Finite-time generalization of the thermodynamic uncertainty relation

Steady-state solution of Markov jump processes in terms of arrival probabilities

Several dynamical processes can be modeled as Markov jump processes among a finite number N of sites (the distinct physical states). Here we consider strongly connected networks with time-independent site-to-site jump rate constants, and focus on the steady-state occupation probabilities of the sites. We provide a physically framed expression of the steady-state distribution in terms of arrival probabilities, here defined as the probabilities of going from starting sites to target sites with a given number of jumps (regardless of the time required). In particular, the full set of return probabilities (for all the sites of the network) up to N-1 jumps is necessary and sufficient. A few examples illustrate the outcomes, including the case of stochastic chemical kinetics.

Dissipation Bounds Precision of Current Response to Kinetic Perturbations

The precision of currents in Markov networks is bounded by dissipation via the so-called thermodynamic uncertainty relation (TUR). In our Letter, we demonstrate a similar inequality that bounds the precision of the static current response to perturbations of kinetic barriers. Perturbations of such type, which affect only the system kinetics but not the thermodynamic forces, are highly important in biochemistry and nanoelectronics. We prove that our inequality cannot be derived from the standard TUR. Instead, it implies the standard TUR and provides an even tighter bound for dissipation. We also provide a procedure for obtaining the optimal response precision for a given model.

Stochastic differential equation for a system coupled to a thermostatic bath via an arbitrary interaction Hamiltonian

The conventional Langevin equation offers a mathematically convenient framework for investigating open stochastic systems interacting with their environment or a bath. However, it is not suitable for a wide variety of systems whose dynamics rely on the nature of the environmental interaction, as the equation does not incorporate any specific information regarding that interaction. Here, we present a stochastic differential equation (SDE) for an open system coupled to a thermostatic bath via an arbitrary interaction Hamiltonian. This SDE encodes the interaction information to a fictitious potential (mean force) and a position-dependent damping coefficient. Surprisingly, we find that the conventional Langevin equation can be recovered in the presence of arbitrary strong interactions given two conditions: translational invariance of the potential and mutual independence of baths. Our results provide a comprehensive framework for studying open stochastic systems with an arbitrary interaction Hamiltonian and yield deeper insight into why various experiments fit the conventional Langevin description regardless of the strength or type of interaction.

Steady-state probabilities for Markov jump processes in terms of powers of the transition rate matrix

Several types of dynamics at stationarity can be described in terms of a Markov jump process among a finite number N of representative sites. Before dealing with the dynamical aspects, one basic problem consists in expressing the a priori steady-state occupation probabilities of the sites. In particular, one wishes to go beyond the mere black-box computational tools and find expressions in which the jump rate constants appear explicitly, therefore allowing for a potential design/control of the network. For strongly connected networks admitting a unique stationary state with all sites populated, here we express the occupation probabilities in terms of a formula that involves powers of the transition rate matrix up to order N - 1. We also provide an expression of the derivatives with respect to the jump rate constants, possibly useful in sensitivity analysis frameworks. Although we refer to dynamics in (bio)chemical networks at thermal equilibrium or under nonequilibrium steady-state conditions, the results are valid for any Markov jump process under the same assumptions.

Thermodynamic cost for precision of general counting observables

We analytically derive universal bounds that describe the tradeoff between thermodynamic cost and precision in a sequence of events related to some internal changes of an otherwise hidden physical system. The precision is quantified by the fluctuations in either the number of events counted over time or the waiting times between successive events. Our results are valid for the same broad class of nonequilibrium driven systems considered by the thermodynamic uncertainty relation, but they extend to both time-symmetric and asymmetric observables. We show how optimal precision saturating the bounds can be achieved. For waiting-time fluctuations of asymmetric observables, a phase transition in the optimal configuration arises, where higher precision can be achieved by combining several signals.

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.

