Stochastic Thermodynamics: Experiment and Theory

We present a thermodynamic theory for material chains made up of nanoscopic subunits with symbolic meaning in the presence of memory. This framework is based on the characterization of single sequences of symbols constructed under a protocol and is used to derive the behavior of ensembles of sequences similarly constructed. We apply our formalism to DNA replication and RNA transcription finding that Watson and Crick hybridization energies with which nucleotides are branched to the template strand during the copying process are optimal to regulate the fidelity in proofreading. We also explain from first principles the fidelity of these biomachines and their efficiency, the latter conjugating mechanochemical and information components. Finally, we provide an explanation for the polymerase Brownian ratchet translocation mechanism, which involves the DNA/RNA flexibility. Then, we expand our theory to purely symbolic systems conveying information, that is, to systems where information content is not determined by thermodynamic laws, but able to respond to ad hoc, imposed operational rules, as is the case for human languages. Inspired by how information is managed in biomolecular systems, we introduce writing, entailing any bit string generation, and revision, as comprising proofreading and editing, in Information Theory. We derive necessary and sufficient conditions for both effective proofreading and editing. Our results underlie any communication system, ranging from human languages and computer science to gene evolution.

Transient compartments have been recently shown to be able to maintain functional replicators in the context of prebiotic studies. Here, we show that a broad class of selection dynamics is able to achieve this goal. We identify two key parameters, the relative amplification of nonactive replicators (parasites) and the size of compartments. These parameters account for competition and diversity, and the results are relevant to similar multilevel selection problems, such as those found in virus-host ecology and trait group selection.

Motivated by the recent development of fast and ultra-sensitive thermometry in nanoscale systems, we investigate quantum calorimetric detection of individual heat pulses in the sub-meV energy range. We propose a hybrid superconducting injector-calorimeter set-up, with the energy of injected pulses carried by tunneling electrons. Treating all heat transfer events microscopically, we analyse the statistics of the calorimeter temperature fluctuations and derive conditions for an accurate measurement of the heat pulse energies. Our results pave the way for novel, fundamental quantum thermodynamics experiments, including calorimetric detection of single microwave photons.

Countless works in the literature have investigated how coarse-graining influences our prediction of the physical properties of a system. In systems out of equilibrium, it is still unclear what is the role of the entropy production, and for systems described by a Master Equation, it can be estimated using Schnakenberg's formula. On the other hand, some years ago Seifert derived an analogous formula for dynamics described by a Fokker-Planck. In this work, we aim at connecting both formulations, and starting from a Master-Equation system we calculate how Schnackenberg's entropy production is influenced by coarse-graining of the system. We show that such a value can be reduced to the Seifert's formula for some simple choices of the dynamics, but, surprisingly enough, we demonstrate that, in general, microscopic fluxes circulating in the system can give a macroscopic contribution to the entropy production. In consequence, neglecting information leads to an underestimation of the entropy production, and only a lower bound can be provided when the dynamics is coarse-grained.

Currents play an prominent role in stochastic systems: many observables like heat, work, entropy production, electrical currents and mass flows can all be expressed as stochastic currents. The need to characterize the fluctuating behavior of currents led, in the recent years, to the discovery of uncertainty relations. Such relations state that the entropy production rate of a system sets a minimum amount of fluctuations on every stochastic current. Uncertainty relations were believed to be general and independent of the system characteristics. However, it was recently discovered that such lower bounds depend on the nature of time: discrete or continuous. To understand the physical reason, we compare here current fluctuations in discrete-time Markov chains and continuous-time master equations. We prove that the random timing of transitions in the master equations makes current fluctuations always more likely. From this analysis we derive a rule to map the moments of a current between discrete and continuous time. We then exploit this mapping to obtain new uncertainty bounds. Our findings show that the quests for uncertainty bounds in discrete and continuous time reduce to a single problem.

Biochemical oscillations are ubiquitous in biology and allow organisms to properly time their biological functions. Underlying the oscillations are series of chemical reactions, which are inherently stochastically timed; this affects the accuracy of the period of oscillations and they eventually lose coherence. The simplest model of a stochastic reaction network is a Markov state process on a single ring that is biased to have a net current. For this class of models, we use a perturbation theory to obtain analytical expressions for the number of coherent oscillations and the period of oscillations in the presence of disorder in the hopping rates. Our main finding is that both of these quantities can be robustly tuned, even in the presence of high disorder in the network, when the chemical affinity is sufficiently high. Our results confirm the recently postulated bound on the coherence of biochemical oscillations set by the chemical affinity of the network (Phys. Rev. E 95, 062409). While recent work has established that increasing energy dissipation improves the coherence of oscillations, our findings suggest that it plays the additional role of making the coherence and the period of oscillations robust to changes in the rates resulting from noisy environment of the cell.

