Stochastic thermodynamics under coarse graining

Thermodynamic cost for precision of general counting observables

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

Nonequilibrium dynamics and entropy production of a trapped colloidal particle in a complex nonreciprocal medium

We discuss the two-dimensional motion of a Brownian particle that is confined to a harmonic trap and driven by a shear flow. The surrounding medium induces memory effects modeled by a linear, typically nonreciprocal coupling of the particle coordinates to an auxiliary (hidden) variable. The system's behavior resulting from the microscopic Langevin equations for the three variables is analyzed by means of exact moment equations derived from the Fokker-Planck representation, and numerical Brownian dynamics simulations. Increasing the shear rate beyond a critical value we observe, for suitable coupling scenarios with nonreciprocal elements, a transition from a stationary to a nonstationary state, corresponding to an escape from the trap. We analyze this behavior, analytically and numerically, in terms of the associated moments of the probability distribution, and from the perspective of nonequilibrium thermodynamics. Intriguingly, the entropy production rate remains finite when crossing the stability threshold.

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

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

Milestoning estimators of dissipation in systems observed at a coarse resolution

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

Emergence of Memory in Equilibrium versus Nonequilibrium Systems

Experiments often probe observables that correspond to low-dimensional projections of high-dimensional dynamics. In such situations distinct microscopic configurations become lumped into the same observable state. It is well known that correlations between the observable and the hidden degrees of freedom give rise to memory effects. However, how and under which conditions these correlations emerge remain poorly understood. Here we shed light on two fundamentally different scenarios of the emergence of memory in minimal stationary systems, where observed and hidden degrees of freedom either evolve cooperatively or are coupled by a hidden nonequilibrium current. In the reversible setting the strongest memory manifests when the timescales of hidden and observed dynamics overlap, whereas, strikingly, in the driven setting maximal memory emerges under a clear timescale separation. Our results hint at the possibility of fundamental differences in the way memory emerges in equilibrium versus driven systems that may be utilized as a "diagnostic" of the underlying hidden transport mechanism.

Functional hierarchies in brain dynamics characterized by signal reversibility in ferret cortex

Brain signal irreversibility has been shown to be a promising approach to study neural dynamics. Nevertheless, the relation with cortical hierarchy and the influence of different electrophysiological features is not completely understood. In this study, we recorded local field potentials (LFPs) during spontaneous behavior, including awake and sleep periods, using custom micro-electrocorticographic (μECoG) arrays implanted in ferrets. In contrast to humans, ferrets remain less time in each state across the sleep-wake cycle. We deployed a diverse set of metrics in order to measure the levels of complexity of the different behavioral states. In particular, brain irreversibility, which is a signature of non-equilibrium dynamics, captured by the arrow of time of the signal, revealed the hierarchical organization of the ferret's cortex. We found different signatures of irreversibility and functional hierarchy of large-scale dynamics in three different brain states (active awake, quiet awake, and deep sleep), showing a lower level of irreversibility in the deep sleep stage, compared to the other. Irreversibility also allowed us to disentangle the influence of different cortical areas and frequency bands in this process, showing a predominance of the parietal cortex and the theta band. Furthermore, when inspecting the embedded dynamic through a Hidden Markov Model, the deep sleep stage was revealed to have a lower switching rate and lower entropy production. These results suggest functional hierarchies in organization that can be revealed through thermodynamic features and information theory metrics.

Violations of the fluctuation-dissipation theorem reveal distinct nonequilibrium dynamics of brain states

The brain is a nonequilibrium system whose dynamics change in different brain states, such as wakefulness and deep sleep. Thermodynamics provides the tools for revealing these nonequilibrium dynamics. We used violations of the fluctuation-dissipation theorem to describe the hierarchy of nonequilibrium dynamics associated with different brain states. Together with a whole-brain model fitted to empirical human neuroimaging data, and deriving the appropriate analytical expressions, we were able to capture the deviation from equilibrium in different brain states that arises from asymmetric interactions and hierarchical organization.

Optimal Power Extraction from Active Particles with Hidden States

We identify generic protocols achieving optimal power extraction from a single active particle subject to continuous feedback control under the assumption that its spatial trajectory, but not its instantaneous self-propulsion force, is accessible to direct observation. Our Bayesian approach draws on the Onsager-Machlup path integral formalism and is exemplified in the cases of free run-and-tumble and active Ornstein-Uhlenbeck dynamics in one dimension. Such optimal protocols extract positive work even in models characterized by time-symmetric positional trajectories and thus vanishing informational entropy production rates. We argue that the theoretical bounds derived in this work are those against which the performance of realistic active matter engines should be compared.

