Unified Thermodynamic Uncertainty Relations in Linear Response

Stochastic thermodynamic bounds on logical circuit operation

Using a thermodynamically consistent, mesoscopic model for modern complementary metal-oxide-semiconductor transistors, we study an array of logical circuits and explore how their function is constrained by recent thermodynamic uncertainty relations when operating near thermal energies. For a single NOT gate, we find operating direction-dependent dynamics and a trade-off between dissipated heat and operation time certainty. For a memory storage device, we find an exponential relationship between the memory retention time and energy required to sustain that memory state. For a clock, we find that the certainty in the cycle time is maximized at biasing voltages near thermal energy, as is the trade-off between this certainty and the heat dissipated per cycle. We identify a control mechanism that can increase the cycle time certainty without an offsetting increase in heat dissipation by working at a resonance condition for the clock. These results provide a framework for assessing the thermodynamic costs of realistic computing devices, allowing for circuits to be designed and controlled for thermodynamically optimal operation.

Precisely controlled colloids: a playground for path-wise non-equilibrium physics

We investigate path-wise observables in experiments on driven colloids in a periodic light field to dissect selected intricate transport features, kinetics, and transition-path time statistics out of thermodynamic equilibrium. These observables directly reflect the properties of individual paths in contrast to the properties of an ensemble of particles, such as radial distribution functions or mean-squared displacements. In particular, we present two distinct albeit equivalent formulations of the underlying stochastic equation of motion, highlight their respective practical relevance, and show how to interchange between them. We discuss conceptually different notions of local velocities and interrogate one- and two-sided first-passage and transition-path time statistics in and out of equilibrium. Our results reiterate how path-wise observables may be employed to systematically assess the quality of experimental data and demonstrate that, given sufficient control and sampling, one may quantitatively verify subtle theoretical predictions.

Trade-off between coherence and dissipation for excitable phase oscillators

Thermodynamic uncertainty relation (TUR) bounds coherence in stochastic oscillatory systems. In this paper, we show that both dynamical and thermodynamic bounds play important roles for the excitable oscillators, e.g., neurons. Firstly, we investigate the trade-off between coherence and dissipation both in the sub- and superthreshold regions for a single excitable unit, where both the TUR and the saddle-node on an invariant circle (SNIC) bounds constrain the fluctuation of interspike intervals. Secondly, we show that the widely studied phenomenon called coherence resonance, where there exists a noise strength to make the oscillatory responses of the system most coherent, is also bounded by the TUR in the one-dimensional excitable phase model. Finally, we study the coherence-dissipation relation in ensembles of strongly coupled excitable oscillators.

Thermodynamic uncertainty relation for quantum entropy production

In quantum thermodynamics, entropy production is usually defined in terms of the quantum relative entropy between two states. We derive a lower bound for the quantum entropy production in terms of the mean and variance of quantum observables, which we refer to as a thermodynamic uncertainty relation (TUR) for the entropy production. In the absence of coherence between the states, our result reproduces classic TURs in stochastic thermodynamics. For the derivation of the TUR, we introduce a lower bound for a quantum generalization of the χ^{2} divergence between two states and discuss its implications for stochastic and quantum thermodynamics, as well as the limiting case where it reproduces the quantum Cramér-Rao inequality.

Thermodynamic Geometry of Nonequilibrium Fluctuations in Cyclically Driven Transport

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.

Variance sum rule for entropy production

Entropy production is the hallmark of nonequilibrium physics, quantifying irreversibility, dissipation, and the efficiency of energy transduction processes. Despite many efforts, its measurement at the nanoscale remains challenging. We introduce a variance sum rule (VSR) for displacement and force variances that permits us to measure the entropy production rate σ in nonequilibrium steady states. We first illustrate it for directly measurable forces, such as an active Brownian particle in an optical trap. We then apply the VSR to flickering experiments in human red blood cells. We find that σ is spatially heterogeneous with a finite correlation length, and its average value agrees with calorimetry measurements. The VSR paves the way to derive σ using force spectroscopy and time-resolved imaging in living and active matter.