In small size systems, the relevant physical quantities are subjected to fluctuations. Over recent years progress in experiments along with new theoretical insights contributed to a rapid development of stochastic thermodynamics. The aim of this framework is to generalize the laws of thermodynamics to small systems. Not long ago a new approach to stochastic thermodynamics was proposed [1], which is based on the extension of local equilibrium hypothesis to mesoscopic systems. Starting from macroscopic non-equilibrium thermodynamics, this approach enables to derive stochastic differential equations for thermodynamic quantities of interest such as heat or entropy. This formalism has already been applied in the study of Brownian motion [2] and chemical reactions in small systems [3]. In this talk, our approach will be applied to the study of thermodiffusion in a mesoscopic system connected to matter and heat reservoirs. This system can be seen at once as a separation device, using coupling between heat and matter fluxes to reach better separation and as an engine, using coupling to drive originally non-spontaneous transport (e.g., using matter flux to transfer heat from a cold to a hot body). In both cases the efficiency of the device can be assessed by defining different ratios of stochastic thermodynamic variables (the separation and thermodynamic efficiency, respectively). Previous theoretical studies on thermodynamic efficiency pointed out that its distributions were able to display bimodality [4], suggesting the coexistence of different working regimes caused by the fluctuations. Using our approach, we will show that bimodality of distributions of both efficiencies can be tuned by varying external control parameters like the temperature or density of the reservoirs. The conditions of its appearance will be discussed. Finally, the evolution of the bimodal shape in the thermodynamic limit following the efficiency considered will also be investigated. [1] De Decker, Y. ; Cantú Ros, A. G. ; Nicolis, G. Extended Local Equilibrium Approach to Stochastic Thermodynamics. Eur. Phys. J. Spec. Top. 2015, 224 (5), 947–968. [2] Nicolis, G.; De Decker, Y. Stochastic Thermodynamics of Brownian Motion. Entropy 2017, 19 (9), 434. [3] De Decker, Y.; Derivaux, J.-F.; Nicolis, G. Stochastic Thermodynamics of Reactive Systems: An Extended Local Equilibrium Approach. Phys. Rev. E 2016, 93 (4), 42127. [4] Polettini, M.; Verley, G.; Esposito, M. Efficiency Statistics at All Times: Carnot Limit at Finite Power. Phys. Rev. Lett. 2015, 114 (5), 050601.

We present a feedback protocol that is able to confine a system in contact with a thermal bath to a single micro-state with neither heat dissipation nor performed work. The protocol adjusts the Hamiltonian of the system in such a way that the Bayesian posterior distribution after measurement is in equilibrium. As a result, the whole process satisfies feedback reversibility --the process is indistinguishable from its time reversal— and assures the lowest possible dissipation for confinement. In spite of the whole process being reversible it can surprisingly be implemented in finite time. It can be shown that the protocol can optimally convert all extracted information into work.

In some situations in stochastic thermodynamics not all relevant slow degrees of freedom are accessible. Consequently, one adopts an effective description involving only the visible degrees of freedom. This gives rise to an apparent entropy production that violates standard fluctuation theorems. I present an analytically solvable model illustrating how the fluctuation theorems are modified. Furthermore, I define an alternative to the apparent entropy production: the marginal entropy production which fulfills the fluctuation theorems in the usual form. One can show that the non-Markovianity of the visible process is responsible for the deviations in the fluctuation theorems.

We consider a Brownian particle which, in addition to being in contact with a thermal bath, is driven by active fluctuations. These active fluctuations do not fulfill a fluctuation-dissipation relation and therefore play the role of a non-equilibrium environment. Using an Ornstein-Uhlenbeck process as a model for the active fluctuations, we derive the path probability of the Brownian particle subject to both, thermal and active noise. From the case of passive Brownian motion, it is well-known that the log-ratio of path probabilities for observing a certain particle trajectory forward in time versus observing its time-reserved twin trajectory quantifies the entropy production in the thermal environment. We calculate this path probability ratio for active Brownian motion and derive a generalized ``entropy production'', which fulfills an integral fluctuation theorem. We show that those parts of this ``entropy production’’, which are different from the usual dissipation of heat in the thermal environment, can be associated with the mutual information between the particle trajectory and the history of the non-equilibrium environment.

