Thermodynamic Uncertainty Relation Research Articles

Out-of-Equilibrium Fluctuation-Dissipation Bounds.

We prove a general inequality between the charge current and its fluctuations valid for any weakly interacting coherent electronic conductor and for any stationary out-of-equilibrium condition, thereby going beyond established fluctuation-dissipation relations. The developed fluctuation-dissipation bound saturates at large temperature bias and reveals additional insight for heat engines, since it limits the output power by power fluctuations. It is valid when the thermodynamic uncertainty relations break down due to quantum effects and provides stronger constraints close to thermovoltage.

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 uncertainty relations in the presence of non-linear friction and memory

A new thermodynamic uncertainty relation (TUR) is derived for systems described by linearly coupled Langevin equations in the presence of non-linear frictional forces. In our scheme, the main variable represents the velocity of a particle, while the other coupled variables describe memory effects which may arise from strongly correlated degrees of freedom with several time-scales and, in general, are associated with thermal baths at different temperatures. The new TUR gives a lower bound for the mean-squared displacement of the position of the particle, including its asymptotic diffusion coefficient. This bound, in several examples worked out here, appears to be a good analytical estimate of the real diffusion coefficient. The new TUR can be also applied in the absence of any external force (with or without thermal equilibrium between the baths), a case which usually goes beyond the scope of original TURs. We show applications to non-linear frictional models with memory, such as the Coulomb and the Prandtl-Tomlinson models, usually representative of friction at the nano-scale and within atomic-force microscopy experiments.

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.

Universal thermodynamic bounds on the Fano factor of discriminatory networks with unidirectional transitions

We derive a universal lower bound on the Fano factors of general biochemical discriminatory networks involving irreversible catalysis steps, based on the thermodynamic uncertainty relation, and compare it to a numerically exact Pareto optimal front. This bound is completely general, involving only the reversible entropy production per product formed and the error fraction of the system. We then show that by judiciously choosing which transitions to include in the reversible entropy production, one can derive a family of bounds that can be fine-tuned to include physical observables at hand. Lastly, we test our bound by considering three discriminatory schemes: a multi-stage Michaelis-Menten network, a Michaelis-Menten network with correlations between subsequent products, and a multi-stage kinetic proofreading network, where for the latter application the bound is altered to include the hydrolytic cost of the proofreading steps. We find that our bound is remarkably tight.

Thermodynamic precision of a chain of motors: the difference between phase and noise correlation

Inspired by recent experiments on fluctuations of flagellar beating in sperm and C. reinhardtii, we investigate the precision of phase fluctuations in a system of nearest-neighbor-coupled molecular motors. We model the system as a Kuramoto chain of oscillators with a coupling constant k and noisy driving. The precision p is a Fano-factor-like observable, which obeys the thermodynamic uncertainty relation (TUR), which is an upper bound related to dissipation. We first consider independent motor noises with diffusivity D: in this case, the precision goes as k/D , coherently with the behavior of spatial order. The minimum observed precision is that of the uncoupled oscillator punc ; the maximum observed precision is Npunc , saturating the TUR bound. Then we consider driving noises which are spatially correlated, as may happen in the presence of some direct coupling between adjacent motors. Such a spatial correlation in the noise does not evidently reduce the degree of spatial correlation in the chain, but sensibly reduces the maximum attainable precision p, coherently with experimental observations. The limiting behavior of the precision, in the two opposite cases of negligible interaction and strong interaction, is well reproduced by the precision of the single chain site punc and the precision of the center of mass of the chain Neffpunc with Neff<N : both do not depend on the degree of interaction in the chain, but Neff decreases with the correlation length of the motor noises.

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.

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.

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.

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.

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.

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.

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.

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.

Statistical features of systems driven by non-Gaussian processes: theory & practice

Nowadays many tools, e.g. fluctuation relations, are available to characterize the statistical properties of non-equilibrium systems. However, most of these tools rely on the assumption that the driving noise is normally distributed. Here we consider a class of Markov processes described by Langevin equations driven by a mixture of Gaussian and Poissonian noises, focusing on their non-equilibrium properties. In particular, we prove that detailed balance does not hold even when correlation functions are symmetric under time reversal. In such cases, a breakdown of the time reversal symmetry can be highlighted by considering higher order correlation functions. Furthermore, the entropy production may be different from zero even for vanishing currents. We provide analytical expressions for the average entropy production rate in several cases. We also introduce a scale dependent estimate for entropy production, suitable for inference from experimental signals. The empirical entropy production allows us to discuss the role of spatial and temporal resolutions in characterizing non-equilibrium features. Finally, we revisit the Brownian gyrator introducing an additional Poissonian noise showing that it behaves as a two dimensional linear ratchet. It has also the property that when Onsager relations are satisfied its entropy production is positive although it is minimal. We conclude discussing estimates of entropy production for partially accessible systems, comparing our results with the lower bound provided by the thermodynamic uncertainty relations.

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.

Thermodynamic uncertainty relations for systems with broken time reversal symmetry: The case of superconducting hybrid systems

Thermodynamic uncertainty relations for systems with broken time reversal symmetry: The case of superconducting hybrid systems

Thermodynamic uncertainty relation in nondegenerate and degenerate maser heat engines

Thermodynamic uncertainty relation in nondegenerate and degenerate maser heat engines

Collective synchronization of dissipatively-coupled noise-activated processes

We study the stochastic dynamics of an arbitrary number of noise-activated cyclic processes, or oscillators, that are all coupled to each other via a dissipative coupling. The N coupled oscillators are described by N phase coordinates driven in a tilted washboard potential. At low N and strong coupling, we find synchronization as well as an enhancement in the average speed of the oscillators. In the large N regime, we show that the collective dynamics can be described through a mean-field theory, which predicts a great enhancement in the average speed. In fact, beyond a critical value of the coupling strength, noise activation becomes irrelevant and the dynamics switch to an effectively deterministic ‘running’ mode. Finally, we study the stochastic thermodynamics of the coupled oscillators, in particular their performance with regards to the thermodynamic uncertainty relation.