Thermodynamic uncertainty theorem

Thermodynamic uncertainty relations (TURs) express a fundamental lower bound on the precision (inverse scaled variance) of any thermodynamic charge-e.g., work or heat-by functionals of the average entropy production. Relying on purely variational arguments, we significantly extend TUR inequalities by incorporating and analyzing the impact of higher statistical cumulants of the entropy production itself within the general framework of time-symmetrically-controlled computation. We derive an exact expression for the charge that achieves the minimum scaled variance, for which the TUR bound tightens to an equality that we name the thermodynamic uncertainty theorem (TUT). Importantly, both the minimum scaled variance charge and the TUT are functionals of the stochastic entropy production, thus retaining the impact of its higher moments. In particular, our results show that, beyond the average, the entropy production distribution's higher moments have a significant effect on any charge's precision. This is made explicit via a thorough numerical analysis of "swap" and "reset" computations that quantitatively compares the TUT against previous generalized TURs.

Thermodynamic Bounds on Correlation Times

We derive a variational expression for the correlation time of physical observables in steady-state diffusive systems. As a consequence of this variational expression, we obtain lower bounds on the correlation time, which provide speed limits on the self-averaging of observables. In equilibrium, the bound takes the form of a trade-off relation between the long- and short-time fluctuations of an observable. Out of equilibrium, the trade-off can be violated, leading to an acceleration of self-averaging. We relate this violation to the steady-state entropy production rate, as well as the geometric structure of the irreversible currents, giving rise to two complementary speed limits. One of these can be formulated as a lower estimate on the entropy production from the measurement of time-symmetric observables. Using an illustrating example, we show the intricate behavior of the correlation time out of equilibrium for different classes of observables and how this can be used to partially infer dissipation even if no time-reversal symmetry breaking can be observed in the trajectories of the observable.

Precision bound and optimal control in periodically modulated continuous quantum thermal machines

We use Floquet formalism to study fluctuations in periodically modulated continuous quantum thermal machines. We present a generic theory for such machines, followed by specific examples of sinusoidal, optimal, and circular modulations, respectively. The thermodynamic uncertainty relations (TUR) hold for all modulations considered. Interestingly, in the case of sinusoidal modulation, the TUR ratio assumes a minimum at the heat engine to refrigerator transition point, while the chopped random basis optimization protocol allows us to keep the ratio small for a wide range of modulation frequencies. Furthermore, our numerical analysis suggests that TUR can show signatures of heat engine to refrigerator transition, for more generic modulation schemes. We also study bounds in fluctuations in the efficiencies of such machines; our results indicate that fluctuations in efficiencies are bounded from above for a refrigerator and from below for an engine. Overall, this study emphasizes the crucial role played by different modulation schemes in designing practical quantum thermal machines.

Probability inequalities for direct and inverse dynamical outputs in driven fluctuating systems

When a fluctuating system is subjected to a time-dependent drive or nonconservative forces, the direct-inverse symmetry of the dynamics can be broken so inducing an average bias. Here we start from the fluctuation theorem, a cornerstone of stochastic thermodynamics, for inspecting the unbalancing between direct and inverse dynamical outputs, here called "events," in a bidirectional forward-backward setup. The occurrence of an event might correspond to the realization of a quantitative output, or to the realization of a sequence of acts that compose a complex "narrative." The focus is on mutual bounds between the probabilities of occurrence of direct and inverse events in the forward and backward mode. The inspection is made for systems in contact with a thermal bath, and by assuming Markov dynamics on the uncontrolled degrees of freedom. The approach comprises both the case of systems under a time-dependent drive and time-independent external forces. The general formulation is then used to derive (or re-derive) specialized results valid for finite-time processes, and for systems taken into steady conditions (either periodic steady states or steady states) starting from equilibrium. Among the results, we find already known forms of "generalized" thermodynamic uncertainty relations, and derive useful constraints concerning the work distribution function for systems in steady conditions.

Thermodynamic Uncertainty Relation in Interacting Many-Body Systems

The thermodynamic uncertainty relation (TUR) has been well studied for systems with few degrees of freedom. While, in principle, the TUR holds for more complex systems with many interacting degrees of freedom as well, little is known so far about its behavior in such systems. We analyze the TUR in the thermodynamic limit for mixtures of driven particles with short-range interactions. Our main result is an explicit expression for the optimal estimate of the total entropy production in terms of single-particle currents and correlations between two-particle currents. Quantitative results for various versions of a driven lattice gas demonstrate the practical implementation of this approach.