Speed limit, dissipation bound, and dissipation-time trade-off in thermal relaxation processes

We investigate bounds on speed, nonadiabatic entropy production, and the trade-off relation between them for classical stochastic processes with time-independent transition rates. Our results show that the time required to evolve from an initial to a desired target state is bounded from below by the information-theoretical ∞-Rényi divergence between these states, divided by the total rate. Furthermore, we conjecture and provide extensive numerical evidence for an information-theoretical bound on the nonadiabatic entropy production and a dissipation-time trade-off relation that outperforms previous bounds in some cases..

Nonequilibrium Thermodynamics of the Majority Vote Model

The majority vote model is one of the simplest opinion systems yielding distinct phase transitions and has garnered significant interest in recent years. This model, as well as many other stochastic lattice models, are formulated in terms of stochastic rules with no connection to thermodynamics, precluding the achievement of quantities such as power and heat, as well as their behaviors at phase transition regimes. Here, we circumvent this limitation by introducing the idea of a distinct and well-defined thermal reservoir associated to each local configuration. Thermodynamic properties are derived for a generic majority vote model, irrespective of its neighborhood and lattice topology. The behavior of energy/heat fluxes at phase transitions, whether continuous or discontinuous, in regular and complex topologies, is investigated in detail. Unraveling the contribution of each local configuration explains the nature of the phase diagram and reveals how dissipation arises from the dynamics.

Energy Conversion and Entropy Production in Biased Random Walk Processes-From Discrete Modeling to the Continuous Limit

We considered discrete and continuous representations of a thermodynamic process in which a random walker (e.g., a molecular motor on a molecular track) uses periodically pumped energy (work) to pass N sites and move energetically downhill while dissipating heat. Interestingly, we found that, starting from a discrete model, the limit in which the motion becomes continuous in space and time (N→∞) is not unique and depends on what physical observables are assumed to be unchanged in the process. In particular, one may (as usually done) choose to keep the speed and diffusion coefficient fixed during this limiting process, in which case, the entropy production is affected. In addition, we also studied processes in which the entropy production is kept constant as N→∞ at the cost of a modified speed or diffusion coefficient. Furthermore, we also combined this dynamics with work against an opposing force, which made it possible to study the effect of discretization of the process on the thermodynamic efficiency of transferring the power input to the power output. Interestingly, we found that the efficiency was increased in the limit of N→∞. Finally, we investigated the same process when transitions between sites can only happen at finite time intervals and studied the impact of this time discretization on the thermodynamic variables as the continuous limit is approached.

Time-Resolved Statistics of Snippets as General Framework for Model-Free Entropy Estimators

Irreversibility is commonly quantified by entropy production. An external observer can estimate it through measuring an observable that is antisymmetric under time reversal like a current. We introduce a general framework that allows us to infer a lower bound on entropy production through measuring the time-resolved statistics of events with any symmetry under time reversal, in particular, time-symmetric instantaneous events. We emphasize Markovianity as a property of certain events rather than of the full system and introduce an operationally accessible criterion for this weakened Markov property. Conceptually, the approach is based on snippets as particular sections of trajectories between two Markovian events, for which a generalized detailed balance relation is discussed.

The lack of temporal brain dynamics asymmetry as a signature of impaired consciousness states

Life is a constant battle against equilibrium. From the cellular level to the macroscopic scale, living organisms as dissipative systems require the violation of their detailed balance, i.e. metabolic enzymatic reactions, in order to survive. We present a framework based on temporal asymmetry as a measure of non-equilibrium. By means of statistical physics, it was discovered that temporal asymmetries establish an arrow of time useful for assessing the reversibility in human brain time series. Previous studies in human and non-human primates have shown that decreased consciousness states such as sleep and anaesthesia result in brain dynamics closer to the equilibrium. Furthermore, there is growing interest in the analysis of brain symmetry based on neuroimaging recordings and since it is a non-invasive technique, it can be extended to different brain imaging modalities and applied at different temporo-spatial scales. In the present study, we provide a detailed description of our methodological approach, paying special attention to the theories that motivated this work. We test, for the first time, the reversibility analysis in human functional magnetic resonance imaging data in patients suffering from disorder of consciousness. We verify that the tendency of a decrease in the asymmetry of the brain signal together with the decrease in non-stationarity are key characteristics of impaired consciousness states. We expect that this work will open the way for assessing biomarkers for patients' improvement and classification, as well as motivating further research on the mechanistic understanding underlying states of impaired consciousness.

