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The performance of equilibrium sensors is restricted by the laws of equilibrium thermodynamics. We here investigate the physical limits on nonequilibrium sensing in bipartite systems with nonreciprocal coupling. We show that one of the subsystems, acting as a Maxwell's demon, can significantly suppress the fluctuations of the other subsystem relative to its response to an external perturbation. Such negative violation of the fluctuation-dissipation relation can considerably improve the signal-to-noise ratio above its corresponding equilibrium value, allowing the subsystem to operate as an enhanced sensor. We find that the nonequilibrium signal-to-noise ratio of linear systems may be arbitrary large at low frequencies, even at a fixed overall amount of dissipation., Comment: 14 pages, 3 figures

Giant diffusion, where the diffusion coefficient of a Brownian particle in a periodic potential with an external force is significantly enhanced by the external force, is a non-trivial non-equilibrium phenomenon. We propose a simple stochastic model of giant diffusion, which is based on a biased continuous-time random walk (CTRW). In this model, we introduce a flight time in the biased CTRW. We derive the diffusion coefficients of this model by the renewal theory and find that there is a maximum diffusion coefficient when the bias is changed. Giant diffusion is universally observed in the sense that there is a peak of the diffusion coefficient for any tilted periodic potentials and the degree of the diffusivity is greatly enhanced especially for low-temperature regimes. The biased CTRW models with flight times are applied to diffusion under three tilted periodic potentials. Furthermore, the temperature dependence of the maximum diffusion coefficient and the external force that attains the maximum are presented for diffusion under a tilted sawtooth potential., Comment: 13 pages, 7 figures

Based on a variational expression for the steady-state entropy production rate in overdamped Langevin dynamics, we derive concrete upper bounds on the entropy production rate in various physical settings. For particles in a thermal environment and driven by non-conservative forces, we show that the entropy production rate can be upper bounded by considering only the statistics of the driven particles. We use this finding to argue that the presence of non-driven, passive degrees of freedom generally leads to decreased dissipation. Another upper bound can be obtained only in terms of the variance of the non-conservative force, which leads to a universal upper bound for particles that are driven by a constant force that is applied in a certain region of space. Extending our results to systems attached to multiple heat baths or with spatially varying temperature and/or mobility, we show that the temperature difference between the heat baths or the gradient of the temperature can be used to upper bound the entropy production rate. We show that most of these results extend in a straightforward way to underdamped Langevin dynamics and demonstrate them in three concrete examples., Comment: 16 pages, 4 figures

The power spectral density of an observable quantifies the amount of fluctuations at a given frequency and can reveal the influence of different timescales on the observable's dynamics. Here, we show that the spectral density in a continuous-time Markov process can be both lower and upper bounded by an expression involving two constants that depend on the observable and the properties of the system. In equilibrium, we identify these constants with the low- and high-frequency limit of the spectral density, respectively; thus, the spectrum at arbitrary frequency is bounded by the short- and long-time behavior of the observable. Out of equilibrium, on the other hand, the constants can no longer be identified with the limiting behavior of the spectrum, allowing for peaks that correspond to oscillations in the dynamics. We show that the height of these peaks is related to dissipation, allowing to infer the degree to which the system is out of equilibrium from the measured spectrum., Comment: 13 pages, 4 figures

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 tradeoff relation between the long- and short-time fluctuations of an observable. Out of equilibrium, the tradeoff 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., Comment: 16 pages, 2 figures

A system can be driven out of equilibrium by both time-dependent and nonconservative forces, which gives rise to a decomposition of the dissipation into two non-negative components, called the excess and housekeeping entropy productions. We derive thermodynamic uncertainty relations for the excess and housekeeping entropy. These can be used as tools to estimate the individual components, which are in general difficult to measure directly. We introduce a decomposition of an arbitrary current into excess and housekeeping parts, which provide lower bounds on the respective entropy production. Furthermore, we also provide a geometric interpretation of the decomposition, and show that the uncertainties of the two components are not independent, but rather have to obey a joint uncertainty relation, which also yields a tighter bound on the total entropy production. We apply our results to two examples that illustrate the physical interpretation of the components of the current and how to estimate the entropy production., Comment: 14 pages, 10 figures