Thermodynamic skewness relation from detailed fluctuation theorem

The detailed fluctuation theorem (DFT) is a statement about the asymmetry in the statistics of the entropy production. Consequences of the DFT are the second law of thermodynamics and the thermodynamics uncertainty relation, which translate into lower bounds for the mean and variance of currents, respectively. However, far from equilibrium, mean and variance are not enough to characterize the underlying distribution of the entropy production. The fluctuations are not necessarily Gaussian (nor symmetric), which means their skewness could be nonzero. We prove that the DFT imposes a negative tight lower bound for the skewness of the entropy production as a function of the mean. As application, we check the bound in the heat exchange problem between two thermal reservoirs mediated by a qubit swap engine.

Virial-like thermodynamic uncertainty relation in the tight-binding regime

We presented a methodology to approximate the entropy production for Brownian motion in a tilted periodic potential. The approximation stems from the well known thermodynamic uncertainty relation. By applying a virial-like expansion, we provided a tighter lower limit solely in terms of the drift velocity and diffusion. The approach presented is systematically analyzed in the tight-binding regime. We also provide a relative simple rule to validate using the tight-binding approach based on drift and diffusion relations rather than energy barriers and forces. We also discuss the implications of our results outside the tight-binding regime.

Mathematical, Thermodynamical, and Experimental Necessity for Coarse Graining Empirical Densities and Currents in Continuous Space

We present general results on fluctuations and spatial correlations of the coarse-grained empirical density and current of Markovian diffusion in equilibrium or nonequilibrium steady states on all timescales. We unravel a deep connection between current fluctuations and generalized time-reversal symmetry, providing new insight into time-averaged observables. We highlight the essential role of coarse graining in space from mathematical, thermodynamical, and experimental points of view. Spatial coarse graining is required to uncover salient features of currents that break detailed balance, and a thermodynamically "optimal" coarse graining ensures the most precise inference of dissipation. Defined without coarse graining, the fluctuations of empirical density and current are proven to diverge on all timescales in dimensions higher than one, which has far-reaching consequences for the central-limit regime in continuous space. We apply the results to examples of irreversible diffusion. Our findings provide new intuition about time-averaged observables and allow for a more efficient analysis of single-molecule experiments.

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.

A model of minimal entropy generation for cytoskeletal transport systems with multiple interacting motors

We study the steady-state rate of entropy generation for multiple interacting particles. The description used is based on the partially asymmetric exclusion process in a lattice with periodic boundary conditions. Our methodology shows that in the steady-state, the rate of entropy generation is directly proportional to the bulk drift and the applied driving force. Since in many cases the driving force is unknown or hard to determine. We circumvent this by deriving a lower bound for the entropy, resulting in an extended thermodynamic uncertainty relation for the asymmetric simple exclusion process. We systematically compared this bound with the actual entropy generation. Thus, we identify the force regimes, and particles' density conditions where the entropy bound derived from this extended thermodynamic uncertainty relation is meaningful.

Universal bounds on cooling power and cooling efficiency for autonomous absorption refrigerators

For steady-state autonomous absorption refrigerators operating in the linear response regime, we show that there exists a hierarchy between the relative fluctuation of currents for cold, hot, and work terminals. Our proof requires the Onsager reciprocity relation along with the refrigeration condition that sets the direction of the mean currents for each terminal. As a consequence, the universal bounds on the mean cooling power, obtained following the thermodynamic uncertainty relations, follow a hierarchy. Interestingly, within this hierarchy, the tightest bound is given in terms of the work current fluctuation. Furthermore, the relative uncertainty hierarchy introduces a bound on cooling efficiency that is tighter than the bound obtained from the thermodynamic uncertainty relations. Interestingly, all of these bounds saturate in the tight-coupling limit. We test the validity of our results for two paradigmatic absorption refrigerator models: (i) a four-level working fluid and (ii) a two-level working fluid, operating in the weak (additive) and strong (multiplicative) system-bath interaction regimes, respectively.

Bounds on fluctuations for machines with broken time-reversal symmetry: A linear response study

For a generic class of machines with broken time-reversal symmetry we show that in the linear response regime the relative fluctuation of the sum of output currents for time-forward and time-reversed processes is always lower bounded by the corresponding relative fluctuation of the sum of input currents. This bound is received when the same operating condition, for example, engine, refrigerator, or pump, is imposed on both the forward and the reversed processes. As a consequence, universal upper and lower bounds for the ratio between fluctuations of output and input current are obtained. Furthermore, we establish an important connection between our results and the recently obtained generalized thermodynamic uncertainty relations for time-reversal symmetry-broken systems. We illustrate these findings for two different types of machines: (1) a steady-state three-terminal quantum thermoelectric setup in presence of an external magnetic field and (2) a periodically driven classical Brownian heat engine.

