Thermodynamic uncertainty relation for biomolecular processes

Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay

Many natural and artificial systems are subject to some sort of delay, which can be in the form of a single discrete delay or distributed over a range of times. Here, we discuss the impact of this distribution on (thermo-)dynamical properties of time-delayed stochastic systems. To this end, we study a simple classical model with white and colored noise, and focus on the class of Gamma-distributed delays which includes a variety of distinct delay distributions typical for feedback experiments and biological systems. A physical application is a colloid subject to time-delayed feedback control, which is, in principle, experimentally realizable by co-moving optical traps. We uncover several unexpected phenomena in regard to the system's linear stability and its thermodynamic properties. First, increasing the mean delay time can destabilize or stabilize the process, depending on the distribution of the delay. Second, for all considered distributions, the heat dissipated by the controlled system (e.g., the colloidal particle) can become negative, which implies that the delay force extracts energy and entropy of the bath. As we show here, this refrigerating effect is particularly pronounced for exponential delay. For a specific non-reciprocal realization of a control device, we find that the entropic costs, measured by the total entropy production of the system plus controller, are the lowest for exponential delay. The exponential delay further yields the largest stable parameter regions. In this sense, exponential delay represents the most effective and robust type of delayed feedback.

Thermodynamic Uncertainty Relation in Slowly Driven Quantum Heat Engines

Thermodynamic uncertainty relations express a trade-off between precision, defined as the noise-to-signal ratio of a generic current, and the amount of associated entropy production. These results have deep consequences for autonomous heat engines operating at steady state, imposing an upper bound for their efficiency in terms of the power yield and its fluctuations. In the present Letter we analyze a different class of heat engines, namely, those which are operating in the periodic slow-driving regime. We show that an alternative TUR is satisfied, which is less restrictive than that of steady-state engines: it allows for engines that produce finite power, with small power fluctuations, to operate close to reversibility. The bound further incorporates the effect of quantum fluctuations, which reduces engine efficiency relative to the average power and reliability. We finally illustrate our findings in the experimentally relevant model of a single-ion heat engine.

Stochastic Thermodynamics of a Piezoelectric Energy Harvester Model

We experimentally study a piezoelectric energy harvester driven by broadband random vibrations. We show that a linear model, consisting of an underdamped Langevin equation for the dynamics of the tip mass, electromechanically coupled with a capacitor and a load resistor, can accurately describe the experimental data. In particular, the theoretical model allows us to define fluctuating currents and to study the stochastic thermodynamics of the system, with focus on the distribution of the extracted work over different time intervals. Our analytical and numerical analysis of the linear model is succesfully compared to the experiments.

Optimal Rectification without Forward-Current Suppression by Biological Molecular Motor

We experimentally show that biological molecular motor F_{1}-ATPase (F_{1}) implements an optimal rectification mechanism. The rectification mechanism hardly suppresses the synthesis of adenosine triphosphate by F_{1}, which is F_{1}'s physiological role, while inhibiting the unfavorable hydrolysis of adenosine triphosphate. This optimal rectification contrasts highly with that of a simple ratchet model, where the inhibition of the backward current is inevitably accompanied by the suppression of the forward current. Our detailed analysis of single-molecule trajectories demonstrates a novel but simple rectification mechanism of F_{1} with parallel landscapes and asymmetric transition rates.

Local detailed balance across scales: From diffusions to jump processes and beyond

Diffusive dynamics in presence of deep energy minima and weak nongradient forces can be coarse grained into a mesoscopic jump process over the various basins of attraction. Combining standard weak-noise results with a path integral expansion around equilibrium, we show that the emerging transition rates satisfy local detailed balance (LDB). Namely, the log ratio of the transition rates between nearby basins of attractions equals the free-energy variation appearing at equilibrium, supplemented by the work done by the nonconservative forces along the typical transition path. When the mesoscopic dynamics possesses a large-size deterministic limit, it can be further reduced to a jump process over macroscopic states satisfying LDB. The persistence of LDB under coarse graining of weakly nonequilibrium states is a generic consequence of the fact that only dissipative effects matter close to equilibrium.