A nonequilibrium system is characterized by a set of thermodynamic forces and fluxes which give rise to entropy production (EP). We show that these forces and fluxes have an information-geometric structure, which allows us to decompose EP into contributions from different types of forces in general (linear and nonlinear) discrete systems. We focus on the excess and housekeeping decomposition, which separates contributions from conservative and nonconservative forces. Unlike the Hatano-Sasa decomposition, our housekeeping/excess terms are always well-defined, including in systems with odd variables and nonlinear systems without steady states. Our decomposition leads to far-from-equilibrium thermodynamic uncertainty relations and speed limits. As an illustration, we derive a thermodynamic bound on the time necessary for one cycle in a chemical oscillator.

We propose a housekeeping/excess decomposition of entropy production for general nonlinear dynamics in a discrete space, including chemical reaction networks and discrete stochastic systems. We exploit the geometric structure of thermodynamic forces to define the decomposition; this does not rely on the notion of a steady state, and even applies to systems that exhibit multistability, limit cycles, and chaos. In the decomposition, distinct aspects of the dynamics contribute separately to entropy production: the housekeeping part stems from a cyclic mode that arises from external driving, generalizing Schnakenberg's cyclic decomposition to non-steady states, while the excess part stems from an instantaneous relaxation mode that arises from conservative forces. Our decomposition refines previously known thermodynamic uncertainty relations and speed limits. In particular, it not only improves an optimal-transport-theoretic speed limit, but also extends the optimal transport theory of discrete systems to nonlinear and nonconservative settings., Comment: 27 pages, 7 figures

We derive bounds on the precision of fluctuating currents, which are valid for the steady state of underdamped Langevin dynamics. These bounds provide a generalization of the overdamped thermodynamic uncertainty relation to the finite-mass regime. In the overdamped case, the precision of a current is bounded by the entropy production. By contrast, the underdamped bound acquires two additional positive terms, which characterize the local mean acceleration and the fluctuations of the velocity. We generalize the bound to the cases of a magnetic field and anisotropic temperature, and derive a joint bound for several observables. Finally, we apply our results to biased free diffusion and the Brownian gyrator (with and without magnetic field), as well as to diffusion in periodic potentials. In the latter case, we show that the underdamped bound can be tight when taking into account the correlations between the current and other observables., Comment: 23 pages, 11 figures

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 Neto\v{c}ny. 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 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-Neto\v{c}ny-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., Comment: 26 pages, 7 figures

We investigate the problem of minimizing the entropy production for a physical process that can be described in terms of a Markov jump dynamics. We show that, without any further constraints, a given time-evolution may be realized at arbitrarily small entropy production, yet at the expense of diverging activity. For a fixed activity, we find that the dynamics that minimizes the entropy production is given in terms of conservative forces. The value of the minimum entropy production is expressed in terms of the graph-distance based Wasserstein distance between the initial and final configuration. This yields a new kind of speed limit relating dissipation, the average number of transitions and the Wasserstein distance. It also allows us to formulate the optimal transport problem on a graph in term of a continuous-time interpolating dynamics, in complete analogy to the continuous space setting. We demonstrate our findings for simple state networks, a time-dependent pump and for spin flips in the Ising model., Comment: 23 pages, 6 figures

Two qualitatively different ways of driving a physical system out of equilibrium, time-dependent and non-conservative forcing, are reflected by the decomposition of the system's entropy production into excess and housekeeping parts. We show that the difference between these two types of driving gives rise to a geometric formulation in terms of two orthogonal contributions to the currents in the system. This geometric picture in a natural way leads to variational expressions for both the excess and housekeeping entropy, which allow calculating both contributions independently from the trajectory data of the system. We demonstrate this by calculating the excess and housekeeping entropy of a particle in a time-dependent, tilted periodic potential., Comment: 7 pages, 2 figures