Stochastic thermodynamics of opinion dynamics models

We show that models of opinion formation and dissemination in a community of individuals can be framed within stochastic thermodynamics from which we can build a nonequilibrium thermodynamics of opinion dynamics. This is accomplished by decomposing the original transition rate that defines an opinion model into two or more transition rates, each representing the contact with heat reservoirs at different temperatures, and postulating an energy function. As the temperatures are distinct, heat fluxes are present even at the stationary state and linked to the production of entropy, the fundamental quantity that characterizes nonequilibrium states. We apply the present framework to a generic-vote model including the majority-vote model in a square lattice and in a cubic lattice. The fluxes and the rate of entropy production are calculated by numerical simulation and by the use of a pair approximation.

The time scales of irreversibility in spontaneous brain activity are altered in obsessive compulsive disorder

We study how obsessive-compulsive disorder (OCD) affects the complexity and time-reversal symmetry-breaking (irreversibility) of the brain resting-state activity as measured by magnetoencephalography (MEG). Comparing MEG recordings from OCD patients and age/sex matched control subjects, we find that irreversibility is more concentrated at faster time scales and more uniformly distributed across different channels of the same hemisphere in OCD patients than in control subjects. Furthermore, the interhemispheric asymmetry between homologous areas of OCD patients and controls is also markedly different. Some of these differences were reduced by 1-year of Kundalini Yoga meditation treatment. Taken together, these results suggest that OCD alters the dynamic attractor of the brain's resting state and hint at a possible novel neurophysiological characterization of this psychiatric disorder and how this therapy can possibly modulate brain function.

First and second laws of information processing by nonequilibrium dynamical states

The averaged steady-state surprisal links a driven stochastic system's information processing to its nonequilibrium thermodynamic response. By explicitly accounting for the effects of nonequilibrium steady states, a decomposition of the surprisal results in an information processing first law that extends and tightens-to strict equalities-various information processing second laws. Applying stochastic thermodynamics' integral fluctuation theorems then shows that the decomposition reduces to the second laws under appropriate limits. In unifying them, the first law paves the way to identifying the mechanisms by which nonequilibrium steady-state systems leverage information-bearing degrees of freedom to extract heat. To illustrate, we analyze an autonomous Maxwellian information ratchet that tunably violates detailed balance in its effective dynamics. This demonstrates how the presence of nonequilibrium steady states qualitatively alters an information engine's allowed functionality.

Inferring Subsystem Efficiencies in Bipartite Molecular Machines

Molecular machines composed of coupled subsystems transduce free energy between different external reservoirs, in the process internally transducing energy and information. While subsystem efficiencies of these molecular machines have been measured in isolation, less is known about how they behave in their natural setting when coupled together and acting in concert. Here, we derive upper and lower bounds on the subsystem efficiencies of a bipartite molecular machine. We demonstrate their utility by estimating the efficiencies of the F_{o} and F_{1} subunits of ATP synthase and that of kinesin pulling a diffusive cargo.

Beat of a current

The fluctuation relation, a milestone of modern thermodynamics, is only established when a set of fundamental currents can be measured. Here we prove that it also holds for systems with hidden transitions if observations are carried "at their own beat," that is, by stopping the experiment after a fixed number of visible transitions, rather than the elapse of an external clock time. This suggests that thermodynamic symmetries are more resistant to the loss of information when described in the space of transitions.

Stochastic Thermodynamics of a Finite Quantum System Coupled to Two Heat Baths

We consider a situation where an N-level system (NLS) is coupled successively to two heat baths with different temperatures without being necessarily thermalized and approaches a steady state. For this situation we apply a general Jarzynski-type equation and conclude that heat and entropy is flowing from the hot bath to the cold one. The Clausius relation between increase of entropy and transfer of heat divided by a suitable temperature assumes the form of two inequalities. Our approach is illustrated by an analytical example. For the linear regime, i.e., for small temperature differences between the two heat baths, we derive an expression for the heat conduction coefficient.