Exploring dissipative sources of non-Markovian biochemical reaction systems

Many biological processes including important intracellular processes are governed by biochemical reaction networks. Usually, these reaction systems operate far from thermodynamic equilibrium, implying free-energy dissipation. On the other hand, single reaction events happen often in a memory manner, leading to non-Markovian kinetics. A question then arises: how do we calculate free-energy dissipation (defined as the entropy production rate) in this physically real case? We derive an analytical formula for calculating the energy consumption of a general reaction system with molecular memory characterized by nonexponential waiting-time distributions. It shows that this dissipation is composed of two parts: one from broken detailed balance of an equivalent Markovian system with the same topology and substrates, and the other from the direction-time dependence of waiting-time distributions. But, if the system is in a detailed balance and the waiting-time distribution is direction-time independent, there is no energy dissipation even in the non-Markovian case. These general results provide insights into the physical mechanisms underlying nonequilibrium processes. A continuous-time random-walk model and a generalized model of stochastic gene expression are chosen to clearly show dissipative sources and the relationship between energy dissipation and molecular memory.

Improved bounds on entropy production in living systems

Living systems maintain or increase local order by working against the second law of thermodynamics. Thermodynamic consistency is restored as they consume free energy, thereby increasing the net entropy of their environment. Recently introduced estimators for the entropy production rate have provided major insights into the efficiency of important cellular processes. In experiments, however, many degrees of freedom typically remain hidden to the observer, and, in these cases, existing methods are not optimal. Here, by reformulating the problem within an optimization framework, we are able to infer improved bounds on the rate of entropy production from partial measurements of biological systems. Our approach yields provably optimal estimates given certain measurable transition statistics. In contrast to prevailing methods, the improved estimator reveals nonzero entropy production rates even when nonequilibrium processes appear time symmetric and therefore may pretend to obey detailed balance. We demonstrate the broad applicability of this framework by providing improved bounds on the energy consumption rates in a diverse range of biological systems including bacterial flagella motors, growing microtubules, and calcium oscillations within human embryonic kidney cells.

Full counting statistics of the particle currents through a Kitaev chain and the exchange fluctuation theorem

Exchange fluctuation theorems (XFTs) describe a fundamental symmetry relation for particle and energy exchange between several systems. Here we study the XFTs of a Kitaev chain connected to two reservoirs at the same temperature but different bias. By varying the parameters in the Kitaev chain model, we calculate analytically the full counting statistics of the transport current and formulate the corresponding XFTs for multiple current components. We also demonstrate the XFTs with numerical results. We find that due to the presence of the U(1) symmetry breaking terms in the Hamiltonian of the Kitaev chain, various forms of the XFTs emerge, and they can be interpreted in terms of various well-known transport processes.

Inertial effects on the Brownian gyrator

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

Coherences and the thermodynamic uncertainty relation: Insights from quantum absorption refrigerators

The thermodynamic uncertainty relation, originally derived for classical Markov-jump processes, provides a tradeoff relation between precision and dissipation, deepening our understanding of the performance of quantum thermal machines. Here, we examine the interplay of quantum system coherences and heat current fluctuations on the validity of the thermodynamics uncertainty relation in the quantum regime. To achieve the current statistics, we perform a full counting statistics simulation of the Redfield quantum master equation. We focus on steady-state quantum absorption refrigerators where nonzero coherence between eigenstates can either suppress or enhance the cooling power, compared with the incoherent limit. In either scenario, we find enhanced relative noise of the cooling power (standard deviation of the power over the mean) in the presence of system coherence, thereby corroborating the thermodynamic uncertainty relation. Our results indicate that fluctuations necessitate consideration when assessing the performance of quantum coherent thermal machines.

