Dissipation Bounds All Steady-State Current Fluctuations

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.

Information geometry approach to quantum stochastic thermodynamics

Recent advancements have revealed new links between information geometry and classical stochastic thermodynamics, particularly through the Fisher information (FI) with respect to time. Recognizing the nonuniqueness of the quantum Fisher metric in Hilbert space, we exploit the fact that any quantum Fisher information (QFI) can be decomposed into a metric-independent incoherent part and a metric-dependent coherent contribution. We demonstrate that the incoherent component of any QFI can be directly linked to entropic acceleration, and for GKSL dynamics with local detailed balance, to the rate of change of generalized thermodynamic forces and entropic flow, paralleling the classical results. Furthermore, we tighten a classical uncertainty relation between the geometric uncertainty of a path in state space and the time-averaged rate of information change and demonstrate that it also holds for quantum systems. We generalize a classical geometric bound on the entropy rate for far-from-equilibrium processes by incorporating a nonnegative quantum contribution that arises from the geometric action due to coherent dynamics. Finally, we apply an information-geometric analysis to the recently proposed quantum-thermodynamic Mpemba effect, demonstrating this framework's ability to capture thermodynamic phenomena.

Random walk with horizontal and cyclic currents

We construct a minimal two-chain random walk model and study the information that fluctuations of the flux and higher cumulants can reveal about the model: its structure, parameters, and whether it operates under nonequilibrium conditions. The two coupled chains allow for both horizontal and cyclic transport. We capture these processes by deriving the cumulant generating function of the system, which characterizes both horizontal and cyclic transport in the long time limit. First, we show that either the horizontal or the cyclic currents, along with their higher-order cumulants, can be used to unravel the intrinsic structure and parameters of the model. Second, we investigate the "zero current" situation, in which the horizontal current vanishes. We find that fluctuations of the horizontal current reveal the nonequilibrium condition at intermediate bias, while the cyclic current remains nonzero throughout. We also show that in nonequilibrium scenarios close to the zero horizontal current limit, the entropy production rate is more tightly lower-bounded by the relative noise of the cyclic current, and vice versa. Finally, simulations of transport before the steady state sets in allow for the extraction of the interchain hopping rate. Our study, illustrating the information concealed in fluctuations, could see applications in chemical networks, cellular processes, and charge and energy transport materials.

Universal energy-speed-accuracy trade-offs in driven nonequilibrium systems

The connection between measure theoretic optimal transport and dissipative nonequilibrium dynamics provides a language for quantifying nonequilibrium control costs, leading to a collection of thermodynamic speed limits, which rely on the assumption that the target probability distribution is perfectly realized. This is almost never the case in experiments or numerical simulations, so here we address the situation in which the external controller is imperfect. We obtain a lower bound for the dissipated work in generic nonequilibrium control problems that (1) is asymptotically tight and (2) matches the thermodynamic speed limit in the case of optimal driving. Along with analytically solvable examples, we refine this imperfect driving notion to systems in which the controlled degrees of freedom are slow relative to the nonequilibrium relaxation rate, and identify independent energy contributions from fast and slow degrees of freedom. Furthermore, we develop a strategy for optimizing minimally dissipative protocols based on optimal transport flow matching, a generative machine learning technique. This latter approach ensures the scalability of both the theoretical and computational framework we put forth. Crucially, we demonstrate that we can compute the terms in our bound numerically using efficient algorithms from the computational optimal transport literature and that the protocols we learn saturate the bound.

Stochastic Scovil-Schulz-DuBois machine and its three types of cycles

Three types of cycles are identified in the quantum jump trajectories of the Scovil-Schulz-DuBois (SSDB) machine: An R cycle as refrigeration, an H cycle as a heat engine, and an N cycle in which the machine is neutral. The statistics of these cycles are investigated via a semi-Markov process method. We find that in the large time limit, whether the machine operates as a heat engine or refrigerator depends on the ratio between the numbers of R cycles and H cycles per unit time. Further increasing the hot bath temperature above a certain threshold does not increase the machine's power output. The cause is that, in this situation, the N cycle has a greater probability than the H cycle and R cycle. Although the SSDB machine operates by randomly switching between these three cycles, at the level of a single quantum jump trajectory, its heat engine efficiency and the refrigerator's coefficient of performance remain constant.