A universal method for analyzing copolymer growth

Polymers consisting of more than one type of monomer, known as copolymers, are vital to both living and synthetic systems. Copolymerization has been studied theoretically in a number of contexts, often by considering a Markov process in which monomers are added or removed from the growing tip of a long copolymer. To date, the analysis of the most general models of this class has necessitated simulation. We present a general method for analyzing such processes without resorting to simulation. Our method can be applied to models with an arbitrary network of sub-steps prior to addition or removal of a monomer, including non-equilibrium kinetic proofreading cycles. Moreover, the approach allows for a dependency of addition and removal reactions on the neighboring site in the copolymer and thermodynamically self-consistent models in which all steps are assumed to be microscopically reversible. Using our approach, thermodynamic quantities such as chemical work; kinetic quantities such as time taken to grow; and statistical quantities such as the distribution of monomer types in the growing copolymer can be directly derived either analytically or numerically from the model definition.

Brownian particles in periodic potentials: Coarse-graining versus fine structure

We study the motion of an overdamped particle connected to a thermal heat bath in the presence of an external periodic potential in one dimension. When we coarse-grain, i.e., bin the particle positions using bin sizes that are larger than the periodicity of the potential, the packet of spreading particles, all starting from a common origin, converges to a normal distribution centered at the origin with a mean-squared displacement that grows as 2D^{*}t, with an effective diffusion constant that is smaller than that of a freely diffusing particle. We examine the interplay between this coarse-grained description and the fine structure of the density, which is given by the Boltzmann-Gibbs (BG) factor e^{-V(x)/k_{B}T}, the latter being nonnormalizable. We explain this result and construct a theory of observables using the Fokker-Planck equation. These observables are classified as those that are related to the BG fine structure, like the energy or occupation times, while others, like the positional moments, for long times, converge to those of the large-scale description. Entropy falls into a special category as it has a coarse-grained and a fine structure description. The basic thermodynamic formula F=TS-E is extended to this far-from-equilibrium system. The ergodic properties are also studied using tools from infinite ergodic theory.

Fluctuations of cell geometry and their nonequilibrium thermodynamics in living epithelial tissue

We measure different contributions to entropy production in a living functional epithelial tissue. We do this by extracting the functional dynamics of development while at the same time quantifying fluctuations. Using the translucent Drosophila melanogaster pupal epithelium as an ideal tissue for high-resolution live imaging, we measure the entropy associated with the stochastic geometry of cells in the epithelium. This is done using a detailed analysis of the dynamics of the shape and orientation of individual cells which enables separation of local and global aspects of the tissue behavior. Intriguingly, we find that we can observe irreversible dynamics in the cell geometries but without a change in the entropy associated with those degrees of freedom, showing that there is a flow of energy into those degrees of freedom. Hence, the living system is controlling how the entropy is being produced and partitioned into its different parts.

Thermodynamics and efficiency of sequentially collisional Brownian particles: The role of drivings

Brownian particles placed sequentially in contact with distinct thermal reservoirs and subjected to external driving forces are promising candidates for the construction of reliable engine setups. In this contribution, we address the role of driving forces for enhancing the collisional machine performance. Analytical expressions for thermodynamic quantities such as power output and efficiency are obtained for general driving schemes. A proper choice of these driving schemes substantially increases both power output and efficiency and extends the working regime. Maximizations of power and efficiency, whether with respect to the strength of the force, driving scheme, or both have been considered and exemplified for two kind of drivings: generic power-law and harmonic (sinusoidal) drivings.

Homeostatic and adaptive energetics: Nonequilibrium fluctuations beyond detailed balance in voltage-gated ion channels

Stochastic thermodynamics has largely succeeded in characterizing both equilibrium and far-from-equilibrium phenomena. Yet many opportunities remain for application to mesoscopic complex systems-especially biological ones-whose effective dynamics often violate detailed balance and whose microscopic degrees of freedom are often unknown or intractable. After reviewing excess and housekeeping energetics-the adaptive and homeostatic components of a system's dissipation-we extend stochastic thermodynamics with a trajectory class fluctuation theorem for nonequilibrium steady-state, nondetailed-balanced complex systems. We then take up the neurobiological examples of voltage-gated sodium and potassium ion channels to apply and illustrate the theory, elucidating their nonequilibrium behavior under a biophysically plausible action potential drive. These results uncover challenges for future experiments and highlight the progress possible understanding the thermodynamics of complex systems-without exhaustive knowledge of every underlying degree of freedom.