Origin of loose bound of the thermodynamic uncertainty relation in a dissipative two-level quantum system

Thermodynamic uncertainty relations (TURs), originally discovered for classical systems, dictate the tradeoff between dissipation and fluctuations of irreversible current, specifying a minimal bound that constrains the two quantities. In a series of efforts to extend the relation to the one under more generalized conditions, it has been noticed that the bound is less tight in open quantum processes. To study the origin of the loose bounds, we consider an external field-driven transition dynamics of a two-level quantum system weakly coupled to the bosonic bath as a model of an open quantum system. The model makes it explicit that the imaginary part of quantum coherence, which contributes to dissipation to the environment, is responsible for loosening the TUR bound by suppressing the relative fluctuations in the irreversible current of transitions, whereas the real part of the coherence tightens it. Our study offers a better understanding of how quantum nature affects the TUR bound.

Universal form of thermodynamic uncertainty relation for Langevin dynamics

The thermodynamic uncertainty relation (TUR) provides a stricter bound for entropy production (EP) than that of the thermodynamic second law. This stricter bound can be utilized to infer the EP and derive other tradeoff relations. Though the validity of the TUR has been verified in various stochastic systems, its application to general Langevin dynamics has not been successfully unified, especially for underdamped Langevin dynamics, where odd parity variables in time-reversal operation such as velocity get involved. Previous TURs for underdamped Langevin dynamics are neither experimentally accessible nor reduced to the original form of the overdamped Langevin dynamics in the zero-mass limit. Here, we find a TUR for underdamped Langevin dynamics with an arbitrary time-dependent protocol, which is operationally accessible when all mechanical forces are controllable. We show that the original TUR is a consequence of our underdamped TUR in the zero-mass limit. This indicates that the TUR formulation presented here can be regarded as the universal form of the TUR for general Langevin dynamics.

Detailed fluctuation theorem bound for apparent violations of the second law

The second law of thermodynamics is a statement about the statistics of the entropy production, 〈Σ〉≥0. For small systems, it is known that the entropy production is a random variable and negative values (Σ<0) might be observed in some experiments. This situation is sometimes called apparent violation of the second law. In this sense, how often is the second law violated? For a given average 〈Σ〉, we show that the strong detailed fluctuation theorem implies a lower tight bound for the apparent violations of the second law. As applications, we verify that the bound is satisfied for the entropy produced in the heat exchange problem between two reservoirs mediated by a bosonic mode in the weak-coupling approximation, a levitated nanoparticle, and a classical particle in a box.

Thermodynamic uncertainty relations for many-body systems with fast jump rates and large occupancies

A universal large N theory of nonequilibrium fluctuations emerges in the limit of fast jump rates and large occupancies. We use this theory to derive a set of coarse-grained thermodynamic uncertainty relations-one of them being an activity bound. Importantly, the activity serves as a tighter bound for the entropy production in 1D systems. These results are particularly useful in the many-body regime, where typically a coarse-grained approach is required to handle the large microscopic state space.

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.

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.

Thermodynamic Uncertainty Relation Bounds the Extent of Anomalous Diffusion

In a finite system driven out of equilibrium by a constant external force the thermodynamic uncertainty relation (TUR) bounds the variance of the conjugate current variable by the thermodynamic cost of maintaining the nonequilibrium stationary state. Here we highlight a new facet of the TUR by showing that it also bounds the timescale on which a finite system can exhibit anomalous kinetics. In particular, we demonstrate that the TUR bounds subdiffusion in a single file confined to a ring as well as a dragged Gaussian polymer chain even when detailed balance is satisfied. Conversely, the TUR bounds the onset of superdiffusion in the active comb model. Remarkably, the fluctuations in a comb model evolving from a steady state behave anomalously as soon as detailed balance is broken. Our work establishes a link between stochastic thermodynamics and the field of anomalous dynamics that will fertilize further investigations of thermodynamic consistency of anomalous diffusion models.

Bounds on fluctuations for finite-time quantum Otto cycle

For quantum Otto engine driven quasistatically, we provide exact full statistics of heat and work for a class of working fluids that follow a scale-invariant energy eigenspectra under driving. Equipped with the full statistics we go on to derive a universal expression for the ratio of nth cumulant of output work and input heat in terms of the mean Otto efficiency. Furthermore, for nonadiabatic driving of quantum Otto engine with working fluid consisting of either a (i) qubit or (ii) a harmonic oscillator, we show that the relative fluctuation of output work is always greater than the corresponding relative fluctuation of input heat absorbed from the hot bath. As a result, the ratio between the work fluctuation and the input heat fluctuation receives a lower bound in terms of the square value of the average efficiency of the engine. The saturation of the lower bound is received in the quasistatic limit of the engine.