We discuss how to use correlations between different physical observables to improve recently obtained thermodynamics bounds, notably the fluctuation-response inequality (FRI) and the thermodynamic uncertainty relation (TUR). We show that increasing the number of measured observables will always produce a tighter bound. This tighter bound becomes particularly useful if one of the observables is a conserved quantity, whose expectation is invariant under a given perturbation of the system. For the case of the TUR, we show that this applies to any function of the state of the system. The resulting correlation-TUR takes into account the correlations between a current and a non-current observable, thereby tightening the TUR. We demonstrate our finding on a model of the $\text{F}_1$-ATPase molecular motor, a Markov jump model consisting of two rings and transport through a two-dimensional channel. We find that the correlation-TUR is significantly tighter than the TUR and can be close to an equality even far from equilibrium., Comment: 18 pages, 7 figures

We introduce a continuous time-reversal operation which connects the time-forward and time-reversed trajectories in the steady state of an irreversible Markovian dynamics via a continuous family of stochastic dynamics. This continuous time-reversal allows us to derive a tighter version of the thermodynamic uncertainty relation (TUR) involving observables evaluated relative to their local mean value. Moreover, the family of dynamics realizing the continuous time-reversal contains an equilibrium dynamics halfway between the time-forward and time-reversed dynamics. We show that this equilibrium dynamics, together with an appropriate choice of the observable, turns the inequality in the TUR into an equality. We demonstrate our findings for the example of a particle diffusing in a tilted periodic potential., Comment: 18 pages, 2 figures

Thermodynamic uncertainty relations (TURs) are the inequalities which give lower bounds on the entropy production rate using only the mean and the variance of fluctuating currents. Since the TURs do not refer to the full details of the stochastic dynamics, it would be promising to apply the TURs for estimating the entropy production rate from a limited set of trajectory data corresponding to the dynamics. Here we investigate a theoretical framework for estimation of the entropy production rate using the TURs along with machine learning techniques without prior knowledge of the parameters of the stochastic dynamics. Specifically, we derive a TUR for the short-time region and prove that it can provide the exact value, not only a lower bound, of the entropy production rate for Langevin dynamics, if the observed current is optimally chosen. This formulation naturally includes a generalization of the TURs with the partial entropy production of subsystems under autonomous interaction, which reveals the hierarchical structure of the estimation. We then construct estimators on the basis of the short-time TUR and machine learning techniques such as the gradient ascent. By performing numerical experiments, we demonstrate that our learning protocol performs well even in nonlinear Langevin dynamics. We also discuss the case of Markov jump processes, where the exact estimation is shown to be impossible in general. Our result provides a platform that can be applied to a broad class of stochastic dynamics out of equilibrium, including biological systems.

We derive a relation between the dissipation in a stochastic dynamics and the Wasserstein distance. We show that the minimal amount of dissipation required to transform an initial state to a final state during a diffusion process is given by the Wasserstein distance between the two states, divided by the total time of the process. This relation implies a lower bound on the dissipation for any diffusion process in terms of its initial and final state. Using a lower bound on the Wasserstein distance, we further show that we can give a lower bound on the dissipation in terms of only the mean and convariance matrix of the initial and final state. We apply this result to derive the optimal forces that minimize the dissipation for given initial and final mean and covariance., Comment: 5 pages

Continuous-time random walks offer powerful coarse-grained descriptions of transport processes. We here microscopically derive such a model for a Brownian particle diffusing in a deep periodic potential. We determine both the waiting-time and the jump-length distributions in terms of the parameters of the system, from which we analytically deduce the non-Gaussian characteristic function. We apply this continuous-time random walk model to characterize the underdamped diffusion of single Cesium atoms in a one-dimensional optical lattice. We observe excellent agreement between experimental and theoretical characteristic functions, without any free parameter., Comment: 10 pages, 7 figures