A unicellular walker controlled by a microtubule-based finite-state machine

Cells are complex biochemical systems whose behaviors emerge from interactions among myriad molecular components. Computation is often invoked as a general framework for navigating this cellular complexity. However, it is unclear how cells might embody computational processes such that the theories of computation, including finite-state machine models, could be productively applied. Here, we demonstrate finite-state-machine-like processing embodied in cells using the walking behavior of Euplotes eurystomus, a ciliate that walks across surfaces using fourteen motile appendages (cirri). We found that cellular walking entails regulated transitions among a discrete set of gait states. The set of observed transitions decomposes into a small group of high-probability, temporally irreversible transitions and a large group of low-probability, time-symmetric transitions, thus revealing stereotypy in the sequential patterns of state transitions. Simulations and experiments suggest that the sequential logic of the gait is functionally important. Taken together, these findings implicate a finite-state-machine-like process. Cirri are connected by microtubule bundles (fibers), and we found that the dynamics of cirri involved in different state transitions are associated with the structure of the fiber system. Perturbative experiments revealed that the fibers mediate gait coordination, suggesting a mechanical basis of gait control.

Emergent second law for non-equilibrium steady states

The Gibbs distribution universally characterizes states of thermal equilibrium. In order to extend the Gibbs distribution to non-equilibrium steady states, one must relate the self-information \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{\mathcal{I}}}}}}}}(x)=-\!\log ({P}_{{{{{{{{\rm{ss}}}}}}}}}(x))$$\end{document}I(x)=−log(Pss(x)) of microstate x to measurable physical quantities. This is a central problem in non-equilibrium statistical physics. By considering open systems described by stochastic dynamics which become deterministic in the macroscopic limit, we show that changes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\Delta }}{{{{{{{\mathcal{I}}}}}}}}={{{{{{{\mathcal{I}}}}}}}}({x}_{t})-{{{{{{{\mathcal{I}}}}}}}}({x}_{0})$$\end{document}ΔI=I(xt)−I(x0) in steady state self-information along deterministic trajectories can be bounded by the macroscopic entropy production Σ. This bound takes the form of an emergent second law \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\Sigma }}+{k}_{b}{{\Delta }}{{{{{{{\mathcal{I}}}}}}}}\,\ge \,0$$\end{document}Σ+kbΔI≥0, which contains the usual second law Σ ≥ 0 as a corollary, and is saturated in the linear regime close to equilibrium. We thus obtain a tighter version of the second law of thermodynamics that provides a link between the deterministic relaxation of a system and the non-equilibrium fluctuations at steady state. In addition to its fundamental value, our result leads to novel methods for computing non-equilibrium distributions, providing a deterministic alternative to Gillespie simulations or spectral methods.

Contrary to states of thermal equilibrium, there is no universal characterization of non-equilibrium steady states displaying constant flows of energy and/or matter. Here, the authors make progress in this direction by deriving an emergent and stricter version of the second law of thermodynamics.

Information-driven transitions in projections of underdamped dynamics

Low-dimensional representations of underdamped systems often provide useful insights and analytical tractability. Here, we build such representations via information projections, obtaining an optimal model that captures the most information on observed spatial trajectories. We show that, in paradigmatic systems, the minimization of the information loss drives the appearance of a discontinuous transition in the optimal model parameters. Our results raise serious warnings for general inference approaches, and they unravel fundamental properties of effective dynamical representations impacting several fields, from biophysics to dimensionality reduction.

Information Thermodynamics of the Transition-Path Ensemble

The reaction coordinate describing a transition between reactant and product is a fundamental concept in the theory of chemical reactions. Within transition-path theory, a quantitative definition of the reaction coordinate is found in the committor, which is the probability that a trajectory initiated from a given microstate first reaches the product before the reactant. Here we develop an information-theoretic origin for the committor and show how selecting transition paths from a long ergodic equilibrium trajectory induces entropy production which exactly equals the information that system dynamics provide about the reactivity of trajectories. This equality of entropy production and dynamical information generation also holds at the level of arbitrary individual coordinates, providing parallel measures of the coordinate's relevance to the reaction, each of which is maximized by the committor.

Simulating a chemically fueled molecular motor with nonequilibrium molecular dynamics

Most computer simulations of molecular dynamics take place under equilibrium conditions-in a closed, isolated system, or perhaps one held at constant temperature or pressure. Sometimes, extra tensions, shears, or temperature gradients are introduced to those simulations to probe one type of nonequilibrium response to external forces. Catalysts and molecular motors, however, function based on the nonequilibrium dynamics induced by a chemical reaction's thermodynamic driving force. In this scenario, simulations require chemostats capable of preserving the chemical concentrations of the nonequilibrium steady state. We develop such a dynamic scheme and use it to observe cycles of a particle-based classical model of a catenane-like molecular motor. Molecular motors are frequently modeled with detailed-balance-breaking Markov models, and we explicitly construct such a picture by coarse graining the microscopic dynamics of our simulations in order to extract rates. This work identifies inter-particle interactions that tune those rates to create a functional motor, thereby yielding a computational playground to investigate the interplay between directional bias, current generation, and coupling strength in molecular information ratchets.