Geometric phaselike effects of driven transport in presence of reservoir squeezing

In a bare bosonic site coupled to two reservoirs, we explore the statistics of boson exchange in the presence of two simultaneous processes: squeezing the two reservoirs and driving the two reservoirs. The squeezing parameters compete with the geometric phaselike effect or geometricity to alter the nature of the steady-state flux and noise. The even (odd) geometric cumulants and the total minimum entropy are found to be symmetric (antisymmetric) with respect to exchanging the left and right squeezing parameters. Upon increasing the strength of the squeezing parameters, loss of geometricity is observed. Under maximum squeezing, one can recover a standard steady-state fluctuation theorem even in the presence of phase-different driving protocol. A recently proposed modified geometric thermodynamic uncertainty principle is found to be robust.

Thermodynamic precision in the nonequilibrium exchange scenario

We discuss exchange scenario thermodynamic uncertainty relations for the work done on a two-qubit entangled nonequilibrium steady state obtained by coupling the two qubits and putting each of them in weak contact with a thermal bath. In this way we investigate the use of entangled nonequilibrium steady states as end points of thermodynamic cycles. In this framework we prove analytically that for a paradigmatic unitary it is possible to construct an exchange scenario thermodynamic uncertainty relation. However, despite holding in many cases, we also show that such a relation ceases to be valid when considering other suitable unitary quenches. Furthermore, this paradigmatic example allows us to shed light on the role of the entanglement between the two qubits for precise work absorption. By considering the projection of the entangled steady state onto the set of separable states, we provide examples where such projection implies an increase of the relative uncertainty, showing the usefulness of entanglement.

Persistence time bound for subdiffusion based on multidimensional thermodynamic uncertainty relation: Application to an analytically solvable model

The thermodynamic uncertainty relation (TUR) is an inequality showing the tradeoff relationship between the relative fluctuation of current observables and thermodynamic costs. It is one of the most important results of stochastic thermodynamics. There are various applications for TUR, one of which is the recent finding of thermodynamic constraints on the time window in which anomalous diffusion of Brownian particles can occur, including subdiffusion and superdiffusion, which are slower and faster than normal diffusion, respectively. These constraints are quite nontrivial because they are not generally derived from the asymptotic normal-diffusive behavior of the anomalous diffusion itself. In this study, we applied multidimensional TUR to the subdiffusion of Brownian particles obeying multivariate Langevin dynamics with a translationally invariant Hamiltonian in equilibrium. Multidimensional TUR is an improved TUR that includes information on another observable in addition to the one currently being considered. The use of an additional observable yields tighter bounds on the current fluctuation than those obtained using TUR. As an example, we demonstrated our theory using the one-dimensional Rouse model, which is known as a simple and analytically tractable model of polymer chains. We demonstrated that we improved the bounds for the persistence time of subdiffusion of the Rouse model, which became tighter as a more correlated observable with the current was used.

General Upper Bounds on Fluctuations of Trajectory Observables

Thermodynamic uncertainty relations (TURs) are general lower bounds on the size of fluctuations of dynamical observables. They have important consequences, one being that the precision of estimation of a current is limited by the amount of entropy production. Here, we prove the existence of general upper bounds on the size of fluctuations of any linear combination of fluxes (including all time-integrated currents or dynamical activities) for continuous-time Markov chains. We obtain these general relations by means of concentration bound techniques. These "inverse TURs" are valid for all times and not only in the long time limit. We illustrate our analytical results with a simple model, and discuss wider implications of these new relations.

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.

Arbitrary-time thermodynamic uncertainty relation from fluctuation theorem

The thermodynamic uncertainty relation (TUR) provides a universal entropic bound for the precision of the fluctuation of the charge transfer, for example, for a class of continuous-time stochastic processes. However, its extension to general nonequilibrium dynamics is still an unsolved problem. We derive TUR for an arbitrary finite time from exchange fluctuation theorem under a geometric necessary and sufficient condition. We also generally show a necessary and sufficient condition of multidimensional TUR in a unified manner. As a nontrivial practical consequence, we obtain universal scaling relations among the mean and variance of the charge transfer in short time regime. In this manner, we can deepen our understanding of a link between two important rigorous relations, i.e., the fluctuation theorem and the thermodynamic uncertainty relation.