Thermodynamic Uncertainty Relation in Slowly Driven Quantum Heat Engines

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.

Information bound for entropy production from the detailed fluctuation theorem

Fluctuation theorems impose fundamental bounds in the statistics of the entropy production with the second law of thermodynamics being the most famous. Using information theory, we quantify the information of entropy production and find an upper tight bound as a function of its mean from the strong detailed fluctuation theorem. The bound is given in terms of a maximal distribution, a member of the exponential family with nonlinear argument. We show that the entropy produced by heat transfer using a bosonic mode at weak coupling reproduces the maximal distribution in a limiting case. The upper bound is extended to the continuous domain and verified for the heat transfer using a levitated nanoparticle. Finally, we show that a composition of qubit swap engines satisfies a particular case of the maximal distribution regardless of its size.

Thermodynamics of precision in quantum nanomachines

Fluctuations strongly affect the dynamics and functionality of nanoscale thermal machines. Recent developments in stochastic thermodynamics have shown that fluctuations in many far-from-equilibrium systems are constrained by the rate of entropy production via so-called thermodynamic uncertainty relations. These relations imply that increasing the reliability or precision of an engine's power output comes at a greater thermodynamic cost. Here we study the thermodynamics of precision for small thermal machines in the quantum regime. In particular, we derive exact relations between the power, power fluctuations, and entropy production rate for several models of few-qubit engines (both autonomous and cyclic) that perform work on a quantized load. Depending on the context, we find that quantum coherence can either help or hinder where power fluctuations are concerned. We discuss design principles for reducing such fluctuations in quantum nanomachines and propose an autonomous three-qubit engine whose power output for a given entropy production is more reliable than would be allowed by any classical Markovian model.

Thermodynamic uncertainty relation for energy transport in a transient regime: A model study

We investigate a transient version of the recently discovered thermodynamic uncertainty relation (TUR) which provides a precision-cost trade-off relation for certain out-of-equilibrium thermodynamic observables in terms of net entropy production. We explore this relation in the context of energy transport in a bipartite setting for three exactly solvable toy model systems (two coupled harmonic oscillators, two coupled qubits, and a hybrid coupled oscillator-qubit system) and analyze the role played by the underlying statistics of the transport carriers in the TUR. Interestingly, for all these models, depending on the statistics, the TUR ratio can be expressed as a sum or a difference of a universal term which is always greater than or equal to 2 and a corresponding entropy production term. We find that the generalized version of the TUR, originating from the universal fluctuation symmetry, is always satisfied. However, interestingly, the specialized TUR, a tighter bound, is always satisfied for the coupled harmonic oscillator system obeying Bose-Einstein statistics. Whereas, for both the coupled qubit, obeying Fermi-like statistics, and the hybrid qubit-oscillator system with mixed Fermi-Bose statistics, violation of the tighter bound is observed in certain parameter regimes. We have provided conditions for such violations. We also provide a rigorous proof following the nonequilibrium Green's function approach that the tighter bound is always satisfied in the weak-coupling regime for generic bipartite systems.

Thermodynamic uncertainty relations for bosonic Otto engines

We study two-mode bosonic engines undergoing an Otto cycle. The energy exchange between the two bosonic systems is provided by a tunable unitary bilinear interaction in the mode operators modeling frequency conversion, whereas the cyclic operation is guaranteed by relaxation to two baths at different temperatures after each interacting stage. By means of a two-point-measurement approach we provide the joint probability of the stochastic work and heat. We derive exact expressions for work and heat fluctuations, identities showing the interdependence among average extracted work, fluctuations, and efficiency, along with thermodynamic uncertainty relations between the signal-to-noise ratio of observed work and heat and the entropy production. We outline how the presented approach can be suitably applied to derive thermodynamic uncertainty relations for quantum Otto engines with alternative unitary strokes.

Thermodynamic Uncertainty Relation for Time-Dependent Driving

Thermodynamic uncertainty relations yield a lower bound on entropy production in terms of the mean and fluctuations of a current. We derive their general form for systems under arbitrary time-dependent driving from arbitrary initial states and extend these relations beyond currents to state variables. The quality of the bound is discussed for various types of observables for an interacting pair of colloidal particles in a moving laser trap and for the dynamical unfolding of a small protein. Since the input for evaluating these bounds does not require specific knowledge of the system or its coupling to the time-dependent control, they should become widely applicable tools for thermodynamic inference in time-dependently driven systems.