In contrast with systems driven by an external field, energy dissipation in active matter is local and independent for each particle. This leads to new dynamics and phases, such as clustering with purely repulsive interactions and collective directed motion. While these phenomena have been studied extensively, understanding how the local dissipation affects the collective dynamics, and its connection with entropy production [1, 2], has remained an elusive goal. Based on methods of large deviations, we explore how tuning the dissipation, as an independent parameter, modifies the emerging collective behavior. This amounts to a change of ensemble where individual trajectories are biased in terms of their dissipation. By deriving an auxiliary dynamics which effectively realizes the dissipation bias, we put forward a microscopic mechanism which promotes clustering at low dissipation [3]. Moreover, the direct sampling of the biased ensemble reveals the emergence of a collective moving state at high dissipation, despite the absence of aligning interactions [4]. We combine heuristic and analytic arguments to rationalize the dynamical phase transitions between these states. Overall, our results shed a new light on the control of collective properties by local dissipation. They open the door to the search of new phases and dynamics, as well as unexpected transitions between them, in biased ensembles of active matter. [1] ÉF, C. Nardini, M. E. Cates, J. Tailleur, P. Visco, and F. van Wijland, 'How far from equilibrium is active matter?', Phys. Rev. Lett. 117, 038103 (2016) [2] C. Nardini, ÉF, E. Tjhung, F. van Wijland, J. Tailleur, and M. E. Cates, 'Entropy production in field theories without time-reversal symmetry: Quantifying the non-equilibrium character of active matter', Phys. Rev. X 7, 021007 (2017) [3] ÉF, T. Nemoto, and S. Vaikuntanathan, 'Collisional efficiency controls transport and clustering in active fluids', in preparation [4] T. Nemoto, ÉF, M. E. Cates, R. L. Jack, and J. Tailleur, 'Optimizing active work: dynamical phase transitions, collective motion and jamming', arXiv:1805.02887

We present a non-equilibrium thermodynamical approach to the turbulent cascade with respect to the evolution process of velocity increments u_r towards smaller scales. Evidence of Markov property for the turbulent cascade process in scale down to the so-called Einstein-Markov length [1], which corresponds to a three-point (two-scale) closure of general joint multi-scale statistics, has been shown in previous studies [1,5]. Based on an estimation method by Kramers-Moyal coefficients D^{(k)}(u_r,r), a Fokker-Planck equation for the cascade process can be estimated directly from the measured data, thus the whole multi-scale statistics is expressed by a differential equation. As found in many experimental data [2,4,5,6] we find a linear function for D^{(1)}(u_r,r)=d_{11}(r)u_r and a parabolic function for D^{(2)}(u_r,r)=d_{22}(r)u_r^2+d_{21}(r)u_r+d_{20}(r) (we define d_{ij}\neq0 as D^{(1,2)}≠0). As the often discussed multiscaling models for turbulence like Kolmogorov 41, 62, etc. can be related to explicit forms of the Fokker-Planck equation, we are able to reconstruct all structure functions, even the third order one, accurately. Since structure functions are 2-point (one scale) quantities, there is a wide range of Fokker-Planck equations that can reproduce these structure functions. This statistical approach is very general and can be linked to the non-equilibrium thermodynamics of microscopic systems [7]. This analogy enables to apply concepts of stochastic thermodynamics to turbulent flows [3,4] and to define the thermodynamical quantity of entropy. For every individual cascade trajectory u(.), which we define as the complete evolution of a velocity increment from the integral length to the Taylor length, the entropy change ΔS_{tot} can be determined. As a new feature we find that the entropy fluctuations of the turbulent cascade fulfill in high precision the rigorous integral fluctuation theorem = 1, which is a fundamental entropy law of non-equilibrium thermodynamics [7]. Furthermore, we present first experimental evidence that both the integral and detailed fluctuation theorem is fulfilled for a macroscopic system by analyzing experimental measurements of turbulent flows for the first time. So far experimental studies of stochastic thermodynamics has been made using particle systems (mainly biological systems and other nano-systems). Taking the IFT as a valid law for turbulence it can be used to quantify the applicability of the estimated Fokker-Planck equation for describing the turbulent cascade process. Using the same functional form of the Fokker-Planck equation as used for the reconstruction of structure function, solely the parametrization D^{(1,2)}≠0 show a good convergence of the integral fluctuation theorem to the theoretical value 1. Using d_{20}(r)=0 (K62) or d_{22}(r)=0 the empirical average diverges. In this contribution we will investigate these features of the turbulent cascade for different turbulent flows like free jets, wake flows of regular grid as well as low-temperature, large size forced von Karman liquid helium flows (performed in the SHREK experiment). [1] Lueck, S., Renner, C., Peinke, J., Friedrich, R., et al., (2006), The markov-einstein coherence length a new meaning for the taylor length in turbulence, Physics Letters A, 359(5), 335-338; [2] Nawroth, A., Peinke, J., Kleinhans, D., Friedrich, R., (2007), Improved estimation of fokker-planck equations through optimization, Physical Review E 76(5),056, 102; [3] Nickelsen D. and Engel A., (2013), Probing small-scale intermittency with a fluctuation theorem, Physical Review Letters, 110, 214501; [4] Reinke N., Fuchs A., Nickelsen D., Peinke J., (2018), On universal features of the turbulent cascade and its non-equilibrium thermodynamic process, J. Fluid Mech.; [5] Renner, C., Peinke, J., Friedrich, R., (2001), Experimental indications for markov properties of smallscale turbulence, Journal of Fluid Mechanics, 433, 383-409; [6] Renner C., Peinke J., Friedrich R., Chanal O. and Chabaud B., (2002), Universality of small scale turbulence, Physical Review Letters, 89, 124502; [7] Seifert, U., (2012), Stochastic thermodynamics, fluctuation theorems and molecular machines, Reports on Progress in Physics, 75(12)