Scaling of entropy production under coarse graining in active disordered media

Entropy production plays a fundamental role in the study of nonequilibrium systems by offering a quantitative handle on the degree of time-reversal symmetry breaking. It depends crucially on the degree of freedom considered as well as on the scale of description. How the entropy production at one resolution of the degrees of freedom is related to the entropy production at another resolution is a fundamental question which has recently attracted interest. This relationship is of particular relevance to coarse-grained and continuum descriptions of a given phenomenon. In this work, we derive the scaling of the entropy production under iterative coarse graining on the basis of the correlations of the underlying microscopic transition rates for noninteracting particles in active disordered media. Our approach unveils a natural criterion to distinguish equilibrium-like and genuinely nonequilibrium macroscopic phenomena based on the sign of the scaling exponent of the entropy production per mesostate.

Inverse linear versus exponential scaling of work penalty in finite-time bit reset

A bit reset is a basic operation in irreversible computing. This costs work and dissipates energy in the computer, creating a limit on speeds and energy efficiency of future irreversible computers. It was recently shown by Zhen et al. [Phys. Rev. Lett. 127, 190602 (2021)0031-900710.1103/PhysRevLett.127.190602] that for a finite-time reset protocol, the additional work on top of the quasistatic protocol can always be minimized by considering a two-level system, and then be lower bounded through a thermodynamical speed limit. An important question is to understand under what protocol parameters, including a bit reset error and maximum energy shift, this penalty decreases exponentially vs inverse linearly in the protocol time. Here we provide several analytical results to address this question, as well as numerical simulations of specific examples of protocols.

Insights from an information thermodynamics analysis of a synthetic molecular motor

Information is physical, a realization that has transformed the physics of measurement and communication. However, the flow between information, energy and mechanics in chemical systems remains largely unexplored. Here we analyse a minimalist autonomous chemically driven molecular motor in terms of information thermodynamics, a framework that quantitatively relates information to other thermodynamic parameters. The treatment reveals how directional motion is generated by free energy transfer from chemical to mechanical (conformational and/or co-conformational) processes by 'energy flow' and 'information flow'. It provides a thermodynamic level of understanding of molecular motors that is general, complements previous analyses based on kinetics and has practical implications for machine design. In line with kinetic analysis, we find that power strokes do not affect the directionality of chemically driven machines. However, we find that power strokes can modulate motor velocity, the efficiency of free energy transfer and the number of fuel molecules consumed per cycle. This may help explain the role of such (co-)conformational changes in biomachines and illustrates the interplay between energy and information in chemical systems.

Modeling of biomolecular machines in non-equilibrium steady states

Numerical computations have become a pillar of all modern quantitative sciences. Any computation involves modeling-even if often this step is not made explicit-and any model has to neglect details while still being physically accurate. Equilibrium statistical mechanics guides both the development of models and numerical methods for dynamics obeying detailed balance. For systems driven away from thermal equilibrium, such a universal theoretical framework is missing. For a restricted class of driven systems governed by Markov dynamics and local detailed balance, stochastic thermodynamics has evolved to fill this gap and to provide fundamental constraints and guiding principles. The next step is to advance stochastic thermodynamics from simple model systems to complex systems with tens of thousands or even millions of degrees of freedom. Biomolecules operating in the presence of chemical gradients and mechanical forces are a prime example for this challenge. In this Perspective, we give an introduction to isothermal stochastic thermodynamics geared toward the systematic multiscale modeling of the conformational dynamics of biomolecular and synthetic machines, and we outline some of the open challenges.

Broken detailed balance and entropy production in the human brain

To perform biological functions, living systems must break detailed balance by consuming energy and producing entropy. At microscopic scales, broken detailed balance enables a suite of molecular and cellular functions, including computations, kinetic proofreading, sensing, adaptation, and transportation. But do macroscopic violations of detailed balance enable higher-order biological functions, such as cognition and movement? To answer this question, we adapt tools from nonequilibrium statistical mechanics to quantify broken detailed balance in complex living systems. Analyzing neural recordings from hundreds of human subjects, we find that the brain violates detailed balance at large scales and that these violations increase with physical and cognitive exertion. Generally, we provide a flexible framework for investigating broken detailed balance at large scales in complex systems.