Power, efficiency, and fluctuations in steady-state heat engines

We consider the quality factor Q, which quantifies the trade-off between power, efficiency, and fluctuations in steady-state heat engines modeled by dynamical systems. We show that the nonlinear scattering theory, in both classical and quantum mechanics, sets the bound Q=3/8 when approaching the Carnot efficiency. On the other hand, interacting, nonintegrable, and momentum-conserving systems can achieve the value Q=1/2, which is the universal upper bound in linear response. This result shows that interactions are necessary to achieve the optimal performance of a steady-state heat engine.

Free diffusion bounds the precision of currents in underdamped dynamics

The putative generalization of the thermodynamic uncertainty relation (TUR) to underdamped dynamics is still an open problem. So far, bounds that have been derived for such a dynamics are not particularly transparent and they do not converge to the known TUR in the overdamped limit. Furthermore, it was found that there are restrictions for a TUR to hold such as the absence of a magnetic field. In this article we first analyze the properties of driven free diffusion in the underdamped regime and show that it inherently violates the overdamped TUR for finite times. Based on numerical evidence, we then conjecture a bound for one-dimensional driven diffusion in a potential which is based on the result for free diffusion. This bound converges to the known overdamped TUR in the corresponding limit. Moreover, the conjectured bound holds for observables that involve higher powers of the velocity as long as the observable is odd under time reversal. Finally, we address the applicability of this bound to underdamped dynamics in higher dimensions.

Quantum Thermodynamic Uncertainty Relation for Continuous Measurement

We use quantum estimation theory to derive a thermodynamic uncertainty relation in Markovian open quantum systems, which bounds the fluctuation of continuous measurements. The derived quantum thermodynamic uncertainty relation holds for arbitrary continuous measurements satisfying a scaling condition. We derive two relations; the first relation bounds the fluctuation by the dynamical activity and the second one does so by the entropy production. We apply our bounds to a two-level atom driven by a laser field and a three-level quantum thermal machine with jump and diffusion measurements. Our result shows that there exists a universal bound upon the fluctuations, regardless of continuous measurements.

Entropy production estimation with optimal current

Entropy production characterizes the thermodynamic irreversibility and reflects the amount of heat dissipated into the environment and free energy lost in nonequilibrium systems. According to the thermodynamic uncertainty relation, we propose a deterministic method to estimate the entropy production from a single trajectory of system states. We explicitly and approximately compute an optimal current that yields the tightest lower bound using predetermined basis currents. Notably, the obtained tightest lower bound is intimately related to the multidimensional thermodynamic uncertainty relation. By proving the saturation of the thermodynamic uncertainty relation in the short-time limit, the exact estimate of the entropy production can be obtained for overdamped Langevin systems, irrespective of the underlying dynamics. For Markov jump processes, because the attainability of the thermodynamic uncertainty relation is not theoretically ensured, the proposed method provides the tightest lower bound for the entropy production. When entropy production is the optimal current, a more accurate estimate can be further obtained using the integral fluctuation theorem. We illustrate the proposed method using three systems: a four-state Markov chain, a periodically driven particle, and a multiple bead-spring model. The estimated results in all examples empirically verify the effectiveness and efficiency of the proposed method.

Thermodynamic uncertainty relations including measurement and feedback

Thermodynamic uncertainty relations quantify how the signal-to-noise ratio of a given observable is constrained by dissipation. Fluctuation relations generalize the second law of thermodynamics to stochastic processes. We show that any fluctuation relation directly implies a thermodynamic uncertainty relation, considerably increasing their range of applicability. In particular, we extend thermodynamic uncertainty relations to scenarios which include measurement and feedback. Since feedback generally breaks time-reversal invariance, the uncertainty relations involve quantities averaged over the forward and the backward experiment defined by the associated fluctuation relation. This implies that the signal-to-noise ratio of a given experiment can in principle become arbitrarily large as long as the corresponding backward experiment compensates, e.g., by being sufficiently noisy. We illustrate our results with the Szilard engine as well as work extraction by free energy reduction in a quantum dot.