We investigate the connection between the time-evolution of averages of stochastic quantities and the Fisher information and its induced statistical length. As a consequence of the Cramer-Rao bound, we find that the rate of change of the average of any observable is bounded from above by its variance times the temporal Fisher information. As a consequence of this bound, we obtain a speed limit on the evolution of stochastic observables: Changing the average of an observable requires a minimum amount of time given by the change in the average squared, divided by the fluctuations of the observable times the thermodynamic cost of the transformation. In particular for relaxation dynamics, which do not depend on time explicitly, we show that the Fisher information is a monotonically decreasing function of time and that this minimal required time is determined by the initial preparation of the system. We further show that the monotonicity of the Fisher information can be used to detect hidden variables in the system and demonstrate our findings for simple examples of continuous and discrete random processes., Comment: 25 pages, 4 figures

We extend a class of recently derived thermodynamic uncertainty relations to vector-valued observables. In contrast to the scalar-valued observables examined previously, this multidimensional thermodynamic uncertainty relation provides a natural way to study currents in high-dimensional systems and to obtain relations between different observables. Our proof is based on the generalized Cr{\'a}mer-Rao inequality, which we interpret as a relation between physical observables and the Fisher information. This allows us to develop high-dimensional versions of both the original, steady state uncertainty relation and the more recently obtained generalized uncertainty relation for time-periodic systems. We apply the multidimensional uncertainty relation to obtain a new constraint on the performance of steady-state heat engines, which is tighter than previous bounds and reveals the role of heat-work correlations. As a second application, we show that the uncertainty relation is connected to a bound on the differential mobility. As a result of this connection, we find that a necessary condition for equality in the uncertainty relation is that the system obeys the equilibrium fluctuation-dissipation relation., Comment: 18 pages, 1 figure

We show that observing the trajectories of confined particles in a thermal equilibrium state yields an estimate on the free-space diffusion coefficient. For generic trapping potentials and interactions between particles, the estimate comes in the form of a lower bound on the true diffusion coefficient. For non-interacting particles in harmonic trapping potentials, which approximately describes many experimental situations, the estimate is asymptotically exact. This allows to determine the diffusion coefficient from an equilibrium measurement, as opposed to a direct observation of diffusion, which necessarily starts from a non-equilibrium state. We explicitly demonstrate that the estimate remains quantitatively accurate in the presence of weak interactions and anharmonic corrections., Comment: 5 pages

We present a new approach to response around arbitrary out-of-equilibrium states in the form of a fluctuation-response inequality (FRI). We study the response of an observable to a perturbation of the underlying stochastic dynamics. We find that magnitude of the response is bounded from above by the fluctuations of the observable in the unperturbed system and the Kullback-Leibler divergence between the probability densities describing the perturbed and unperturbed system. This establishes a connection between linear response and concepts of information theory. We show that in many physical situations, the relative entropy may be expressed in terms of physical observables. As a direct consequence of this FRI, we show that for steady state particle transport, the differential mobility is bounded by the diffusivity. For a virtual perturbation proportional to the local mean velocity, we recover the thermodynamic uncertainty relation (TUR) for steady state transport processes. Finally, we use the FRI to derive a generalization of the uncertainty relation to arbitrary dynamics, which involves higher-order cumulants of the observable. We provide an explicit example, in which the TUR is violated but its generalization is satisfied with equality., Comment: 14 pages, 1 figure