Living systems break detailed balance at small scales, consuming energy and producing entropy in the environment to perform molecular and cellular functions. However, it remains unclear how broken detailed balance manifests at macroscopic scales and how such dynamics support higher-order biological functions. Here we present a framework to quantify broken detailed balance by measuring entropy production in macroscopic systems. We apply our method to the human brain, an organ whose immense metabolic consumption drives a diverse range of cognitive functions. Using whole-brain imaging data, we demonstrate that the brain nearly obeys detailed balance when at rest, but strongly breaks detailed balance when performing physically and cognitively demanding tasks. Using a dynamic Ising model, we show that these large-scale violations of detailed balance can emerge from fine-scale asymmetries in the interactions between elements, a known feature of neural systems. Together, these results suggest that violations of detailed balance are vital for cognition and provide a general tool for quantifying entropy production in macroscopic systems.

Efficiency gain and bidirectional operation of quantum engines with decoupled internal levels

We present a mechanism for efficiency increase in quantum heat engines containing internal energy levels that do not couple to the external work sink. The gain is achieved by using these levels to channel heat in a direction opposite to the one dictated by the second law. No quantum coherence, quantum correlations or ergotropy are required. A similar mechanism allows the engine to run "in reverse" and still produce useful work. We illustrate these ideas using a simple quantum Otto cycle in a coupled-spin system. We find this engine also exhibits other counterintuitive phenomenology. For example, its efficiency may increase as the temperature difference between the heat baths decreases. Conversely, it may cease to operate if the hotter bath becomes too hot or the colder bath too cold.

Algorithmic Approaches for Assessing Irreversibility in Time Series: Review and Comparison

The assessment of time irreversibility, i.e., of the lack of invariance of the statistical properties of a system under the operation of time reversal, is a topic steadily gaining attention within the research community. Irreversible dynamics have been found in many real-world systems, with alterations being connected to, for instance, pathologies in the human brain, heart and gait, or to inefficiencies in financial markets. Assessing irreversibility in time series is not an easy task, due to its many aetiologies and to the different ways it manifests in data. It is thus not surprising that several numerical methods have been proposed in the last decades, based on different principles and with different applications in mind. In this contribution we review the most important algorithmic solutions that have been proposed to test the irreversibility of time series, their underlying hypotheses, computational and practical limitations, and their comparative performance. We further provide an open-source software library that includes all tests here considered. As a final point, we show that "one size does not fit all", as tests yield complementary, and sometimes conflicting views to the problem; and discuss some future research avenues.

Estimating Entropy Production from Waiting Time Distributions

Living systems operate far from thermal equilibrium by converting the chemical potential of ATP into mechanical work to achieve growth, replication, or locomotion. Given time series observations of intra-, inter-, or multicellular processes, a key challenge is to detect nonequilibrium behavior and quantify the rate of free energy consumption. Obtaining reliable bounds on energy consumption and entropy production directly from experimental data remains difficult in practice, as many degrees of freedom typically are hidden to the observer, so that the accessible coarse-grained dynamics may not obviously violate detailed balance. Here, we introduce a novel method for bounding the entropy production of physical and living systems which uses only the waiting time statistics of hidden Markov processes and, hence, can be directly applied to experimental data. By determining a universal limiting curve, we infer entropy production bounds from experimental data for gene regulatory networks, mammalian behavioral dynamics, and numerous other biological processes. Further considering the asymptotic limit of increasingly precise biological timers, we estimate the necessary entropic cost of heartbeat regulation in humans, dogs, and mice.

Universal Bound on Energy Cost of Bit Reset in Finite Time

We consider how the energy cost of bit reset scales with the time duration of the protocol. Bit reset necessarily takes place in finite time, where there is an extra penalty on top of the quasistatic work cost derived by Landauer. This extra energy is dissipated as heat in the computer, inducing a fundamental limit on the speed of irreversible computers. We formulate a hardware-independent expression for this limit in the framework of stochastic processes. We derive a closed-form lower bound on the work penalty as a function of the time taken for the protocol and bit reset error. It holds for discrete as well as continuous systems, assuming only that the master equation respects detailed balance.

Nonequilibrium brain dynamics as a signature of consciousness

The cognitive functions of human and nonhuman primates rely on the dynamic interplay of distributed neural assemblies. As such, it seems unlikely that cognition can be supported by macroscopic brain dynamics at the proximity of equilibrium. We confirmed this hypothesis by investigating electrocorticography data from nonhuman primates undergoing different states of unconsciousness (sleep, and anesthesia with propofol, ketamine, and ketamine plus medetomidine), and functional magnetic resonance imaging data from humans, both during deep sleep and under propofol anesthesia. Systematically, all states of reduced consciousness unfolded at higher proximity to equilibrium compared to conscious wakefulness, as demonstrated by the computation of entropy production and the curl of probability flux in phase space. Our results establish nonequilibrium macroscopic brain dynamics as a robust signature of consciousness, opening the way for the characterization of cognition and awareness using tools from statistical mechanics.