Thermodynamic uncertainty relation to assess biological processes

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

Information bound for entropy production from the detailed fluctuation theorem

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

Dissipation-recurrence inequalities at the steady state

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

Role of thermal fluctuations in biological copying mechanisms

During transcription, translation, or self-replication of DNA or RNA, information is transferred to the newly formed species from its predecessor. These processes can be interpreted as (generalized) biological copying mechanism as the new biological entities like DNA, RNA, or proteins are representing the information of their parent bodies uniquely. The accuracy of these copying processes is essential, since errors in the copied code can reduce the functionality of the next generation. Such errors might result from perturbations on these processes. Most important in this context is the temperature of the medium, i.e., thermal noise. Although a reasonable amount of experimental studies have been carried out on this important issue, theoretical understanding is truly sparse. In the present work, we illustrate a model study which is able to focus on the effect of the temperature on the process of biological copying mechanisms, as well as on mutation. We find for our paradigmatic models that, in a quite general scenario, the copying processes are most accurate at an intermediate temperature range; i.e., there exists an optimum temperature where mutation is most unlikely. This allows us to interpret the observations for some biological species with the aid of our model study.

Statistics of Reaction Flux and Dynamical Activity Associated with a Diffusion-Influenced Ligand-Binding Reaction

In this paper, we consider a macromolecule with two competitive binding sites where a ligand can bind to and gives rise to a unicyclic reaction network consisting of four states-(i) a single state with both binding sites vacant, (ii) two states with one bound site and one free binding site, and (iii) an another single state with both sites occupied. We obtain probability densities of the time-integrated current along the clockwise direction and the dynamical activity or mean number of jumps between different states for finite times at a fast diffusion limit. On the other hand, in the diffusion-limited case, ligand diffusion between the two binding sites directly connects the mono-ligated states-changing the reaction scheme. Addition of the new reaction channel alters the precision of ligand occupancy to a single site, the mean dynamical activity, and the mean entropy production rate. All of these quantities are calculated with varying degrees of competition between the two sites for ligands, and we find that increase in the competition between the two sites decreases all above-mentioned additive functionals. The upper bound of precision associated with a single-site ligand occupancy for a diffusion-influenced reaction network is set by the mean dynamical activity (mean entropy production rate) at small (large) ligand concentrations at the steady-state limit.

Nonequilibrium Thermodynamics in Biochemical Systems and Its Application

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

Entropy production fluctuations encode collective behavior in active matter

We derive a general lower bound on distributions of entropy production in interacting active matter systems. The bound is tight in the limit that interparticle correlations are small and short-ranged, which we explore in four canonical active matter models. In all models studied, the bound is weak where collective fluctuations result in long-ranged correlations, which subsequently links the locations of phase transitions to enhanced entropy production fluctuations. We develop a theory for the onset of enhanced fluctuations and relate it to specific phase transitions in active Brownian particles. We also derive optimal control forces that realize the dynamics necessary to tune dissipation and manipulate the system between phases. In so doing, we uncover a general relationship between entropy production and pattern formation in active matter, as well as ways of controlling it.

Thermodynamics of precision in quantum nanomachines

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

Energetics of stochastic BCM type synaptic plasticity and storing of accurate information

Excitatory synaptic signaling in cortical circuits is thought to be metabolically expensive. Two fundamental brain functions, learning and memory, are associated with long-term synaptic plasticity, but we know very little about energetics of these slow biophysical processes. This study investigates the energy requirement of information storing in plastic synapses for an extended version of BCM plasticity with a decay term, stochastic noise, and nonlinear dependence of neuron’s firing rate on synaptic current (adaptation). It is shown that synaptic weights in this model exhibit bistability. In order to analyze the system analytically, it is reduced to a simple dynamic mean-field for a population averaged plastic synaptic current. Next, using the concepts of nonequilibrium thermodynamics, we derive the energy rate (entropy production rate) for plastic synapses and a corresponding Fisher information for coding presynaptic input. That energy, which is of chemical origin, is primarily used for battling fluctuations in the synaptic weights and presynaptic firing rates, and it increases steeply with synaptic weights, and more uniformly though nonlinearly with presynaptic firing. At the onset of synaptic bistability, Fisher information and memory lifetime both increase sharply, by a few orders of magnitude, but the plasticity energy rate changes only mildly. This implies that a huge gain in the precision of stored information does not have to cost large amounts of metabolic energy, which suggests that synaptic information is not directly limited by energy consumption. Interestingly, for very weak synaptic noise, such a limit on synaptic coding accuracy is imposed instead by a derivative of the plasticity energy rate with respect to the mean presynaptic firing, and this relationship has a general character that is independent of the plasticity type. An estimate for primate neocortex reveals that a relative metabolic cost of BCM type synaptic plasticity, as a fraction of neuronal cost related to fast synaptic transmission and spiking, can vary from negligible to substantial, depending on the synaptic noise level and presynaptic firing.