Thermodynamic uncertainty relation for underdamped Langevin systems driven by a velocity-dependent force

Recently, it has been shown that there is a trade-off relation between thermodynamic cost and current fluctuations, referred to as the thermodynamic uncertainty relation (TUR). The TUR has been derived for various processes, such as discrete-time Markov jump processes and overdamped Langevin dynamics. For underdamped dynamics, it has recently been reported that some modification is necessary for application of the TUR. However, the previous TUR for underdamped dynamics is not applicable to a system driven by a velocity-dependent force. In this study, we present a TUR, applicable to a system driven by a velocity-dependent force in the context of underdamped Langevin dynamics, by extending the theory of Vu and Hasegawa [Phys. Rev. E 100, 032130 (2019)2470-004510.1103/PhysRevE.100.032130]. We show that our TUR accurately describes the trade-off properties of a molecular refrigerator (cold damping), Brownian dynamics in a magnetic field, and an active particle system.

Thermodynamic uncertainty relation in thermal transport

We use the fundamental nonequilibrium steady-state fluctuation symmetry and derive a condition on the validity of the thermodynamic uncertainty relation (TUR) in thermal transport problems, classical and quantum alike. We test this condition and study the breakdown of the TUR in different thermal transport junctions of bosonic and electronic degrees of freedom. We prove that the TUR is valid in harmonic oscillator junctions. In contrast, in the nonequilibrium spin-boson model, which realizes many-body effects, it is satisfied in the Markovian limit, but violations arise as we tune (reduce) the cutoff frequency of the thermal baths, thus observing non-Markovian dynamics. We consider heat transport by noninteracting electrons in a tight-binding chain model. We show that the TUR is feasibly violated by tuning, e.g., the hybridization energy of the chain to the metal leads. These results manifest that the validity of the TUR relies on the statistics of the participating carriers, their interaction, and the nature of their couplings to the macroscopic contacts (metal electrodes and phonon baths).

Uncertainty relations for underdamped Langevin dynamics

A trade-off between the precision of an arbitrary current and the dissipation, known as the thermodynamic uncertainty relation, has been investigated for various Markovian systems. Here, we study the thermodynamic uncertainty relation for underdamped Langevin dynamics. By employing information inequalities, we prove that for such systems, the relative fluctuation of a current at a steady state is constrained by both the entropy production and the average dynamical activity. We find that unlike what is the case for overdamped dynamics, the dynamical activity plays an important role in the bound. We illustrate our results with two systems, a single-well potential system and a periodically driven Brownian particle model, and numerically verify the inequalities.

Fluctuation Theorem Uncertainty Relation

The fluctuation theorem is the fundamental equality in nonequilibrium thermodynamics that is used to derive many important thermodynamic relations, such as the second law of thermodynamics and the Jarzynski equality. Recently, the thermodynamic uncertainty relation was discovered, which states that the fluctuation of observables is lower bounded by the entropy production. In the present Letter, we derive a thermodynamic uncertainty relation from the fluctuation theorem. We refer to the obtained relation as the fluctuation theorem uncertainty relation, and it is valid for arbitrary dynamics, stochastic as well as deterministic, and for arbitrary antisymmetric observables for which a fluctuation theorem holds. We apply the fluctuation theorem uncertainty relation to an overdamped Langevin dynamics for an antisymmetric observable. We demonstrate that the antisymmetric observable satisfies the fluctuation theorem uncertainty relation but does not satisfy the relation reported for current-type observables in continuous-time Markov chains. Moreover, we show that the fluctuation theorem uncertainty relation can handle systems controlled by time-symmetric external protocols, in which the lower bound is given by the work exerted on the systems.

Thermodynamic Uncertainty Relations from Exchange Fluctuation Theorems

Thermodynamic uncertainty relations (TURs) place strict bounds on the fluctuations of thermodynamic quantities in terms of the associated entropy production. In this Letter, we identify the tightest (and saturable) matrix-valued TUR that can be derived from the exchange fluctuation theorems describing the statistics of heat and particle flow between multiple systems of arbitrary dimensions. Our result holds for both quantum and classical systems, undergoing general finite-time nonstationary processes. Moreover, it provides bounds not only for the variances, but also for the correlations between thermodynamic quantities. To demonstrate the relevance of TURs to the design of nanoscale machines, we consider the operation of a 2-qubit swap engine undergoing an Otto cycle and show how our results can be used to place strict bounds on the correlations between heat and work.

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.