We derive a bound on generalized currents for Langevin systems in terms of the total entropy production in the system and its environment. For overdamped dynamics, any generalized current is bounded by the total rate of entropy production. We show that this entropic bound on the magnitude of generalized currents imposes power-efficiency tradeoff relations for ratchets in contact with a heat bath: Maximum efficiency---Carnot efficiency for a Smoluchowski-Feynman ratchet and unity for a flashing or rocking ratchet---can only be reached at vanishing power output. For underdamped dynamics, while there may be reversible currents that are not bounded by the entropy production rate, we show that the output power and heat absorption rate are irreversible currents and thus obey the same bound. As a consequence, a power-efficiency tradeoff relation holds not only for underdamped ratchets but also for periodically driven heat engines. For weak driving, the bound results in additional constraints on the Onsager matrix beyond those imposed by the Second Law. Finally, we discuss the connection between heat and entropy in a non-thermal situation where the friction and noise intensity are state-dependent., Comment: 18 pages, 2 figures

We derive a universal bound on generalized currents in Langevin systems in terms of the mean-square fluctuations of the current and the total entropy production. This bound generalizes a relation previously found by Barato et al. to arbitrary times and transient states. Using the bound, we define a new efficiency for stochastic transport, which measures how close a given system comes to saturating the bound. The existence of such a bounded efficiency implies that stochastic transport is unavoidably accompanied by a fluctuations and dissipation, which cannot be reduced arbitrarily. We apply the definition of transport efficiency to steady state particle transport and heat engines and show that the transport efficiency may approach unity at finite current, in contrast to the thermodynamic efficiency. Finally, we derive a bound on purely diffusive transport in terms of the Shannon entropy., Comment: 13 pages, 2 figures

We investigate the occurrence of heat leakages in overdamped Brownian harmonic systems. We exactly compute the underdamped and overdamped stochastic heats exchanged with the bath for a sudden frequency or temperature switch. We show that the underdamped heat reduces to the corresponding overdamped expression in the limit of large friction for the isothermal process. However, we establish that this is not the case for the isochoric transformation. We microscopically derive the additionally generated heat leakage and relate its origin to the initial relaxation of the velocity of the system. Our results highlight the limitations of the overdamped approximation for the evaluation of the stochastic heat in systems with changing bath temperature., Comment: 5 pages, 4 figures

We show that uncorrelated Gaussian noise, despite its paradigmatic association with thermal equilibrium, can drive a system out of equilibrium and can serve as a resource from which work can be extracted. We consider an overdamped particle in a periodic potential with an internal degree of freedom and a state-dependent friction, coupled to an equilibrium bath. Applying additional Gaussian white noise drives the system into a non-equilibrium steady state and causes a finite current if the potential is spatially asymmetric. We calculate the current explicitly in the three complementary limits. Since the particle current is driven solely by additive Gaussian white noise, this shows that the latter can potentially be exploited as a work resource to power small engines. By comparing the extracted power to the energy injection due to the noise, we find an expression for the efficiency of such an engine., Comment: 11 pages, 5 figures

The Boltzmann-Gibbs density, a central result of equilibrium statistical mechanics, relates the energy of a system in contact with a thermal bath to its equilibrium statistics. This relation is lost for non-thermal systems such as cold atoms in optical lattices, where the heat bath is replaced by the laser beams of the lattice. We investigate in detail the stationary phase-space probability for Sisyphus cooling under harmonic confinement. In particular, we elucidate whether the total energy of the system still describes its stationary state statistics. We find that this is true for the center part of the phase-space density for deep lattices, where the Boltzmann-Gibbs density provides an approximate description. The relation between energy and statistics also persists for strong confinement and in the limit of high energies, where the system becomes underdamped. However, the phase-space density now exhibits heavy power-law tails. In all three cases we find expressions for the leading order phase-space density and corrections which break the equivalence of probability and energy and violate energy equipartition. The non-equilibrium nature of the steady state is confounded by explicit violations of detailed balance. We complement these analytical results with numerical simulations to map out the intricate structure of the phase-space density., Comment: 25 pages, 18 figures

We investigate the performance of an underdamped stochastic heat engine for a time-dependent harmonic oscillator. We analytically determine the optimal protocol that maximizes the efficiency at fixed power. The maximum efficiency reduces to the Curzon-Ahlborn formula at maximum power and the Carnot formula at zero power. We further establish that the efficiency at maximum power is universally given by the Curzon-Ahlborn efficiency in the weakly damped regime. Finally, we show that even small deviations from operation at maximum power may result in a significantly increased efficiency., Comment: 6 pages, 3 figures