Stochastic Thermodynamics of Multiple Co-Evolving Systems-Beyond Multipartite Processes

Many dynamical systems consist of multiple, co-evolving subsystems (i.e., they have multiple degrees of freedom). Often, the dynamics of one or more of these subsystems will not directly depend on the state of some other subsystems, resulting in a network of dependencies governing the dynamics. How does this dependency network affect the full system's thermodynamics? Prior studies on the stochastic thermodynamics of multipartite processes have addressed this question by assuming that, in addition to the constraints of the dependency network, only one subsystem is allowed to change state at a time. However, in many real systems, such as chemical reaction networks or electronic circuits, multiple subsystems can-or must-change state together. Here, we investigate the thermodynamics of such composite processes, in which multiple subsystems are allowed to change state simultaneously. We first present new, strictly positive lower bounds on entropy production in composite processes. We then present thermodynamic uncertainty relations for information flows in composite processes. We end with strengthened speed limits for composite processes.

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.

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).

Bound for the moment generating function from the detailed fluctuation theorem

A famous consequence of the detailed fluctuation theorem (FT), p(Σ)/p(-Σ)=exp(Σ), is the integral FT 〈exp(-Σ)〉=1 for a random variable Σ and a distribution p(Σ). When Σ represents the entropy production in thermodynamics, the main outcome of the integral FT is the second law, 〈Σ〉≥0. However, a full description of the fluctuations of Σ might require knowledge of the moment generating function (MGF), G(α):=〈exp(αΣ)〉. In the context of the detailed FT, we show the MGF is lower bounded in the form G(α)≥B(α,〈Σ〉) for a given mean 〈Σ〉. As applications, we verify that the bound is satisfied for the entropy produced in the heat exchange problem between two reservoirs mediated by a weakly coupled bosonic mode and a qubit swap engine.

Thermodynamic bounds for diffusion in nonequilibrium systems with multiple timescales

We derive a thermodynamic uncertainty relation bounding the mean squared displacement of a Gaussian process with memory, driven out of equilibrium by unbalanced thermal baths and/or by external forces. Our bound is tighter with respect to previous results and also holds at finite time. We apply our findings to experimental and numerical data for a vibrofluidized granular medium, characterized by regimes of anomalous diffusion. In some cases our relation can distinguish between equilibrium and nonequilibrium behavior, a nontrivial inference task, particularly for Gaussian processes.

Upper bounding the average residence times in partially observed steady-state Markov jump processes

Several types of stochastic dynamics can be modeled as a continuous-time Markov jump process among a finite number of sites. Within such framework, we face the problem of getting an upper bound on the average residence time of the system in a given site β (i.e., the average lifetime of the site) if what we can observe is only the permanence of the system in an adjacent site α and the occurrence of the transitions α→β. Supposing to have a long time record of this partial monitoring of the network under steady-state conditions, we show that an upper bound on the average time spent in the unobserved site can indeed be given. The bound is formally proved, tested by means of simulations, and illustrated for a multicyclic enzymatic reaction scheme.

Thermodynamics of a continuously monitored double-quantum-dot heat engine in the repeated interactions framework

Understanding the thermodynamic role of measurement in quantum mechanical systems is a burgeoning field of study. In this article, we study a double quantum dot (DQD) connected to two macroscopic fermionic thermal reservoirs. We assume that the DQD is continuously monitored by a quantum point contact (QPC), which serves as a charge detector. Starting from a minimalist microscopic model for the QPC and reservoirs, we show that the local master equation of the DQD can alternatively be derived in the framework of repeated interactions and that this framework guarantees a thermodynamically consistent description of the DQD and its environment (including the QPC). We analyze the effect of the measurement strength and identify a regime in which particle transport through the DQD is both assisted and stabilized by dephasing. We also find that in this regime the entropic cost of driving the particle current with fixed relative fluctuations through the DQD is reduced. We thus conclude that under continuous measurement a more constant particle current may be achieved at a fixed entropic cost.