Work belongs to the most basic notions in thermodynamics but it is not well understood in quantum systems, especially in open quantum systems. We introduce a novel concept of the quantum work functional along individual Feynman path and give a framework to calculate the work statistics for a generic open quantum system - quantum Brownian motion. This path integral approach to study quantum work is, though equivalent to, but more powerful and insightful than the two-point measurement approach as follows. First, we show that when the Plank constant approaches zero, the Feynman path converges to the classical trajectories, and the quantum work functional converges to the classical fluctuating work. We therefore analytically show the quantum-classical correspondence principle of the work statistics in open quantum systems for the first time. We further give quantum corrections to the classical fluctuating work, which will help our understanding of work in the quantum regime. Second, our framework can be applied to non-Markovian, non-rotating wave approximation, and strong-coupling regime, and the time-variation of the system Hamiltonian can be arbitrary. This is intriguing since the conventional quantum master equation approach to study quantum thermodynamics has a strong limitation on the applicable parameter regime. In addition, we can utilize the analytical and numerical calculation methods developed in the field of path integral to study quantum work statistics. As an example, we demonstrate the usefulness of our framework by analytically calculating the work statistics for a dragged harmonic oscillator. [1] K. Funo and H. T. Quan, arXiv:1708.05113.

Changes are inevitable in nature, and those that are most dramatic with often drastic consequences are the ones that occur all of a sudden. An example is a diffusing system that, during its temporal evolution, makes a sudden jump (a “reset”) to a fixed state or configuration. This scenario has found widespread applications in e.g. optimal searching problems. In this work, we study the dynamics of overdamped Brownian particles diffusing in conservative force fields and undergoing stochastic resetting to a given location with a generic space-dependent rate of resetting. We present a systematic approach involving path integrals and elements of renewal theory that allows to derive analytical expressions for a variety of statistics of the dynamics such as (i) the propagator prior to first reset; (ii) the distribution of the first-reset time, and (iii) the spatial distribution of the particle at long times. We apply our approach to several representative and hitherto unexplored examples of resetting dynamics. A particularly interesting example for which we find analytical expressions for the statistics of resetting is that of a Brownian particle trapped in a harmonic potential with a rate of resetting that depends on the instantaneous energy of the particle. We find that using energy-dependent resetting processes is more effective in achieving spatial confinement of Brownian particles on a faster timescale than by performing quenches of parameters of the harmonic potential.

We uncover a duality between relaxation and first passage processes in ergodic reversible Markovian dynamics in both discrete and continuous state-space. The duality exists in the form of a spectral interlacing - the respective time scales of relaxation and first passage are shown to interlace. Our canonical theory allows for the first time to determine the full first passage time distribution analytically from the simpler relaxation eigenspectrum. The duality is derived and proven rigorously for both discrete state Markov processes in arbitrary dimension and effectively one-dimensional diffusion processes [1]. The diffusive exploration of a rugged potential landscape is analyzed to demonstrate how the duality allows for an intuitive understanding of first passage trajectories in terms of relaxational eigenmodes. More generally, we provide a comprehensive explanation of the full statistics of reactive trajectories in rugged potentials, incl. the so-called 'few-encounter limit'. [1] D. Hartich and A. Godec, arXiv:1802.10046; arXiv:1802.10049 (2018).

