for time-dependent Fokker-Planck equations

We present a simple thermodynamically consistent method for solving time-dependent Fokker-Planck equations (FPE) for overdamped stochastic processes, also known as Smoluchowski equations. It yields both transition and steady-state behavior and allows for computations of moment-generating and large-deviation functions of observables defined along stochastic trajectories, such as the fluctuating particle current, heat, and work. The key strategy is to approximate the FPE by a master equation with transition rates in configuration space that obey a local detailed balance condition for arbitrary discretization. Its time-dependent solution is obtained by a direct computation of the time-ordered exponential, representing the propagator of the FPE, by summing over all possible paths in the discretized space. The method thus not only preserves positivity and normalization of the solutions but also yields a physically reasonable total entropy production, regardless of the discretization. To demonstrate the validity of the method and to exemplify its potential for applications, we compare it against Brownian-dynamics simulations of a heat engine based on an active Brownian particle trapped in a time-dependent quartic potential.

Path-integral methodology and simulations of quantum thermal transport: Full counting statistics approach

We develop and test a computational framework to study heat exchange in interacting, nonequilibrium open quantum systems. Our iterative full counting statistics path integral (iFCSPI) approach extends a previously well-established influence functional path integral method, by going beyond reduced system dynamics to provide the cumulant generating function of heat exchange. The method is straightforward; we implement it for the nonequilibrium spin boson model to calculate transient and long-time observables, focusing on the steady-state heat current flowing through the system under a temperature difference. Results are compared to perturbative treatments and demonstrate good agreement in the appropriate limits. The challenge of converging nonequilibrium quantities, currents and high order cumulants, is discussed in detail. The iFCSPI, a numerically exact technique, naturally captures strong system-bath coupling and non-Markovian effects of the environment. As such, it is a promising tool for probing fundamental questions in quantum transport and quantum thermodynamics.

Information Thermodynamics of Turing Patterns

We set up a rigorous thermodynamic description of reaction-diffusion systems driven out of equilibrium by time-dependent space-distributed chemostats. Building on the assumption of local equilibrium, nonequilibrium thermodynamic potentials are constructed exploiting the symmetries of the chemical network topology. It is shown that the canonical (resp. semigrand canonical) nonequilibrium free energy works as a Lyapunov function in the relaxation to equilibrium of a closed (resp. open) system, and its variation provides the minimum amount of work needed to manipulate the species concentrations. The theory is used to study analytically the Turing pattern formation in a prototypical reaction-diffusion system, the one-dimensional Brusselator model, and to classify it as a genuine thermodynamic nonequilibrium phase transition.

Arcsine Laws in Stochastic Thermodynamics

We show that the fraction of time that a thermodynamic current spends above its average value follows the arcsine law, a prominent result obtained by Lévy for Brownian motion. Stochastic currents with long streaks above or below their average are much more likely than those that spend similar fractions of time above and below their average. Our result is confirmed with experimental data from a Brownian Carnot engine. We also conjecture that two other random times associated with currents obey the arcsine law: the time a current reaches its maximum value and the last time a current crosses its average value. These results apply to, inter alia, molecular motors, quantum dots, and colloidal systems.

Universal Trade-Off between Power, Efficiency, and Constancy in Steady-State Heat Engines

Heat engines should ideally have large power output, operate close to Carnot efficiency and show constancy, i.e., exhibit only small fluctuations in this output. For steady-state heat engines, driven by a constant temperature difference between the two heat baths, we prove that out of these three requirements only two are compatible. Constancy enters quantitatively the conventional trade-off between power and efficiency. Thus, we rationalize and unify recent suggestions for overcoming this simple trade-off. Our universal bound is illustrated for a paradigmatic model of a quantum dot solar cell and for a Brownian gyrator delivering mechanical work against an external force.

Large deviation function for a driven underdamped particle in a periodic potential

Employing large deviation theory, we explore current fluctuations of underdamped Brownian motion for the paradigmatic example of a single particle in a one-dimensional periodic potential. Two different approaches to the large deviation function of the particle current are presented. First, we derive an explicit expression for the large deviation functional of the empirical phase space density, which replaces the level 2.5 functional used for overdamped dynamics. Using this approach, we obtain several bounds on the large deviation function of the particle current. We compare these to bounds for overdamped dynamics that have recently been derived, motivated by the thermodynamic uncertainty relation. Second, we provide a method to calculate the large deviation function via the cumulant generating function. We use this method to assess the tightness of the bounds in a numerical case study for a cosine potential.

Stochastic efficiency of an isothermal work-to-work converter engine

We investigate the efficiency of an isothermal Brownian work-to-work converter engine, composed of a Brownian particle coupled to a heat bath at a constant temperature. The system is maintained out of equilibrium by using two external time-dependent stochastic Gaussian forces, where one is called load force and the other is called drive force. Work done by these two forces are stochastic quantities. The efficiency of this small engine is defined as the ratio of stochastic work done against load force to stochastic work done by the drive force. The probability density function as well as large deviation function of the stochastic efficiency are studied analytically and verified by numerical simulations.

General relations between the power, efficiency, and dissipation for the irreversible heat engines in the nonlinear response regime

We derive the general relations between the maximum power, maximum efficiency, and minimum dissipation for the irreversible heat engine in a nonlinear response regime. In this context, we use the minimally nonlinear irreversible model and obtain the lower and upper bounds of the above relations for the asymmetric dissipation limits. These relations can be simplified further when the system possesses the time-reversal symmetry or antisymmetry. We find that our results are the generalization of various such relations obtained earlier for different heat engines.

Liquid-Gas Transitions in Steady Heat Conduction

We study liquid-gas transitions of heat conduction systems in contact with two heat baths under constant pressure in the linear response regime. On the basis of local equilibrium thermodynamics, we propose an equality with a global temperature, which determines the volume near the equilibrium liquid-gas transition. We find that the formation of the liquid-gas interface is accompanied by a discontinuous change in the volume when increasing the mean temperature of the baths. A supercooled gas near the interface is observed as a stable steady state.

Diverging, but negligible power at Carnot efficiency: Theory and experiment

We discuss the possibility of reaching the Carnot efficiency by heat engines (HEs) out of quasistatic conditions at nonzero power output. We focus on several models widely used to describe the performance of actual HEs. These models comprise quantum thermoelectric devices, linear irreversible HEs, minimally nonlinear irreversible HEs, HEs working in the regime of low-dissipation, overdamped stochastic HEs and an underdamped stochastic HE. Although some of these HEs can reach the Carnot efficiency at nonzero and even diverging power, the magnitude of this power is always negligible compared to the maximum power attainable in these systems. We provide conditions for attaining the Carnot efficiency in the individual models and explain practical aspects connected with reaching the Carnot efficiency at large power output. Furthermore, we show how our findings can be tested in practice using a standard Brownian HE realizable with available micromanipulation techniques.