Second Law of Thermodynamics without Einstein Relation

Materials that are constantly driven out of thermodynamic equilibrium, such as active and living systems, typically violate the Einstein relation. This may arise from active contributions to particle fluctuations which are unrelated to the dissipative resistance of the surrounding medium. We show that in these cases the widely used relation between informatic entropy production and heat dissipation does not hold. Consequently, fluctuation relations for the mechanical work, such as the Jarzynski and Crooks theorems, are invalid. We relate the breaking of the correspondence between entropy production and heat dissipation to departure from the fluctuation-dissipation theorem. We propose a temperaturelike variable that restores this correspondence and gives rise to a generalized second law of thermodynamics, whereby the dissipated heat is necessarily non-negative and vanishes at equilibrium. The Clausius inequality, Carnot maximum efficiency theorem, and relation between the extractable work and the change of free energy are recovered as well.

Thermodynamic Concentration Inequalities and Trade-Off Relations

Thermodynamic tradeoff relations quantify the fundamental concept of "no free lunch" in the physical world, suggesting that faster and more precise physical processes come at a higher thermodynamic cost. The key elements in these trade-off relations are the thermodynamic uncertainty relation and speed limit, which are closely tied to information inequalities from which other trade-off relations are derived. Concentration inequalities are relations that complement information inequalities in statistical analyses and have been widely used in various fields. However, their role in thermodynamic trade-off relations remains unclear. This Letter develops thermodynamic concentration inequalities that provide bounds for the distribution of observables in quantum and classical Markov processes. We derive a set of trade-off relations that generalize speed limits and thermodynamic uncertainty relations from the developed thermodynamic concentration inequalities. The derived trade-off relations hold under minimal assumptions of the underlying physical processes. This Letter clarifies the role of concentration inequalities in thermodynamics, paving the way for deriving new trade-off relations.

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.

A local-global principle for nonequilibrium steady states

The global steady state of a system in thermal equilibrium exponentially favors configurations with lesser energy. This principle is a powerful explanation of self-organization because energy is a local property of configurations. For nonequilibrium systems, there is no such property for which an analogous principle holds, hence no common explanation of the diverse forms of self-organization they exhibit. However, a flurry of recent empirical results has shown that a local property of configurations called "rattling" predicts the steady states of some nonequilibrium systems, leading to claims of a far-reaching principle of nonequilibrium self-organization. But for which nonequilibrium systems is rattling accurate, and why? We develop a theory of rattling in terms of Markov processes that gives simple and precise answers to these key questions. Our results show that rattling predicts a broader class of nonequilibrium steady states than has been claimed and for different reasons than have been suggested. Its predictions hold to an extent determined by the relative variance of, and correlation between, the local and global "parts" of a steady state. We show how these quantities characterize the local-global relationships of various random walks on random graphs, spin-glass dynamics, and models of animal collective behavior. Surprisingly, we find that the core idea of rattling is so general as to apply to equilibrium and nonequilibrium systems alike.

Unified hierarchical relationship between thermodynamic tradeoff relations

Recent years have witnessed a surge of discoveries in the studies of thermodynamic inequalities: the thermodynamic uncertainty relation (TUR) and the entropic bound (EB) provide a lower bound on the entropy production (EP) in terms of nonequilibrium currents; the classical speed limit (CSL) expresses the lower bound on the EP using the geometry of probability distributions; the power-efficiency (PE) tradeoff dictates the maximum power achievable for a heat engine given the level of its thermal efficiency. In this study, we show that there exists a unified hierarchical structure encompassing all of these bounds, with the fundamental inequality given by an extension of the TUR (XTUR) that incorporates the most general range of currentlike and state-dependent observables. By selecting more specific observables, the TUR and the EB follow from the XTUR, and the CSL and the PE tradeoff follow from the EB. Our derivations cover both Langevin and Markov jump systems, with the first proof of the EB for the Markov jump systems and a more generalized form of the CSL. We also present concrete examples of the EB for the Markov jump systems and the generalized CSL.