Fluctuation theorems are powerful equalities that hold far from equilibrium. However, the standard approach to include measurement and feedback schemes may become inapplicable in certain situations, including stochastic continuous measurements, precise measurements of continuous variables, and feedback induced irreversibility. Here we overcome these shortcomings by providing a recipe for producing detailed fluctuation theorems. Based on this recipe, we derive a fluctuation theorem which holds for arbitrary measurement and feedback protocols. The key insight is that the fluctuations which can be inferred from the measurement outcomes can be suppressed by post-selection. Our detailed fluctuation theorem results in a stringent and experimentally accessible inequality on the extractable work, which is saturated when the full entropy production can be inferred from the data.

Recently, a mesoscopic device has been proposed which consists of a voltage biased Josephson Junction coupled to microcavities that exchange hot and cold photons with their respective thermal baths. By absorbing photons from the hot cavity and emitting photons to the cold cavity, cooper pairs use the temperature gradient to tunnel against the voltage bias, effectively converting heat flow into work. Steady state calculations have shown that this system can operate as a quantum heat engine exhibiting both high power and high efficiency. Going beyond the steady state regime, fluctuations of various thermodynamic quantities of the system are investigated by means of full counting statistics. Because the device is experimentally viable, results can be tested and allow for a deeper understanding in fluctuation theorems and thermodynamic fluctuations in general. Current progress and preliminary results are presented.

Inferring the directionality of interactions between cellular processes is a major challenge in systems biology. Time-lagged correlations allow to discriminate between alternative models, but they still rely on assumed underlying interactions. Here, we show that an information-theoretic quantity, the transfer entropy (TE), quantifies the directional influence between fluctuating variables in a model-free way. We present a theoretical approach to compute the transfer entropy, even when the noise has an extrinsic component or in the presence of feedback. We re-analyze the experimental data from Kiviet et al. (2014) [1], where fluctuations in gene expression of metabolic enzymes and growth rate have been measured in single cells of Escherichia coli. We confirm the formerly detected modes between growth and gene expression, while prescribing more stringent conditions on the structure of noise sources [2]. [1] Stochasticity of metabolism and growth at the single-cell level, D. J. Kiviet et al., Nature, 514, 376 (2014). [2] Information-theoretic analysis of the directional influence between cellular processes, Lahiri S, Nghe P, Tans SJ, Rosinberg ML, Lacoste D (2017), PLoS ONE 12(11): e0187431, https://arxiv.org/abs/1709.01746

We reformulate stochastic thermodynamics in terms of noise realizations for Langevin systems in contact with multiple reservoirs and investigated the structure of the second laws of thermodynamics. We derive a hierarchy of fluctuation theorems when one degree of freedom of the system is affected by multiple reservoirs simultaneously, that is, when noise mixing occurs. These theorems and the associated second laws of thermodynamics put stricter bounds on the thermodynamics of Langevin systems. We apply our results to a stochastic machine in noise-mixing environments and demonstrate that our new bounds play a crucial role in determining the potential function and performance of the machine.†. † J. S. Lee and H. Park, arXiv:1804.06582.

Circadian clocks can be viewed as biochemical phase estimators for the periodic day-night light signal. Many organisms use orbital attractors ('free-running clocks') to anticipate the day-night cycle. However, others organisms use simple stimulus-response point attractors (‘hourglass clocks’), and it is not clear when such strategies are sufficient or even preferable to free running clocks. Here, we find that free running clocks, such as those found in the cyanobacterium Synechococcus elongatus and humans, can efficiently project out light intensity noises due to weather patterns (‘external noise’) by exploiting their limit cycle attractor. However, such limit cycles are necessarily vulnerable to ‘internal noise’. Hence, at a sufficiently high internal noise level, point attractor-based ‘hourglass’ clocks, such as those found in a smaller cyanobacterium with low protein copy number, Prochlorococcus marinus, outperform free running clocks. By interpolating between these two regimes, we demonstrate a trade-off between internal and external noise resistance in various forms of biological oscillators.