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.

Direct Route to Thermodynamic Uncertainty Relations and Their Saturation

Thermodynamic uncertainty relations (TURs) bound the dissipation in nonequilibrium systems from below by fluctuations of an observed current. Contrasting the elaborate techniques employed in existing proofs, we here prove TURs directly from the Langevin equation. This establishes the TUR as an inherent property of overdamped stochastic equations of motion. In addition, we extend the transient TUR to currents and densities with explicit time dependence. By including current-density correlations we, moreover, derive a new sharpened TUR for transient dynamics. Our arguably simplest and most direct proof, together with the new generalizations, allows us to systematically determine conditions under which the different TURs saturate and thus allows for a more accurate thermodynamic inference. Finally, we outline the direct proof also for Markov jump dynamics.

Improving estimation of entropy production rate for run-and-tumble particle systems by high-order thermodynamic uncertainty relation

Entropy production plays an important role in the regulation and stability of active matter systems, and its rate quantifies the nonequilibrium nature of these systems. However, entropy production is hard to experimentally estimate even in some simple active systems like molecular motors or bacteria, which may be modeled by the run-and-tumble particle (RTP), a representative model in the study of active matters. Here we resolve this problem for an asymmetric RTP in one dimension, first constructing a finite-time thermodynamic uncertainty relation (TUR) for a RTP, which works well in the short observation time regime for entropy production estimation. Nevertheless, when the activity dominates, i.e., the RTP is far from equilibrium, the lower bound for entropy production from TUR turns out to be trivial. We address this issue by introducing a recently proposed high-order thermodynamic uncertainty relation (HTUR), in which the cumulant generating function of current serves as a key ingredient. To exploit the HTUR, we adopt a method to analytically obtain the cumulant generating function of the current we study, with no need to explicitly know the time-dependent probability distribution. The HTUR is demonstrated to be able to estimate the steady state energy dissipation rate accurately because the cumulant generating function covers higher-order statistics of the current, including rare and large fluctuations besides its variance. Compared to the conventional TUR, the HTUR could give significantly improved estimation of energy dissipation, which can work well even in the far from equilibrium regime. We also provide a strategy based on the improved bound to estimate the entropy production from a moderate amount of trajectory data for experimental feasibility.

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.

Tight uncertainty relations for cycle currents

Several recent inequalities bound the precision of a current, i.e., a counter of the net number of transitions in a system, by a thermodynamic measure of dissipation. However, while currents may be defined locally, dissipation is a global property. Inspired by the fact that, ever since Carnot, cycles are the unit elements of thermodynamic processes, we prove similar bounds tailored to cycle currents, counting net cycle completions, in terms of their conjugate affinities. We show that these inequalities are stricter than previous ones, even far from equilibrium, and that they allow us to tighten those on transition currents. We illustrate our results with a simple model and discuss some technical and conceptual issues related to shifting attention from transition to cycle observables.

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.

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.

Bounds on skewness and kurtosis of steady-state currents

Current fluctuations are a powerful tool to unravel the underlying physics of the observed transport process. This work discusses some general properties of the third and the fourth current cumulant (skewness and kurtosis) related to dynamics and thermodynamics of a transport setup. Specifically, several distinct bounds on these quantities are either analytically derived or numerically conjectured, which are applicable to (1) noninteracting fermionic systems, (2) noninteracting bosonic systems, (3) thermally driven classical Markovian systems, and (4) unicyclic Markovian networks. Finally, it is demonstrated that violation of the obtained inequalities can provide a broad spectrum of information about the physics of the analyzed system; e.g., it can enable one to infer the presence of interactions or unitary dynamics, unravel the topology of the Markovian network, or characterize the nature of thermodynamic forces driving the system. In particular, relevant information about the microscopic dynamics can be gained even at equilibrium when the current variance-a standard measure of current fluctuations-is determined mostly by the thermal noise.