Network analysis for the steady-state thermodynamic uncertainty relation

We perform network analysis of a system described by the master equation to estimate the lower bound of the steady-state current noise, starting from the level 2.5 large deviation function and using the graph theory approach. When the transition rates are uniform, and the system is driven to a nonequilibrium steady state by unidirectional transitions, we derive a noise lower bound, which accounts for fluctuations of sojourn times at all states and is expressed using mesh currents. This bound is applied to the uncertainty in the signal-to-noise ratio of the fluctuating computation time of a schematic Brownian computation plus reset process described by a graph containing one cycle. Unlike the mixed and pseudo-entropy bounds that increase logarithmically with the length of the intended computation path, this bound depends on the number of extraneous predecessors and thus captures the logical irreversibility.

General theory for localizing the where and when of entropy production meets single-molecule experiments

The laws of thermodynamics apply to biophysical systems on the nanoscale as described by the framework of stochastic thermodynamics. This theory provides universal, exact relations for quantities like work, which have been verified in experiments where a fully resolved description allows direct access to such quantities. Complementary studies consider partially hidden, coarse-grained descriptions, in which the mean entropy production typically is not directly accessible but can be bounded in terms of observable quantities. Going beyond the mean, we introduce a fluctuating entropy production that applies to individual trajectories in a coarse-grained description under time-dependent driving. Thus, this concept is applicable to the broad and experimentally significant class of driven systems in which not all relevant states can be resolved. We provide a paradigmatic example by studying an experimentally verified protein unfolding process. As a consequence, the entire distribution of the coarse-grained entropy production rather than merely its mean retains spatial and temporal information about the microscopic process. In particular, we obtain a bound on the distribution of the physical entropy production of individual unfolding events.

Thermodynamic Bounds on Generalized Transport: From Single-Molecule to Bulk Observables

We prove that the transport of any differentiable scalar observable in d-dimensional nonequilibrium systems is bounded from above by the total entropy production scaled by the amount the observation "stretches" microscopic coordinates. The result-a time-integrated generalized speed limit-reflects the thermodynamic cost of transport of observables, and places underdamped and overdamped stochastic dynamics on equal footing with deterministic motion. Our work allows for stochastic thermodynamics to make contact with bulk experiments, and fills an important gap in thermodynamic inference, since microscopic dynamics is, at least for short times, underdamped. Requiring only averages but not sample-to-sample fluctuations, the proven transport bound is practical and applicable not only to single-molecule but also bulk experiments where only averages are observed, which we demonstrate by examples. Our results may facilitate thermodynamic inference on molecular machines without an obvious directionality from bulk observations of transients probed, e.g., in time-resolved x-ray scattering.

Information-theoretical limit on the estimates of dissipation by molecular machines using single-molecule fluorescence resonance energy transfer experiments

Single-molecule fluorescence resonance energy transfer (FRET) experiments are commonly used to study the dynamics of molecular machines. While in vivo molecular processes often break time-reversal symmetry, the temporal directionality of cyclically operating molecular machines is often not evident from single-molecule FRET trajectories, especially in the most common two-color FRET studies. Solving a more quantitative problem of estimating the energy dissipation/entropy production by a molecular machine from single-molecule data is even more challenging. Here, we present a critical assessment of several practical methods of doing so, including Markov-model-based methods and a model-free approach based on an information-theoretical measure of entropy production that quantifies how (statistically) dissimilar observed photon sequences are from their time reverses. The Markov model approach is computationally feasible and may outperform model free approaches, but its performance strongly depends on how well the assumed model approximates the true microscopic dynamics. Markov models are also not guaranteed to give a lower bound on dissipation. Meanwhile, model-free, information-theoretical methods systematically underestimate entropy production at low photoemission rates, and long memory effects in the photon sequences make these methods demanding computationally. There is no clear winner among the approaches studied here, and all methods deserve to belong to a comprehensive data analysis toolkit.

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.

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.

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.