We present an experimental realization of a so-called "optimal process" introduced by V. Cavina et al., Sci. Rep. (2016), that allows to extract large work values above the free energy difference associated to the process. Our experiment uses the two-level charge state of a sub-micronic metallic island weakly coupled through tunnel junctions to external superconducting leads acting as electron reservoirs. The electrostatic energy of the island is controlled with an external gate voltage. The optimal sequence then consists of a quasi-static ramp of this control parameter, followed by an instantaneous quench and finally another quasi-static ramp that brings back the system to its initial potential energy. During the sequence heat is exchanged between the system and the reservoirs through single electron tunneling. By monitoring these stochastic tunneling events with another single-electron transistor, we can compute the work done by the system along a single thermodynamic trajectory and thus the work distribution obtained for many repetitions of the process. We experimentally demonstrate that counter-intuitively, the quench enables less likely but extreme work production above the free energy difference, with probabilities on the work bounds derived from Jarzynski equality and work values depending on the quench amplitude.

We consider a model where two heavy probes locally interact with diffusing colloids driven by periodic time-dependent forces. For a slow frequency of the driving we show that there emerges an algebraically decaying effective interaction between the probes, below a frequency-dependent threshold which diverges in the adiabatic limit. Various other driving regimes are discussed.

Biochemical oscillations are ubiquitous in living organisms. In an autonomous system, not influenced by an external signal, they can only occur out of equilibrium. We show that they emerge through a generic nonequilibrium phase transition, with a characteristic qualitative behavior at criticality. The control parameter is the thermodynamic force which must be above a certain threshold for the onset of biochemical oscillations. This critical behavior is characterized by the thermodynamic flux associated with the thermodynamic force, its diffusion coefficient, and the stationary distribution of the oscillating chemical species. We discuss metrics for the precision of biochemical oscillations by comparing two observables, the Fano factor associated with the thermodynamic flux and the number of coherent oscillations. Since the Fano factor can be small even when there are no biochemical oscillations, we argue that the number of coherent oscillations is more appropriate to quantify the precision of biochemical oscillations. Our results are obtained with three thermodynamically consistent versions of known models: the Brusselator, the activator-inhibitor model, and a model for KaiC oscillations. [1] B. Nguyen, U. Seifert and A.C. Barato, J. Chem. Phys. 149, 045101 (2018)

Nonequilibrium processes of small systems such as molecular machines are ubiquitous in biology, chemistry, and physics, but are often challenging to comprehend. In the past two decades, several exact thermodynamic relations of nonequilibrium processes, collectively known as fluctuation theorems, have been discovered and provided critical insights. These fluctuation theorems are generalizations of the second law and can be unified by a differential fluctuation theorem. We perform the first experimental test of the differential fluctuation theorem using an optically levitated nanosphere in both underdamped and overdamped regimes and in both spatial and velocity spaces. We also test several theorems that can be obtained from it directly, including a generalized Jarzynski equality that is valid for arbitrary initial states, and the Hummer-Szabo relation. Our study experimentally verifies these fundamental theorems and initiates the experimental study of stochastic energetics with the instantaneous velocity measurement.

A Brownian information engine extracts work from a single heat bath at constant temperature by utilizing the information about the microscopic state of a Brownian particle. Much of the experimental studies to date are limited to the realization of such information engine whose initial state is in thermal equilibrium; however, no experimental work has been done to investigate the optimal operating condition for finite cycle period  during which the engine may not have relaxed fully before next cycle begins. Here, we report on the performance of a cyclic information engine as a function of . We found that the extracted work increases with increasing  and is maximum when  reaches infinity, while the extracted power is maximum at finite  or when  approaches zero. We have also measured the efficiency of this engine in general and the efficiency at maximum power.

For fluctuating thermodynamic currents in non-equilibrium steady states, the thermodynamic uncertainty relation expresses a fundamental trade-off between precision, i.e. small fluctuations, and dissipation. Using large deviation theory, we show that this relation follows from a universal bound on current fluctuations that is valid beyond the Gaussian regime and in which only the total rate of entropy production enters. Variants and refinements of this bound hold for fluctuations on finite time scales and for Markovian networks with known topology and cycle affinities. Applied to molecular motors and heat engines, the bound on current fluctuations imposes constraints on the efficiency and power.

We apply our recent theory of the effective thermodynamics for marginal observers to chemical networks, finding a connection between fluctuation-dissipation and the so-called deficiency.

We calculate the large deviation function of the end-to-end distance and the corresponding extension-versus-force relation for random walks. For off-lattice random walks with persistence in higher dimensions, the large-deviation function undergoes a first order phase transition. In 4 dimensions the system is perfectly harmonic.