Stochastic currents and efficiency in an autonomous heat engine

We experimentally demonstrate that a Brownian gyrator of a colloidal particle confined in a two-dimensional harmonic potential with different effective temperatures on orthogonal axes can work as an autonomous heat engine capable of extracting work from the heat bath, generated by an optical feedback trap. The results confirm the theoretically predicted thermodynamic currents and validate the attainability of Carnot efficiency as well as the trade-off relation between power and efficiency. We further show that current fluctuations and the entropy production rate are time independent in the steady state and their product near the Carnot efficiency is close to the lower bound of the thermodynamic uncertainty relation.

Thermodynamic uncertainty relation for Langevin dynamics by scaling time

The thermodynamic uncertainty relation (TUR) quantifies a relationship between current fluctuations and dissipation in out-of-equilibrium overdamped Langevin dynamics, making it a natural counterpart of the fluctuation-dissipation theorem in equilibrium statistical mechanics. For underdamped Langevin dynamics, the situation is known to be more complicated with dynamical activity also playing a role in limiting the magnitude of current fluctuations. Progress on those underdamped TUR-like bounds has largely come from applications of the information-theoretic Cramér-Rao inequality. Here, we present an alternative perspective by employing large deviation theory. The approach offers a general unified treatment of TUR-like bounds for both overdamped and underdamped Langevin dynamics built upon current fluctuations achieved by scaling time. The bounds we derive following this approach are similar to known results but with differences we discuss and rationalize.

Geometric decomposition of entropy production into excess, housekeeping, and coupling parts

For a generic overdamped Langevin dynamics driven out of equilibrium by both time-dependent and nonconservative forces, the entropy production rate can be decomposed into two positive terms, termed excess and housekeeping entropy. However, this decomposition is not unique: There are two distinct decompositions, one due to Hatano and Sasa, the other one due to Maes and Netočný. Here we establish the connection between these two decompositions and provide a simple, geometric interpretation. We show that this leads to a decomposition of the entropy production rate into three positive terms, which we call the excess, housekeeping, and coupling part, respectively. The coupling part characterizes the interplay between the time-dependent and nonconservative forces. We also derive thermodynamic uncertainty relations for the excess and housekeeping entropy in both the Hatano-Sasa and Maes-Netočný decomposition and show that all quantities obey integral fluctuation theorems. We illustrate the decomposition into three terms using a solvable example of a dragged particle in a nonconservative force field.

Universal minimal cost of coherent biochemical oscillations

Biochemical clocks are essential for virtually all living systems. A biochemical clock that is isolated from an external periodic signal and subjected to fluctuations can oscillate coherently only for a finite number of oscillations. Furthermore, such an autonomous clock can oscillate only if it consumes free energy. What is the minimum amount of free-energy consumption required for a certain number of coherent oscillations? We conjecture a universal bound that answers this question. A system that oscillates coherently for N oscillations has a minimal free-energy cost per oscillation of 4π^{2}Nk_{B}T. Our bound is valid for general finite Markov processes, is conjectured based on extensive numerical evidence, is illustrated with numerical simulations of a known model for a biochemical oscillator, and applies to existing experimental data.

Improved estimation for energy dissipation in biochemical oscillations

Biochemical oscillations, regulating the timing of life processes, need consume energy to achieve good performance on crucial functions, such as high accuracy of phase period and high sensitivity to external signals. However, it is a great challenge to precisely estimate the energy dissipation in such systems. Here, based on the stochastic normal form theory (SNFT), we calculate the Pearson correlation coefficient between the oscillatory amplitude and phase, and a trade-off relation between transport efficiency and phase sensitivity can then be derived, which serves as a tighter form than the estimator resulting from the conventional thermodynamic uncertainty relation (TUR). Our findings demonstrate that a more precise energy dissipation estimation can be obtained by enhancing the sensitivity of the biochemical oscillations. Moreover, the internal noise and amplitude power effects have also been discovered.