Uncertainty relation for symmetric Petz-Rényi relative entropy

Holevo introduced a fidelity between quantum states that is symmetric and as effective as the trace distance in evaluating their similarity. This fidelity is bounded by a function of the trace distance, a relationship to which we will refer as Holevo's inequality. More broadly, Holevo's fidelity is part of a one-parameter family of symmetric Petz-Rényi relative entropies, which in turn satisfy a Pinsker's-like inequality with respect to the trace distance. Although Holevo's inequality is tight, Pinsker's inequality is loose for this family. We show that the symmetric Petz-Rényi relative entropies satisfy a tight inequality with respect to the trace distance, improving Pinsker's and reproducing Holevo's as a specific case. Additionally, we show how this result emerges from a symmetric Petz-Rényi uncertainty relation, a result that encompasses several relations in quantum and stochastic thermodynamics.

Estimator of entropy production for partially accessible Markov networks based on the observation of blurred transitions

A central task in stochastic thermodynamics is the estimation of entropy production for partially accessible Markov networks. We establish an effective transition-based description for such networks with transitions that are not distinguishable and therefore blurred for an external observer. We demonstrate that, in contrast to a description based on fully resolved transitions, this effective description is typically non-Markovian at any point in time. Starting from an information-theoretic bound, we derive an operationally accessible entropy estimator for this observation scenario. We illustrate the operational relevance and the quality of this entropy estimator with a numerical analysis of various representative examples.

Milestoning estimators of dissipation in systems observed at a coarse resolution

Many nonequilibrium, active processes are observed at a coarse-grained level, where different microscopic configurations are projected onto the same observable state. Such "lumped" observables display memory, and in many cases, the irreversible character of the underlying microscopic dynamics becomes blurred, e.g., when the projection hides dissipative cycles. As a result, the observations appear less irreversible, and it is very challenging to infer the degree of broken time-reversal symmetry. Here we show, contrary to intuition, that by ignoring parts of the already coarse-grained state space we may-via a process called milestoning-improve entropy-production estimates. We present diverse examples where milestoning systematically renders observations "closer to underlying microscopic dynamics" and thereby improves thermodynamic inference from lumped data assuming a given range of memory, and we hypothesize that this effect is quite general. Moreover, whereas the correct general physical definition of time reversal in the presence of memory remains unknown, we here show by means of physically relevant examples that at least for semi-Markov processes of first and second order, waiting-time contributions arising from adopting a naive Markovian definition of time reversal generally must be discarded.

Exact solution to quantum dynamical activity

The quantum dynamical activity constitutes a thermodynamic cost in trade-off relations such as the quantum speed limit and the quantum thermodynamic uncertainty relation. However, calculating the quantum dynamical activity has been a challenge. In this paper, we present the exact solution for the quantum dynamical activity by deploying the continuous matrix product state method. Moreover, using the derived exact solution, we determine the upper bound of the dynamical activity, which comprises the standard deviation of the system Hamiltonian and jump operators. We confirm the exact solution and the upper bound by performing numerical simulations.

Thermodynamic bounds on the asymmetry of cross-correlations with dynamical activity and entropy production

Entropy production and dynamical activity are two complementary aspects in nonequilibrium physics. The asymmetry of cross-correlation, serving as a distinctive feature of nonequilibrium, also finds widespread utility. In this Letter, we establish two thermodynamic bounds on the normalized asymmetry of cross-correlation in terms of dynamical activity and entropy production rate. These bounds demonstrate broad applicability and offer experimental testability.

Information geometric bound on general chemical reaction networks

We investigate the convergence of chemical reaction networks (CRNs), aiming to establish an upper bound on their reaction rates. The nonlinear characteristics and discrete composition of CRNs pose significant challenges in this endeavor. To circumvent these complexities, we adopt an information geometric perspective, utilizing the natural gradient to formulate a nonlinear system. This system effectively determines an upper bound for the dynamics of CRNs. We corroborate our methodology through numerical simulations, which reveal that our constructed system converges more rapidly than CRNs within a particular class of reactions. This class is defined by the count of chemicals, the highest stoichiometric coefficients in the reactions, and the total number of reactions involved. Further, we juxtapose our approach with traditional methods, illustrating that the latter falls short in providing an upper bound for CRN reaction rates. Although our investigation centers on CRNs, the widespread presence of hypergraphs across various disciplines, ranging from natural sciences to engineering, indicates potential wider applications of our method, including in the realm of information science.