The study analyzes the autonomous quantum Maxwell's demon based on two exchange-coupled quantum dots, each attached to two spin-polarized leads magnetized in antiparallel directions. The system is studied by mean of a local quantum master equation in the Lindblad form. The principle of operation of the device is based on the coherent oscillations between the spin states induced by the exchange-coupling. This spin dynamics effectively acts as a quantum iSWAP gate, due to which one of the dots acts as a feedback controller imposing a specific spin polarization of the other dot. As a result, current in the second dot is pumped against the voltage bias, which leads to locally negative entropy production. Information transfer between the quantum dots is then studied quantitatively by applying a formal duplication of the system, which enables to separate contributions to the rate of change of mutual information associated with different subsystems. The calculated entropy production in a single subsystem and the information flow between the subsystems are shown to obey a local form of the second law of thermodynamics, similar to the one previously derived for classical bipartite systems.

The study shows that the quantum coherent effects can reduce fluctuations of the output power of quantum heat engines, which enables to overcome the recently derived trade-off between efficiency, power and power fluctuations applying to classical Markovian steady-state heat engines. The idea is presented using a model system consisting of two tunnel-coupled orbitals, each attached to a separate electronic reservoir; such a setup can be realized, for example, using quantum dots. Electronic transport is studied using the exact Levitov-Lesovik formula in the case without the Coulomb interaction between electrons, as well as by means of a phenomenological master equation in the interacting case. Power fluctuations are shown to be reduced in the analyzed system due to the fact that tunneling between the orbitals is associated with a unitary evolution of the electron state instead of a stochastic Poissonian transition. Coherent nature of this process reduces stochasticity of the system, thus suppressing the current and power fluctuations. Moreover, it is shown that noise can be further suppressed by the Coulomb interaction between electrons which prevents the double occupancy of the system.

The piston system (particles in a box) is the simplest paradigmatic model in traditional thermodynamics. However, the recently established framework of stochastic thermodynamics (ST) fails to apply to this model system due to the embedded singularity in the potential. In this Letter, we study the ST of a particle in a box by adopting a novel coordinate transformation technique. Through comparing with the exact solution of a breathing harmonic oscillator, we obtain analytical results of work distribution for an arbitrary protocol in the linear response regime and verify various predictions of the fluctuation-dissipation relation. When applying to the Brownian Szilard engine model, we obtain the optimal protocol λt=λ02t/τ for a given sufficiently long total time τ. Our study not only establishes a paradigm for studying ST of a particle in a box but also bridges the long-standing gap in the development of ST. Reference: Physical review letters 11 7, 180603 (2016)

We prove [1] a fluctuation theorem for the currents for small-scale periodically driven systems that implies a fluctuation dissipation relation, symmetry relations for Onsager coefficients, and further relations for nonlinear response coefficients. Our results unify the two classes of small-scale periodically driven systems; heat engines driven by periodic temperature variations and molecular pumps driven by external stimuli. Reference: SR and A. C. Barato, Phys. Rev. E, 96, 052120 (2017).

Stochastic noises in many biological systems (especially in cell metabolic networks) have a nontrivial structure that invalidates a description in terms of bipartite processes. This assumption is often made in the context of stochastic and information thermodynamics (for instance for modeling Maxwell's demons or sensory systems) as it simplies the theoretical analysis and allows the contribution of each components of the overall system to the entropy production to be clearly identied. Although the abandon of the bipartite structure complicates the interpretation of information transfers, a comprehensive description is still available which is addressed in this work. The main focus is on non- equilibrium systems that can be modeled by continuous-time Markov processes (diusions, jump processes, or both) in the presence of additive or multiplicative noises. The objective is to include the information exchanges between two interacting subsystems into a stochastic thermodynamic description. In addition, we also discuss some issues in the calculation of transfer entropy in non-Markov processes. In particular we investigate the ability of the nite-horizon transfer entropy to estimate an interaction delay.

Here we present detailed theoretical analysis of single-particle stochastic machines constructed by manipulating a Brownian particle trapped in time-dependent confining potential. For analysis we use recently developed stochastic thermodynamics and we choose systems, extensively used in single particle experiments with externally controlled optical trap, so that our results can directly be verified in experiments. In our analysis special stress will be on work-extraction, optimization of protocols, different modes of operation depending on extracted work and dissipated heat, efficiency of the machines and finally on the fluctuations of relevant thermodynamic quantities.