Classical Pendulum Clocks Break the Thermodynamic Uncertainty Relation

The thermodynamic uncertainty relation expresses a seemingly universal trade-off between the cost for driving an autonomous system and precision in any output observable. It has so far been proven for discrete systems and for overdamped Brownian motion. Its validity for the more general class of underdamped Brownian motion, where inertia is relevant, was conjectured based on numerical evidence. We now disprove this conjecture by constructing a counterexample. Its design is inspired by a classical pendulum clock, which uses an escapement to couple the motion of an oscillator to regulate the motion of another degree of freedom (a "hand") driven by an external force. Considering a thermodynamically consistent, discrete model for an escapement mechanism, we first show that the oscillations of an underdamped harmonic oscillator in thermal equilibrium are sufficient to break the thermodynamic uncertainty relation. We then show that this is also the case in simulations of a fully continuous underdamped system with a potential landscape that mimics an escaped pendulum.

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.

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.

Higher-order efficiency bound and its application to nonlinear nanothermoelectrics

Power and efficiency of heat engines are two conflicting objectives. A tight efficiency bound is expected to give insights on the fundamental properties of such a power-efficiency tradeoff. Here, we derive an upper bound on the efficiency of steady-state heat engines, which incorporates higher-order fluctuations of power. In a prototypical model of nonlinear nanostructured thermoelectrics, we show that the obtained bound is tighter than a well-established efficiency bound derived from the thermodynamic uncertainty relation, demonstrating that the higher-order terms have rich information about the thermodynamic efficiency in the nonlinear regime. In particular, we find that the higher-order bound is exactly achieved if the tight coupling condition is satisfied. The obtained bound gives a consistent prediction with an observation that nonlinearity enhances the power-efficiency tradeoff, and would also be useful in a variety of nanoscale engines.

Thermodynamic Uncertainty Relation and Thermodynamic Speed Limit in Deterministic Chemical Reaction Networks

We generalize the thermodynamic uncertainty relation (TUR) and thermodynamic speed limit (TSL) for deterministic chemical reaction networks (CRNs). The scaled diffusion coefficient derived by considering the connection between macro- and mesoscopic CRNs plays an essential role in our results. The TUR shows that the product of the entropy production rate and the ratio of the scaled diffusion coefficient to the square of the rate of concentration change is bounded below by two. The TSL states a trade-off relation between speed and thermodynamic quantities, the entropy production, and the time-averaged scaled diffusion coefficient. The results are proved under the general setting of open and nonideal CRNs.

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.

Cost-precision trade-off relation determines the optimal morphogen gradient for accurate biological pattern formation

Spatial boundaries formed during animal development originate from the pre-patterning of tissues by signaling molecules, called morphogens. The accuracy of boundary location is limited by the fluctuations of morphogen concentration that thresholds the expression level of target gene. Producing more morphogen molecules, which gives rise to smaller relative fluctuations, would better serve to shape more precise target boundaries; however, it incurs more thermodynamic cost. In the classical diffusion-depletion model of morphogen profile formation, the morphogen molecules synthesized from a local source display an exponentially decaying concentration profile with a characteristic length λ. Our theory suggests that in order to attain a precise profile with the minimal cost, λ should be roughly half the distance to the target boundary position from the source. Remarkably, we find that the profiles of morphogens that pattern the Drosophila embryo and wing imaginal disk are formed with nearly optimal λ. Our finding underscores the cost-effectiveness of precise morphogen profile formation in Drosophila development.

Inertial effects on the Brownian gyrator

The recent interest into the Brownian gyrator has been confined chiefly to the analysis of Brownian dynamics both in theory and experiment despite the applicability of general cases with definite mass. Considering mass explicitly in the solution of the Fokker-Planck equation and Langevin dynamics simulations, we investigate how inertia can change the dynamics and energetics of the Brownian gyrator. In the Langevin model, the inertia reduces the nonequilibrium effects by diminishing the declination of the probability density function and the mean of a specific angular momentum, j_{θ}, as a measure of rotation. Another unique feature of the Langevin description is that rotation is maximized at a particular anisotropy while the stability of the rotation is minimized at a particular anisotropy or mass. Our results suggest that the Langevin dynamics description of the Brownian gyrator is intrinsically different from that with Brownian dynamics. In addition, j_{θ} is proven to be essential and convenient for estimating stochastic energetics such as heat currents and entropy production even in the underdamped regime.