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.

Limiting flux in quantum thermodynamics

In quantum systems, entropy production is typically defined as the quantum relative entropy between two states. This definition provides an upper bound for any flux (of particles, energy, entropy, etc.) of bounded observables, which proves especially useful near equilibrium. However, this bound tends to be irrelevant in general nonequilibrium situations. We propose a new upper bound for such fluxes in terms of quantum relative entropy, applicable even far from equilibrium and in the strong coupling regime. Additionally, we compare this bound with Monte Carlo simulations of random qubits with coherence, as well as with a model of two interacting nuclear spins.

Thermodynamic Correlation Inequality

Trade-off relations place fundamental limits on the operations that physical systems can perform. This Letter presents a trade-off relation that bounds the correlation function, which measures the relationship between a system's current and future states, in Markov processes. The obtained bound, referred to as the thermodynamic correlation inequality, states that the change in the correlation function has an upper bound comprising the dynamical activity, a thermodynamic measure of the activity of a Markov process. Moreover, by applying the obtained relation to the linear response function, it is demonstrated that the effect of perturbation can be bounded from above by the dynamical activity.

Challenges in molecular dynamics simulations of heat exchange statistics

We study heat exchange in temperature-biased metal-molecule-metal molecular junctions by employing the molecular dynamics simulator LAMMPS. Generating the nonequilibrium steady state with Langevin thermostats at the boundaries of the junction, we show that the average heat current across a gold-alkanedithiol-gold nanojunction behaves physically, with the thermal conductance value matching the literature. In contrast, the full probability distribution function for heat exchange, as generated by the simulator, violates the fundamental fluctuation symmetry for entropy production. We trace this failure back to the implementation of the thermostats and the expression used to calculate the heat exchange. To rectify this issue and produce the correct statistics, we introduce single-atom thermostats as an alternative to conventional many-atom thermostats. Once averaging heat exchange over the hot and cold thermostats, this approach successfully generates the correct probability distribution function, which we use to study the behavior of both the average heat current and its noise. We further examine the thermodynamic uncertainty relation in the molecular junction and show that it holds, albeit demonstrating nontrivial trends. Our study points to the need to carefully implement nonequilibrium molecular dynamics solvers in atomistic simulation software tools for future investigations of noise phenomena in thermal transport.

Adaptive nonequilibrium design of actin-based metamaterials: Fundamental and practical limits of control

The adaptive and surprising emergent properties of biological materials self-assembled in far-from-equilibrium environments serve as an inspiration for efforts to design nanomaterials. In particular, controlling the conditions of self-assembly can modulate material properties, but there is no systematic understanding of either how to parameterize external control or how controllable a given material can be. Here, we demonstrate that branched actin networks can be encoded with metamaterial properties by dynamically controlling the applied force under which they grow and that the protocols can be selected using multi-task reinforcement learning. These actin networks have tunable responses over a large dynamic range depending on the chosen external protocol, providing a pathway to encoding "memory" within these structures. Interestingly, we obtain a bound that relates the dissipation rate and the rate of "encoding" that gives insight into the constraints on control-both physical and information theoretical. Taken together, these results emphasize the utility and necessity of nonequilibrium control for designing self-assembled nanostructures.

Nonequilibrium Response for Markov Jump Processes: Exact Results and Tight Bounds

Generalizing response theory of open systems far from equilibrium is a central quest of nonequilibrium statistical physics. Using stochastic thermodynamics, we develop an algebraic method to study the static response of nonequilibrium steady state to arbitrary perturbations. This allows us to derive explicit expressions for the response of edge currents as well as traffic to perturbations in kinetic barriers and driving forces. We also show that these responses satisfy very simple bounds. For the response to energy perturbations, we straightforwardly recover results obtained using nontrivial graph-theoretical methods.