Diffusion is a central phenomenon in almost all fields of natural science revealing microscopic processes from the observation of macroscopic dynamics. Here, we consider the paradigmatic system of a single atom diffusing in a periodic potential. We engineer microscopic particle-environment interaction to control the ensuing diffusion over a broad range of diffusion constants and from normal to subdiffusion. While one- and two-point properties extracted from single particle trajectories, such as variance or position correlations, indicate apparent Brownian motion, the step size distribution, however, shows exponentially decaying tails. Furthermore non-ergodic dynamics is observed on long time scales. We demonstrate excellent agreement with a model of continuous time random walk with exponential distribution, which applies to various transport phenomena in condensed or soft matter with periodic potentials.

We derive a generalization of the Wiener-Khinchin theorem for nonstationary processes by introducing a time-dependent spectral density that is related to the time-averaged power. We use the nonstationary theorem to investigate aging processes with asymptotically scale-invariant correlation functions. As an application, we analyze the power spectrum of three paradigmatic models of anomalous diffusion: scaled Brownian motion, fractional Brownian motion and diffusion in a logarithmic potential. We moreover elucidate how the nonstationarity of generic subdiffusive processes is related to the infrared catastrophe of 1/f-noise., Comment: 7 pages, 2 figures (including supplemental material)

Cold atoms in dissipative optical lattices have long been known to exhibit anomalous kinetics due to an effective nonlinear friction force. Here we show that confining the spatial motion of the atoms will lead to an anomalous non-Boltzmann-Gibbs equilibrium state, with a power law tail at large energies. Only in the limit of deep optical lattices, do we regain the Boltzmann-Gibbs state. For strong confinement relative to the damping, we find an explicit expression for the equilibrium phase-space distribution, which generally differs from the canonical Boltzmann-Gibbs state at all energies. Both in the low and high energy limits, the equilibrium distribution is a function of the system's Hamiltonian. At intermediate energies, however, the distribution is not a function of energy only and equipartition is violated., Comment: 8 pages, 4 figures

We propose and theoretically investigate a nanomechanical heat engine. We show how a levitated nanoparticle in a harmonic optical trap inside an optical cavity can be used to realize a Stirling cycle in the underdamped regime. The all-optical approach enables fast and exible control of all the thermodynamical parameters and the effcient optimization of the performance of the engine. We develop a systematic optimization procedure to determine optimal driving protocols. We further perform numerical simulations with realistic parameters and evaluate the maximum power and the corresponding effciency., Comment: 5 pages, 3 figures

Infinite densities can describe the long-time properties of systems when ergodicity is broken and the equilibrium Boltzmann-Gibbs distribution fails. We here perform semiclassical Monte Carlo simulations of cold atoms in dissipative optical lattices with realistic parameters. We show that the momentum infinite density, as well as its scale invariance, should be observable in shallow potentials. We further evaluate the momentum autocorrelation function in the stationary and aging regime., Comment: 7 pages, 6 figures

We use a model based on the fractional Langevin equation with external noise to describe the anomalous dynamics observed in microrheology experiments in living cells. This model reproduces both the subdiffusive short-time and the superdiffusive long-time behavior. We show that the former reflects the equilibrium properties of the cell, while the latter is due to the nonequilibrium external noise. This allows to infer the transport properties of the system under active measurements from the transient behavior obtained from passive measurements, extending the connection between active and passive microrheology to the nonequilibrium regime. The active and passive results can be linked via a generalized Stokes-Einstein relation based on an effective time-dependent temperature, which can be determined from the transient passive behavior. In order to reproduce experimental data, we further find that the external noise describing the active components of the cell has to be nonstationary. We establish that the latter leads to a time-dependent noise spectral density., Comment: 12 pages, 5 figures