Thermodynamic uncertainty relation to assess biological processes

We review the trade-offs between speed, fluctuations, and thermodynamic cost involved with biological processes in nonequilibrium states and discuss how optimal these processes are in light of the universal bound set by the thermodynamic uncertainty relation (TUR). The values of the uncertainty product Q of TUR, which can be used as a measure of the precision of enzymatic processes realized for a given thermodynamic cost, are suboptimal when the substrate concentration is at the Michaelis constant, and some of the key biological processes are found to work around this condition. We illustrate the utility of Q in assessing how close the molecular motors and biomass producing machineries are to the TUR bound, and for the cases of biomass production (or biological copying processes), we discuss how their optimality quantified in terms of Q is balanced with the error rate in the information transfer process. We also touch upon the trade-offs in other error-minimizing processes in biology, such as gene regulation and chaperone-assisted protein folding. A spectrum of Q recapitulating the biological processes surveyed here provides glimpses into how biological systems are evolved to optimize and balance the conflicting functional requirements.

Dissipation-recurrence inequalities at the steady state

For Markov jump processes in out-of-equilibrium steady state, we present inequalities which link the average rate of entropy production with the timing of the site-to-site recurrences. Such inequalities are upper bounds on the average rate of entropy production. The combination with the finite-time thermodynamic uncertainty relation (a lower bound) yields inequalities of the pure kinetic kind for the relative precision of a dynamical output. After having derived the main relations for the discrete case, we sketch the possible extension to overdamped Markov dynamics on continuous degrees of freedom, treating explicitly the case of one-dimensional diffusion in tilted periodic potentials; an upper bound on the average velocity is derived, in terms of the average rate of entropy production and the microscopic diffusion coefficient, which corresponds to the finite-time thermodynamic uncertainty relation in the limit of vanishingly small observation time.

Nonequilibrium Thermodynamics in Biochemical Systems and Its Application

Living systems are open systems, where the laws of nonequilibrium thermodynamics play the important role. Therefore, studying living systems from a nonequilibrium thermodynamic aspect is interesting and useful. In this review, we briefly introduce the history and current development of nonequilibrium thermodynamics, especially that in biochemical systems. We first introduce historically how people realized the importance to study biological systems in the thermodynamic point of view. We then introduce the development of stochastic thermodynamics, especially three landmarks: Jarzynski equality, Crooks' fluctuation theorem and thermodynamic uncertainty relation. We also summarize the current theoretical framework for stochastic thermodynamics in biochemical reaction networks, especially the thermodynamic concepts and instruments at nonequilibrium steady state. Finally, we show two applications and research paradigms for thermodynamic study in biological systems.

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.

Sensitivity analysis of the reaction occurrence and recurrence times in steady-state biochemical networks

Continuous-time stationary Markov jump processes among discrete sites are encountered in disparate biochemical ambits. Sites and connecting dynamical events form a 'network' in which the sites are the available system's states, and the events are site-to-site transitions, or even neutral processes in which the system does not change site but the event is however detectable. Examples include conformational transitions in single biomolecules, stochastic chemical kinetics in the space of the molecules copy numbers, and even macroscopic steady-state reactive mixtures if one adopts the viewpoint of a tagged molecule (or even of a molecular moiety) whose state may change when it is involved in a chemical reaction. Among the variety of dynamical descriptors, here we focus on the first occurrence times (starting from a given site) and on the recurrence times of an event of interest. We develop the sensitivity analysis for the lowest moments of the statistical distribution of such times with respect to the rate constants of the network. In particular, simple expressions and inequalities allow us to establish a direct relationship between selective variation of rate constants and effect on average times and variances. As illustrative cases we treat the substrate inhibition in enzymatic catalysis in which a tagged enzyme molecule jumps between three states, and the basic two-site model of stochastic gene expression in which the single gene switches between active and inactive forms.