Limits on the Precision of Catenane Molecular Motors: Insights from Thermodynamics and Molecular Dynamics Simulations

Thermodynamic uncertainty relations (TURs) relate precision to the dissipation rate, yet the inequalities can be far from saturation. Indeed, in catenane molecular motor simulations, we record precision far below the TUR limit. We further show that this inefficiency can be anticipated by four physical parameters: the thermodynamic driving force, fuel decomposition rate, coupling between fuel decomposition and motor motion, and rate of undriven motor motion. The physical insights might assist in designing molecular motors in the future.

Thermokinetic relations

Thermokinetic relations bound thermodynamic quantities, such as entropy production of a physical system over a certain time interval, with statistics of kinetic (or dynamical) observables, such as mean total variation of the observable over the time interval. We introduce a thermokinetic relation to bound the entropy production or the nonadiabatic (or excess) entropy production for overdamped Markov jump processes, possibly with time-varying rates and nonstationary distributions. For stationary cases, this bound is akin to a thermodynamic uncertainty relation, only involving absolute fluctuations rather than the mean square, thereby offering a better lower bound far from equilibrium. For nonstationary cases, this bound generalizes (classical) speed limits, where the kinetic term is not necessarily the activity (number of jumps) but any trajectory observable of interest. As a consequence, in the task of driving a system from a given probability distribution to another, we find a tradeoff between nonadiabatic entropy production and housekeeping entropy production: the latter can be increased to decrease the former, although to a limited extent. We also find constraints specific to constant-rate Markov processes. We illustrate our thermokinetic relations on simple examples from biophysics and computing devices.

Quantum relative entropy uncertainty relation

For classic systems, the thermodynamic uncertainty relation (TUR) states that the fluctuations of a current have a lower bound in terms of the entropy production. Some TURs are rooted in information theory, particularly derived from relations between observations (mean and variance) and dissimilarities, such as the Kullback-Leibler divergence, which plays the role of entropy production in stochastic thermodynamics. We generalize this idea for quantum systems, where we find a lower bound for the uncertainty of quantum observables given in terms of the quantum relative entropy. We apply the result to obtain a quantum thermodynamic uncertainty relation in terms of the quantum entropy production, valid for arbitrary dynamics and nonthermal environments.

Jensen bound for the entropy production rate in stochastic thermodynamics

Bounding and estimating entropy production has long been an important goal of nonequilibrium thermodynamics. We recently derived a lower bound on the total and subsystem entropy production rates of continuous stochastic systems. This "Jensen bound" has led to fundamental limits on the performance of collective transport systems and permitted thermodynamic inference of free-energy transduction between components of bipartite molecular machines. Our original derivation relied on a number of assumptions, which restricted the bound's regime of applicability. Here we derive the Jensen bound far more generally for multipartite overdamped Langevin dynamics. We then consider several extensions, allowing for position-dependent diffusion coefficients, underdamped dynamics, and non-multipartite overdamped dynamics. Our results extend the Jensen bound to a far broader class of systems.

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.

Irreversible Mesoscale Fluctuations Herald the Emergence of Dynamical Phases

We study fluctuating field models with spontaneously emerging dynamical phases. We consider two typical transition scenarios associated with parity-time symmetry breaking: oscillatory instabilities and critical exceptional points. An analytical investigation of the low-noise regime reveals a drastic increase of the mesoscopic entropy production toward the transitions. For an illustrative model of two nonreciprocally coupled Cahn-Hilliard fields, we find physical interpretations in terms of actively propelled interfaces and a coupling of eigenmodes of the linearized dynamics near the critical exceptional point.

Full statistics of nonequilibrium heat and work for many-body quantum Otto engines and universal bounds: A nonequilibrium Green's function approach

We consider a generic four-stroke quantum Otto engine consisting of two unitary and two thermalization strokes with an arbitrary many-body working medium. Using the Schwinger-Keldysh nonequilibrium Green's function formalism, we provide an analytical expression for the cumulant generating function corresponding to the joint probability distribution of nonequilibrium work and heat. The obtained result is valid up to the second order of the external driving amplitude. We then focus on the linear response limit and obtained Onsager's transport coefficients for the generic Otto cycle and show that the traditional fluctuation-dissipation relation for the total work is violated in the quantum domain, whereas for heat it is preserved. This leads to remarkable consequences in obtaining universal constraints on heat and work fluctuations for engine and refrigerator regimes of the Otto cycle and further allows us to make connections to the thermodynamic uncertainty relations. These findings are illustrated using a paradigmatic model that can be feasibly implemented in experiments.