Uncertainty relations in stochastic processes: An information inequality approach

The thermodynamic uncertainty relation is an inequality stating that it is impossible to attain higher precision than the bound defined by entropy production. In statistical inference theory, information inequalities assert that it is infeasible for any estimator to achieve an error smaller than the prescribed bound. Inspired by the similarity between the thermodynamic uncertainty relation and the information inequalities, we apply the latter to systems described by Langevin equations, and we derive the bound for the fluctuation of thermodynamic quantities. When applying the Cramér-Rao inequality, the obtained inequality reduces to the fluctuation-response inequality. We find that the thermodynamic uncertainty relation is a particular case of the Cramér-Rao inequality, in which the Fisher information is the total entropy production. Using the equality condition of the Cramér-Rao inequality, we find that the stochastic total entropy production is the only quantity that can attain equality in the thermodynamic uncertainty relation. Furthermore, we apply the Chapman-Robbins inequality and obtain a relation for the lower bound of the ratio between the variance and the sensitivity of systems in response to arbitrary perturbations.

Thermodynamic uncertainty relation in quantum thermoelectric junctions

Recently, a thermodynamic uncertainty relation (TUR) has been formulated for classical Markovian systems demonstrating trade-off between precision (current fluctuation) and cost (dissipation). Systems that violate the TUR are interesting as they overcome another trade-off relation concerning the efficiency of a heat engine, its power, and its stability (power fluctuations). Here, we analyze the root, extent, and impact on performance of TUR violations in quantum thermoelectric junctions at steady state. Considering noninteracting electrons, first we show that only the "classical" component of the current noise, arising from single-electron transfer events, follows the TUR. The remaining, "quantum" part of current noise is therefore responsible for the potential violation of the TUR in such quantum systems. Next, focusing on the resonant transport regime we determine the parameter range in which the violation of the TUR can be observed-for both voltage-biased junctions and thermoelectric engines. We illustrate our findings with exact numerical simulations of a serial double quantum dot system. Most significantly, we demonstrate that the TUR always holds in noninteracting thermoelectric generators when approaching the thermodynamic efficiency limit.

Unraveling the Large Deviation Statistics of Markovian Open Quantum Systems

We analyze dynamical large deviations of quantum trajectories in Markovian open quantum systems in their full generality. We derive a quantum level-2.5 large deviation principle for these systems, which describes the joint fluctuations of time-averaged quantum jump rates and of the time-averaged quantum state for long times. Like its level-2.5 counterpart for classical continuous-time Markov chains (which it contains as a special case), this description is both explicit and complete, as the statistics of arbitrary time-extensive dynamical observables can be obtained by contraction from the explicit level-2.5 rate functional we derive. Our approach uses an unraveled representation of the quantum dynamics which allows these statistics to be obtained by analyzing a classical stochastic process in the space of pure states. For quantum reset processes we show that the unraveled dynamics is semi-Markovian and derive bounds on the asymptotic variance of the number of quantum jumps which generalize classical thermodynamic uncertainty relations. We finish by discussing how our level-2.5 approach can be used to study large deviations of nonlinear functions of the state, such as measures of entanglement.

Effect of a magnetic field on the thermodynamic uncertainty relation

The thermodynamic uncertainty relation provides a universal lower bound on the product of entropy production and the fluctuations of any current. While proven for Markov dynamics on a discrete set of states and for overdamped Langevin dynamics, its status for underdamped dynamics is still open. We consider a two-dimensional harmonically confined charged particle in a magnetic field under the action of an external torque. We show analytically that, depending on the sign of the magnetic field, the thermodynamic uncertainty relation does not hold for the currents associated with work and heat. A strong magnetic field can effectively localize the particle with concomitant bounded fluctuations and low dissipation. Numerical results for a three-dimensional variant and for further currents suggest that the existence of such a bound depends crucially on the specific current.

Beyond Onsager-Casimir Relations: Shared Dependence of Phenomenological Coefficients on State Variables

Phenomenological coefficients in linear nonequilibrium thermodynamics have been considered to be independent, apart from restrictions due to the Onsager-Casimir reciprocal relations and the requirement to have non-negative entropy production. Recently, it has been shown that functional constraints between these coefficients may hold, restricting their dependence on state variables, especially in the case of coupled phenomena. Here we demonstrate that such restrictions require only mild assumptions on the system of interest and are, in fact, much more constraining than previously reported. Such constraints vastly reduce the set of plausible models for constitutive relations and allow for simpler experimental determinations of dependencies in coupled systems. These results may also clarify inconsistencies in the literature regarding constitutive models used that do not obey these thermodynamic constraints.