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

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

Quantumness and thermodynamic uncertainty relation of the finite-time Otto cycle

To reveal the role of the quantumness in the Otto cycle and to discuss the validity of the thermodynamic uncertainty relation (TUR) in the cycle, we study the quantum Otto cycle and its classical counterpart. In particular, we calculate exactly the mean values and relative error of thermodynamic quantities. In the quasistatic limit, quantumness reduces the productivity and precision of the Otto cycle compared to that in the absence of quantumness, whereas in the finite-time mode, it can increase the cycle's productivity and precision. Interestingly, as the strength (heat conductance) between the system and the bath increases, the precision of the quantum Otto cycle overtakes that of the classical one. Testing the conventional TUR of the Otto cycle, in the region where the entropy production is large enough, we find a tighter bound than that of the conventional TUR. However, in the finite-time mode, both quantum and classical Otto cycles violate the conventional TUR in the region where the entropy production is small. This implies that another modified TUR is required to cover the finite-time Otto cycle. Finally, we discuss the possible origin of this violation in terms of the uncertainty products of the thermodynamic quantities and the relative error near resonance conditions.

Modeling work-speed-accuracy trade-offs in a stochastic rotary machine

Molecular machines are stochastic systems that catalyze the energetic processes keeping living cells alive and structured. Inspired by the examples of F_{1}-ATP synthase and the bacterial flagellum, we present a minimal model of an externally driven stochastic rotary machine. We explore the trade-offs of work, driving speed, and driving accuracy when changing driving strength, speed, and the underlying system dynamics. We find an upper bound on accuracy and work for a particular speed. Our results favor slow driving when tasked with minimizing the work-accuracy ratio and maximizing the rate of successful cycles. Finally, in the parameter regime mapping to the dynamics of F_{1}-ATP synthase, we find a significant decay of driving accuracy at physiological rotation rates, raising questions about how ATP synthase achieves reasonable or even remarkable efficiency in vivo.

Unified approach to classical speed limit and thermodynamic uncertainty relation

The total entropy production quantifies the extent of irreversibility in thermodynamic systems, which is nonnegative for any feasible dynamics. When additional information such as the initial and final states or moments of an observable is available, it is known that tighter lower bounds on the entropy production exist according to the classical speed limits and the thermodynamic uncertainty relations. Here we obtain a universal lower bound on the total entropy production in terms of probability distributions of an observable in the time forward and backward processes. For a particular case, we show that our universal relation reduces to a classical speed limit, imposing a constraint on the speed of the system's evolution in terms of the Hatano-Sasa entropy production. Notably, the obtained classical speed limit is tighter than the previously reported bound by a constant factor. Moreover, we demonstrate that a generalized thermodynamic uncertainty relation can be derived from another particular case of the universal relation. Our uncertainty relation holds for systems with time-reversal symmetry breaking and recovers several existing bounds. Our approach provides a unified perspective on two closely related classes of inequality: classical speed limits and thermodynamic uncertainty relations.

Tighter thermodynamic bound on the speed limit in systems with unidirectional transitions

We consider a general discrete state-space system with both unidirectional and bidirectional links. In contrast to bidirectional links, there is no reverse transition along the unidirectional links. Herein, we first compute the statistical length and the thermodynamic cost function for transitions in the probability space, highlighting contributions from total, environmental, and resetting (unidirectional) entropy production. Then we derive the thermodynamic bound on the speed limit to connect two distributions separated by a finite time, showing the effect of the presence of unidirectional transitions. Uncertainty relationships can be found for the temporal first and second moments of the average resetting entropy production. We derive simple expressions in the limit of slow unidirectional transition rates. Finally, we present a refinement of the thermodynamic bound by means of an optimization procedure. We numerically investigate these results on systems that stochastically reset with constant and periodic resetting rate.