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.

Work Fluctuations in Ergotropic Heat Engines

We study the work fluctuations in ergotropic heat engines, namely two-stroke quantum Otto engines where the work stroke is designed to extract the ergotropy (the maximum amount of work by a cyclic unitary evolution) from a couple of quantum systems at canonical equilibrium at two different temperatures, whereas the heat stroke thermalizes back the systems to their respective reservoirs. We provide an exhaustive study for the case of two qutrits whose energy levels are equally spaced at two different frequencies by deriving the complete work statistics. By varying the values of temperatures and frequencies, only three kinds of optimal unitary strokes are found: the swap operator U1, an idle swap U2 (where one of the qutrits is regarded as an effective qubit), and a non-trivial permutation of energy eigenstates U3, which indeed corresponds to the composition of the two previous unitaries, namely U3=U2U1. While U1 and U2 are Hermitian (and hence involutions), U3 is not. This point has an impact on the thermodynamic uncertainty relations (TURs), which bound the signal-to-noise ratio of the extracted work in terms of the entropy production. In fact, we show that all TURs derived from a strong detailed fluctuation theorem are violated by the transformation U3.

Symmetric-logarithmic-derivative Fisher information for kinetic uncertainty relations

We investigate a symmetric logarithmic derivative (SLD) Fisher information for kinetic uncertainty relations (KURs) of open quantum systems described by the GKSL quantum master equation with and without the detailed balance condition. In a quantum kinetic uncertainty relation derived by Vu and Saito [Phys. Rev. Lett. 128, 140602 (2022)0031-900710.1103/PhysRevLett.128.140602], the Fisher information of probability of quantum trajectory with a time-rescaling parameter plays an essential role. This Fisher information is upper bounded by the SLD Fisher information. For a finite time and arbitrary initial state, we derive a concise expression of the SLD Fisher information, which is a double time integral and can be calculated by solving coupled first-order differential equations. We also derive a simple lower bound of the Fisher information of quantum trajectory. We point out that the SLD Fisher information also appears in the speed limit based on the Mandelstam-Tamm relation by Hasegawa [Nat. Commun. 14, 2828 (2023)2041-172310.1038/s41467-023-38074-8]. When the jump operators connect eigenstates of the system Hamiltonian, we show that the Bures angle in the interaction picture is upper bounded by the square root of the dynamical activity at short times, which contrasts with the classical counterpart.

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.

Impact of Imperfect Timekeeping on Quantum Control

In order to unitarily evolve a quantum system, an agent requires knowledge of time, a parameter that no physical clock can ever perfectly characterize. In this Letter, we study how limitations on acquiring knowledge of time impact controlled quantum operations in different paradigms. We show that the quality of timekeeping an agent has access to limits the circuit complexity they are able to achieve within circuit-based quantum computation. We do this by deriving an upper bound on the average gate fidelity achievable under imperfect timekeeping for a general class of random circuits. Another area where quantum control is relevant is quantum thermodynamics. In that context, we show that cooling a qubit can be achieved using a timer of arbitrary quality for control: timekeeping error only impacts the rate of cooling and not the achievable temperature. Our analysis combines techniques from the study of autonomous quantum clocks and the theory of quantum channels to understand the effect of imperfect timekeeping on controlled quantum dynamics.

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.

Topologically constrained fluctuations and thermodynamics regulate nonequilibrium response

The limits on a system's response to external perturbations inform our understanding of how physical properties can be shaped by microscopic characteristics. Here, we derive constraints on the steady-state nonequilibrium response of physical observables in terms of the topology of the microscopic state space and the strength of thermodynamic driving. Notably, evaluation of these limits requires no kinetic information beyond the state-space structure. When applied to models of receptor binding, we find that sensitivity is bounded by the steepness of a Hill function with a Hill coefficient enhanced by the chemical driving beyond the structural equilibrium limit.