Excess and loss of entropy production for different levels of coarse graining

We investigate the effect of coarse graining on the thermodynamic properties of a system, focusing on entropy production. As a case of study, we consider a one-dimensional colloidal particle in contact with a thermal bath, moving in a sinusoidal potential and driven out of equilibrium by a small constant force. Different levels of coarse graining are evaluated: At first, we compare the results in the underdamped dynamics with those in the overdamped one (first coarse graining). For large values of the friction coefficient, the two dynamics have the same thermodynamics properties, while, for smaller friction values, the overdamped approximation produces an excess of entropy production with respect to that of the underdamped dynamics. Moreover, for further smaller values of the drag coefficient, the excess of entropy production turns into a loss. These regimes are explained by evaluating the jump statistics, observing that the inertia is able to induce multiple jumps and affect the average jump rate. The periodic shape of the potential allows us to approximate the continuous dynamics via a Markov chain after the introduction of a suitable time and space discretization (second level of coarse graining). This discretization procedure is implemented starting both from the underdamped and the overdamped evolution and is analyzed for different values of the friction coefficient.

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.

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.

Thermodynamic nonlinear response relation

The fluctuation-dissipation theorem connects equilibrium to mildly (linearly) perturbed situations in a thermodynamic manner: It involves the observable of interest and the entropy production caused by the perturbation. We derive a relation which connects responses of arbitrary order in perturbation strength to correlations of entropy production of lower order, thereby extending the fluctuation-dissipation theorem to cases far from equilibrium in a thermodynamic way. The relation is validated and studied for a four-state model which is coarse-grained to a non-Markovian two-state model.

Free-energy transduction within autonomous systems

The excess work required to drive a stochastic system out of thermodynamic equilibrium through a time-dependent external perturbation is directly related to the amount of entropy produced during the driving process, allowing excess work and entropy production to be used interchangeably to quantify dissipation. Given the common intuition of biological molecular machines as internally communicating work between components, it is tempting to extend this correspondence to the driving of one component of an autonomous system by another; however, no such relation between the internal excess work and entropy production exists. Here we introduce the "transduced additional free-energy rate" between strongly coupled subsystems of an autonomous system, which is analogous to the excess power in systems driven by an external control parameter that receives no feedback from the system. We prove that this is a relevant measure of dissipation-in that it equals the steady-state entropy production rate due to the downstream subsystem-and demonstrate its advantages with a simple model system.

Inverse Power Law Scaling of Energy Dissipation Rate in Nonequilibrium Reaction Networks

The energy dissipation rate in a nonequilibrium reaction system can be determined by the reaction rates in the underlying reaction network. By developing a coarse-graining process in state space and a corresponding renormalization procedure for reaction rates, we find that energy dissipation rate has an inverse power-law dependence on the number of microscopic states in a coarse-grained state. The dissipation scaling law requires self-similarity of the underlying network, and the scaling exponent depends on the network structure and the probability flux correlation. Existence of the inverse dissipation scaling law is shown in realistic biochemical systems such as biochemical oscillators and microtubule-kinesin active flow systems.

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.

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.

Stochastic approach to entropy production in chemical chaos

Methods are presented to evaluate the entropy production rate in stochastic reactive systems. These methods are shown to be consistent with known results from nonequilibrium chemical thermodynamics. Moreover, it is proved that the time average of the entropy production rate can be decomposed into the contributions of the cycles obtained from the stoichiometric matrix in both stochastic processes and deterministic systems. These methods are applied to a complex reaction network constructed on the basis of Rössler's reinjection principle and featuring chemical chaos.

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.

Exact Coarse Graining Preserves Entropy Production out of Equilibrium

The entropy production rate associated with broken time-reversal symmetry provides an essential characterization of nanosystems out of equilibrium, from driven colloidal particles to molecular motors. Limited access to the dynamical states is generally expected to hinder the correct estimation of this observable. Here we show how memoryless jump processes can be coarse grained, exactly preserving its average and fluctuations at stationarity. This supports univocal applicability of fluctuation theorems for entropy and allows inference of the genuine thermodynamics together with inaccessible process details.