Irreversibility in dynamical phases and transitions

Living and non-living active matter consumes energy at the microscopic scale to drive emergent, macroscopic behavior including traveling waves and coherent oscillations. Recent work has characterized non-equilibrium systems by their total energy dissipation, but little has been said about how dissipation manifests in distinct spatiotemporal patterns. We introduce a measure of irreversibility we term the entropy production factor to quantify how time reversal symmetry is broken in field theories across scales. We use this scalar, dimensionless function to characterize a dynamical phase transition in simulations of the Brusselator, a prototypical biochemically motivated non-linear oscillator. We measure the total energetic cost of establishing synchronized biochemical oscillations while simultaneously quantifying the distribution of irreversibility across spatiotemporal frequencies.

Thermodynamic Uncertainty Relation for General Open Quantum Systems

We derive a thermodynamic uncertainty relation for general open quantum dynamics, described by a joint unitary evolution on a composite system comprising a system and an environment. By measuring the environmental state after the system-environment interaction, we bound the counting observables in the environment by the survival activity, which reduces to the dynamical activity in classical Markov processes. Remarkably, the relation derived herein holds for general open quantum systems with any counting observable and any initial state. Therefore, our relation is satisfied for classical Markov processes with arbitrary time-dependent transition rates and initial states. We apply our relation to continuous measurement and the quantum walk to find that the quantum nature of the system can enhance the precision. Moreover, we can make the lower bound arbitrarily small by employing appropriate continuous measurement.

Thermodynamic uncertainty relations for bosonic Otto engines

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

Thermodynamic Uncertainty Relation for Time-Dependent Driving

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

Geometry of Work Fluctuations versus Efficiency in Microscopic Thermal Machines

When engineering microscopic machines, increasing efficiency can often come at a price of reduced reliability due to the impact of stochastic fluctuations. Here we develop a general method for performing multiobjective optimization of efficiency and work fluctuations in thermal machines operating close to equilibrium in either the classical or quantum regime. Our method utilizes techniques from thermodynamic geometry, whereby we match optimal solutions to protocols parametrized by their thermodynamic length. We characterize the optimal protocols for continuous-variable Gaussian machines, which form a crucial class in the study of thermodynamics for microscopic systems.

Optimizing energetic cost of uncertainty in a driven system with and without feedback

Many biological functions require dynamics to be necessarily driven out of equilibrium. In contrast, in various contexts, a nonequilibrium dynamics at fast timescales can be described by an effective equilibrium dynamics at a slower timescale. In this work, we study two different aspects: (i) the energy-efficiency tradeoff for a specific nonequilibrium linear dynamics of two variables with feedback and (ii) the cost of effective parameters in a coarse-grained theory as given by the "hidden" dissipation and entropy production rate in the effective equilibrium limit of the dynamics. To meaningfully discuss the tradeoff between energy consumption and the efficiency of the desired function, a one-to-one mapping between function(s) and energy input is required. The function considered in this work is the variance of one of the variables. We get a one-to-one mapping by considering the minimum variance obtained for a fixed entropy production rate and vice versa. We find that this minimum achievable variance is a monotonically decreasing function of the given entropy production rate. When there is a timescale separation, in the effective equilibrium limit, the cost of the effective potential and temperature is the associated "hidden" entropy production rate.

Sub-Gaussian and subexponential fluctuation-response inequalities

Sub-Gaussian and subexponential distributions are introduced and applied to study the fluctuation-response relation out of equilibrium. A bound on the difference in expected values of an arbitrary sub-Gaussian or subexponential physical quantity is established in terms of its sub-Gaussian or subexponential norm. Based on that, we find that the entropy difference between two states is bounded by the energy fluctuation in these states. Moreover, we obtain generalized versions of the thermodynamic uncertainty relation in different regimes. Operational issues concerning the application of our results in an experimental setting are also addressed, and nonasymptotic bounds on the errors incurred by using the sample mean instead of the expected value in our fluctuation-response inequalities are derived.