Upper bound for entropy production in Markov processes

The second law of thermodynamics states that entropy production cannot be negative. Recent developments concerning uncertainty relations in stochastic thermodynamics, such as thermodynamic uncertainty relations and speed limits, have yielded refined second laws that provide lower bounds of entropy production by incorporating information from current statistics or distributions. In contrast, in this study we bound the entropy production from above by terms comprising the dynamical activity and maximum transition-rate ratio. We derive two upper bounds: One applies to steady-state conditions, whereas the other applies to arbitrary time-dependent conditions. We verify these bounds through numerical simulation and identify several potential applications.

Thermodynamic variational relation

In systems far from equilibrium, the statistics of observables are connected to entropy production, leading to the thermodynamic uncertainty relation (TUR). However, the derivation of TURs often involves constraining the parity of observables, such as considering asymmetric currents, making it unsuitable for the general case. We propose a thermodynamic variational relation (TVR) between the statistics of general observables and entropy production, based on the variational representation of f divergences. From this result, we derive a universal TUR and other relations for higher-order statistics of observables.

Monitored nonadiabatic and coherent-controlled quantum unital Otto heat engines: First four cumulants

Recently, measurement-based quantum thermal machines have drawn more attention in the field of quantum thermodynamics. However, the previous results on quantum Otto heat engines were either limited to special unital and nonunital channels in the bath stages, or a specific driving protocol at the work strokes and assuming the cycle being time-reversal symmetric, i.e., V^{†}=U (or V=U). In this paper, we consider a single spin-1/2 quantum Otto heat engine, by first replacing one of the heat baths by an arbitrary unital channel, and then we give the exact analytical expression of the characteristic function from which all the cumulants of heat and work emerge. We prove that under the effect of monitoring, ν_{2}>ν_{1} is a necessary condition for positive work, either for a symmetric or asymmetric-driven Otto cycle. Furthermore, going beyond the average we show that the ratio of the fluctuations of work and heat is lower and upper-bounded when the system is working as a heat engine. However, differently from the previous results in the literature, we consider the third and fourth cumulants as well. It is shown that the ratio of the third (fourth) cumulants of work and heat is not upper-bounded by unity nor lower-bounded by the third (fourth) power of the efficiency, as is the case for the ratio of fluctuations. Finally, we consider applying a specific unital map that plays the role of a heat bath in a coherently superposed manner, and we show the role of the initial coherence of the control qubit on efficiency, on the average work and its relative fluctuations.

Entropy production limits all fluctuation oscillations

The oscillation of fluctuation with two state observables is investigated. Following the idea of Ohga et al. [Phys. Rev. Lett. 131, 077101 (2023)10.1103/PhysRevLett.131.077101], we find that the fluctuation oscillation relative to their autocorrelations is bounded from above by the entropy production per characteristic maximum oscillation time. Our result applies to a variety of systems including Langevin systems, chemical reaction systems, and macroscopic systems. In addition, our bound consists of experimentally tractable quantities, which enables us to examine our inequality experimentally.

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.

Random walks on modular chains: Detecting structure through statistics

We study kinetic transport through one-dimensional modular networks consisting of alternating domains using both analytical and numerical methods. We demonstrate that the mean velocity is insensitive to the local structure of the network, and it depends only on global, structural-averaged properties. However, by examining high-order cumulants characterizing the kinetics, we reveal information on the degree of inhomogeneity of blocks and the size of repeating units in the network. Specifically, in unbiased diffusion, the kurtosis is the first transport coefficient that exposes structural information, whereas in biased chains, the diffusion coefficient already reveals structural motifs. Nevertheless, this latter dependence is weak, and it disappears at both low and high biasing. Our study demonstrates that high-order moments of the population distribution over sites provide information about the network structure that is not captured by the first moment (mean velocity) alone. These results are useful towards deciphering mechanisms and determining architectures underlying long-range charge transport in biomolecules and biological and chemical reaction networks.