Circuit Reduction of Heterogeneous Nonequilibrium Systems

Predicting the behavior of heterogeneous nonequilibrium systems is currently analytically intractable. Consequently, complex biological systems have resisted unifying principles. Here, I introduce a mapping from dynamical systems to battery-resistor circuits. I show that in these transformed variables (i) arbitrary numbers of heterogeneous dynamical transitions can be reduced to a Thevenin equivalent resistor which is invariant to driving from equilibrium, (ii) resistors (together with the external driving sources) are sufficient to describe system behavior, and (iii) the resistor's directional symmetry leads to universal theorems of nonequilibrium behavior. This mapping is used to derive two general steady-state relations. First, for any cyclic process, the maximum amplification of any state is tightly bounded by the total dissipation of all states; experimental data are used to show that the master signal protein Ras achieves this bound. Second, for any process, the response of any reaction due to driving any other reaction is identical to the reciprocal response rescaled by the ratio of the corresponding Thevenin resistors. This result generalizes Onsager's reciprocal relation to the strongly driven regime and makes a testable prediction about how systems should be designed or evolved to maximize response. These analytic results represent a new perspective applicable to biological complexity and suggest that this mapping provides the natural variables to study heterogeneous nonequilibrium systems.

Uncertainty Relations and Fluctuation Theorems for Bayes Nets

Recent research has considered the stochastic thermodynamics of multiple interacting systems, representing the overall system as a Bayes net. I derive fluctuation theorems governing the entropy production (EP) of arbitrary sets of the systems in such a Bayes net. I also derive "conditional" fluctuation theorems, governing the distribution of EP in one set of systems conditioned on the EP of a different set of systems. I then derive thermodynamic uncertainty relations relating the EP of the overall system to the precisions of probability currents within the individual systems.

On the definitions and simulations of vibrational heat transport in nanojunctions

Thermal transport through nanosystems is central to numerous processes in chemistry, material sciences, and electrical and mechanical engineering, with classical molecular dynamics as the key simulation tool. Here, we focus on thermal junctions with a molecule bridging two solids that are maintained at different temperatures. The classical steady state heat current in this system can be simulated in different ways, either at the interfaces with the solids, which are represented by thermostats, or between atoms within the conducting molecule. We show that while the latter, intramolecular definition feasibly converges to the correct limit, the molecule-thermostat interface definition is more challenging to converge to the correct result. The problem with the interface definition is demonstrated by simulating heat transport in harmonic and anharmonic one-dimensional chains illustrating unphysical effects such as thermal rectification in harmonic junctions.

Precision and dissipation of a stochastic Turing pattern

Spontaneous pattern formation is a fundamental scientific problem that has received much attention since the seminal theoretical work of Turing on reaction-diffusion systems. In molecular biophysics, this phenomenon often takes place under the influence of large fluctuations. It is then natural to inquire about the precision of such pattern. In particular, spontaneous pattern formation is a nonequilibrium phenomenon, and the relation between the precision of a pattern and the thermodynamic cost associated with it remains largely unexplored. Here, we analyze this relation with a paradigmatic stochastic reaction-diffusion model, i.e., the Brusselator in one spatial dimension. We find that the precision of the pattern is maximized for an intermediate thermodynamic cost, i.e., increasing the thermodynamic cost beyond this value makes the pattern less precise. Even though fluctuations get less pronounced with an increase in thermodynamic cost, we argue that larger fluctuations can also have a positive effect on the precision of the pattern.

Robust oscillations in multi-cyclic Markov state models of biochemical clocks

Organisms often use cyclic changes in the concentrations of chemical species to precisely time biological functions. Underlying these biochemical clocks are chemical reactions and transport processes, which are inherently stochastic. Understanding the physical basis for robust biochemical oscillations in the presence of fluctuations has thus emerged as an important problem. In a previous paper [C. del Junco and S. Vaikuntanathan, Phys. Rev. E 101, 012410 (2020)], we explored this question using the non-equilibrium statistical mechanics of single-ring Markov state models of biochemical networks that support oscillations. Our finding was that they can exploit non-equilibrium driving to robustly maintain the period and coherence of oscillations in the presence of randomness in the rates. Here, we extend our work to Markov state models consisting of a large cycle decorated with multiple small cycles. These additional cycles are intended to represent alternate pathways that the oscillator may take as it fluctuates about its average path. Combining a mapping to single-cycle networks based on first passage time distributions with our previously developed theory, we are able to make analytical predictions for the period and coherence of oscillations in these networks. One implication of our predictions is that a high energy budget can make different network topologies and arrangements of rates degenerate as far as the period and coherence of oscillations are concerned. Excellent agreement between analytical and numerical results confirms that this is the case. Our results suggest that biochemical oscillators can be more robust to fluctuations in the path of the oscillator when they have a high energy budget.

Thermodynamic Uncertainty Relation for Arbitrary Initial States

The thermodynamic uncertainty relation (TUR) describes a trade-off relation between nonequilibrium currents and entropy production and serves as a fundamental principle of nonequilibrium thermodynamics. However, currently known TURs presuppose either specific initial states or an infinite-time average, which severely limits the range of applicability. Here we derive a finite-time TUR valid for arbitrary initial states from the Cramér-Rao inequality. We find that the variance of an accumulated current is bounded from below by the instantaneous current at the final time, which suggests that "the boundary is constrained by the bulk". We apply our results to feedback-controlled processes and successfully explain a recent experiment which reports a violation of a modified TUR with feedback control. We also derive a TUR that is linear in the total entropy production and valid for discrete-time Markov chains with nonsteady initial states. The obtained bound exponentially improves the existing bounds in a discrete-time regime.

Learning Entropy Production via Neural Networks

This Letter presents a neural estimator for entropy production (NEEP), that estimates entropy production (EP) from trajectories of relevant variables without detailed information on the system dynamics. For steady state, we rigorously prove that the estimator, which can be built up from different choices of deep neural networks, provides stochastic EP by optimizing the objective function proposed here. We verify the NEEP with the stochastic processes of the bead spring and discrete flashing ratchet models and also demonstrate that our method is applicable to high-dimensional data and can provide coarse-grained EP for Markov systems with unobservable states.

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

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

Do We Understand the Mechanisms Used by Biological Systems to Correct Their Errors?

Most cellular processes involved in biological information processing display a surprisingly low error rate despite the stochasticity of the underlying biochemical reactions and the presence of competing chemical species. Such high fidelity is the result of nonequilibrium kinetic proofreading mechanisms, i.e., the existence of dissipative pathways for correcting the reactions that went in the wrong direction. While proofreading was often studied from the perspective of error minimization, a number of recent studies have demonstrated that the underlying mechanisms need to consider the interplay of other characteristic properties such as speed, energy dissipation, and noise reduction. Here, we present current views and new insights on the mechanisms of error-correction phenomena and various trade-off scenarios in the optimization of the functionality of biological systems. Existing challenges and future directions are also discussed.

Dissipation-Time Uncertainty Relation

We show that the entropy production rate bounds the rate at which physical processes can be performed in stochastic systems far from equilibrium. In particular, we prove the fundamental tradeoff ⟨S[over ˙]_{e}⟩T≥k_{B} between the entropy flow ⟨S[over ˙]_{e}⟩ into the reservoirs and the mean time T to complete any process whose time-reversed is exponentially rarer. This dissipation-time uncertainty relation is a novel form of speed limit: the smaller the dissipation, the larger the time to perform a process.

Probability of Immobilization on Host Cell Surface Regulates Viral Infectivity

The efficiency of a virus to establish its infection in host cells varies broadly among viruses. It remains unclear if there is a key step in this process that controls viral infectivity. To address this question, we use single-particle tracking and Brownian dynamics simulation to examine human immunodeficiency virus type 1 (HIV-1) infection in cell culture. We find that the frequency of viral-cell encounters is consistent with diffusion-limited interactions. However, even under the most favorable conditions, only 1% of the viruses can become immobilized on cell surface and subsequently enter the cell. This is a result of weak interaction between viral surface gp120 and CD4 receptor, which is insufficient to form a stable complex the majority of the time. We provide the first direct quantitation for efficiencies of these events relevant to measured HIV-1 infectivity and demonstrate that immobilization on host cell surface post-virion-diffusion is the key step in viral infection. Variation of its probability controls the efficiency of a virus to infect its host cells. These results explain the low infectivity of cell-free HIV-1 in vitro and offer a potential rationale for the pervasive high efficiency of cell-to-cell transmission of animal viruses.

Phonon heat transport in cavity-mediated optomechanical nanoresonators

The understanding of heat transport in nonequilibrium thermodynamics is an important research frontier, which is crucial for implementing novel thermodynamic devices, such as heat engines and refrigerators. The convection, conduction, and radiation are the well-known basic ways to transfer thermal energy. Here, we demonstrate a different mechanism of phonon heat transport between two spatially separated nanomechanical resonators coupled by the cavity-enhanced long-range interactions. The single trajectory for thermalization and non-equilibrium dynamics is monitored in real-time. In the strong coupling regime, the instant heat flux spontaneously oscillates back and forth in the nonequilibrium steady states. The universal bound on the precision of nonequilibrium steady-state heat flux, i.e. the thermodynamic uncertainty relation, is verified in such a temperature gradient driven far-off equilibrium system. Our results give more insight into the heat transfer with nanomechanical oscillators, and provide a playground for testing fundamental theories in non-equilibrium thermodynamics.

Heat flux is well understood on macroscopic scales, however when the system size is reduced, novel phenomena are induced by fluctuations. Here, the authors demonstrate phonon heat transport between two nanomechanical resonators coupled by cavity enhanced interactions exhibiting an oscillating heat flux.

Reaching and violating thermodynamic uncertainty bounds in information engines

Thermodynamic uncertainty relations (TURs) set fundamental bounds on the fluctuation and dissipation of stochastic systems. Here, we examine these bounds, in experiment and theory, by exploring the entire phase space of a cyclic information engine operating in a nonequilibrium steady state. Close to its maximal efficiency, we find that the engine violates the original TUR. This experimental demonstration of TUR violation agrees with recently proposed softer bounds: The engine satisfies two generalized TUR bounds derived from the detailed fluctuation theorem with feedback control and another bound linking fluctuation and dissipation to mutual information and Renyi divergence. We examine how the interplay of work fluctuation and dissipation shapes the information conversion efficiency of the engine, and find that dissipation is minimal at a finite noise level, where the original TUR is violated.

Free diffusion bounds the precision of currents in underdamped dynamics

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

Quantifying the non-equilibrium activity of an active colloid

Active matter systems exhibit rich emergent behavior due to constant injection and dissipation of energy at the level of individual agents. Since these systems are far from equilibrium, their dynamics and energetics cannot be understood using the framework of equilibrium statistical mechanics. Recent developments in stochastic thermodynamics extend classical concepts of work, heat, and energy dissipation to fluctuating non-equilibrium systems. We use recent advances in experiment and theory to study the non-thermal dissipation of individual light-activated self-propelled colloidal particles. We focus on characterizing the transition from thermal to non-thermal fluctuations and show that energy dissipation rates on the order of ∼kBT s-1 are measurable from finite time series data.

Chemical Embryology Redux: Metabolic Control of Development

New studies of metabolic reactions and networks in embryos are making important additions to regulatory models of development, so far dominated by genes and signals. Metabolic control of development is not a new idea and can be traced back to Joseph Needham's 'Chemical Embryology', published in the 1930s. Even though Needham's ideas fell by the wayside with the advent of genetic studies of embryogenesis, they demonstrated that embryos provide convenient models for addressing fundamental questions in biochemistry and are now experiencing a comeback, enabled by the powerful merger of detailed mechanistic studies and systems-level techniques. Here we review recent results from studies that quantified the energy budget of embryogenesis in Drosophila and started to untangle the intricate connections between core anabolic processes and developmental transitions. Dynamic coordination of metabolic, genetic, and signaling networks appears to be essential for seamless progression of development.

Quantum Thermodynamic Uncertainty Relation for Continuous Measurement

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

Estimating entropy production by machine learning of short-time fluctuating currents

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.