How quick an operation on a system can be? This is an important problem both from theoretical and practical viewpoints. In the research of heat engines, this problem is asked in the form of incompatibility between high efficiency and large power. Recently, Ref.[1] shows that the linear irreversible thermodynamics does not prohibit the coexistence of the Carnot efficiency and finite power if time-reversal symmetry is broken (e.g., under a magnetic field). However, many studies on various concrete models within the linear response regime have arrived at the same results that the Carnot efficiency and finite power are incompatible in these concrete models [2-5]. In spite of strong expectations, a general bound on efficiency and power has been elusive. Another important research topic is the problem of state transformations, in which how quick a given state can be transformed into another desired state is investigated. In stochastic thermodynamics, state transformations in overdamped Langevin systems have been studied, and interesting relations are obtained [6-8]. However, a general result on state transformations has also been elusive. In this talk, we shall show that the entropy production is the universal key quantity to determine the limit of speed of operations. We first demonstrate this on heat engines, in which we derive a universal trade-off relation between efficiency and power [9,10]. This relation is obtained as a corollary of a trade-off inequality between entropy production and heat flux. This inequality is similar to the thermodynamic uncertainty relation [11,12], while our inequality is applicable to a much broader class of systems including transient processes and systems with broken time-reversal symmetry. We next show a trade-off inequality for state transformations [13]. This inequality bounds the speed of state transformations by the entropy production and dynamical activity, which serves as the time scale of the system [14,15]. Interestingly, in systems with nonzero stationary current, the Hatano-Sasa entropy production [16] bounds the speed of state transformations. [1] G. Benenti, K. Saito, and G. Casati, Phys. Rev. Lett. 106, 230602 (2011). [2] K. Brandner, K. Saito, and U. Seifert, Phys. Rev. Lett. 110, 070603 (2013). [3] V. Balachandran, G. Benenti, and G. Casati, Phys. Rev. B 87, 165419 (2013). [4] K. Brandner, K. Saito, and U. Seifert, Phys. Rev. X 5, 031019 (2015). [5] K. Proesmans and C. Van den Broeck, Phys. Rev. Lett. 115, 090601 (2015). [6] K. Sekimoto and S.-i. Sasa, J. Phys. Soc. Jpn. 66, 3326 (1997). [7] E. Aurell, K. Gawedzki , C. Mejia-Monasterio, R. Mohayaee, P. Muratore-Ginanneschi, J. Stat. Phys. 147, 487 (2012). [8] O. Raz, Y. Subasi, and R. Pugatch, Phys. Rev. Lett. 116, 160601 (2016). [9] N. Shiraishi, K. Saito, and H. Tasaki, Phys. Rev. Lett. 117, 190601 (2016). [10] N. Shiraishi and H. Tajima, Phys. Rev. E 96, 022138 (2017). [11] A. C. Barato and U. Seifert, Phys. Rev. Lett. 114, 158101 (2015). [12] T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Phys. Rev. Lett. 116, 120601 (2016). [13] N. Shiraishi, K. Funo, and K. Saito, arXiv:1802.06554 (2018). [14] J. P. Garrahan, R. L. Jack, V. Lecomte, E. Pitard, K. van Duijvendijk, and F. van Wijland, Phys. Rev. Lett. 98, 195702 (2007). [15] M. Baiesi, C. Maes, and B. Wynants, Phys. Rev. Lett. 103, 010602 (2009). [16] T. Hatano and S.-i. Sasa, Phys. Rev. Lett. 86, 3463 (2001).

We measure extreme fluctuations of stochastic entropy production in an electronic double dot under nonequilibrium steady-state conditions. We find that the cumulative distribution of entropy production’s negative record is bounded at all times by a limiting exponential distribution with a mean value equal to minus the Boltzmann constant. Our experiment reveals an upper bound for the average maximal entropy uptake by a mesoscopic system from its environment in a finite time.

We study the mean velocity and diffusion constant in three related models of molecular Brownian ratchets. Brownian ratchets can be used to describe translocation of biopolymers like DNA through nanopores in cells in the presence of chaperones on the trans side of the pore. Chaperones can bind to the polymer and prevent it from sliding back through the pore. First, we study a simple model that describes the translocation in terms of an asymmetric random walk. It serves as an introductory example but already captures the main features of a Brownian ratchet. We then provide an analytical expression for the diffusion constant in the classical model of a translocation ratchet that was first proposed by Peskin et al.. This model is based on the assumption that the binding and unbinding of the chaperones is much faster than the diffusion of the DNA strand. To remedy this shortcoming, we propose a modified model that is also applicable if the (un)binding rates are finite. We calculate the force dependent mean velocity and diffusivity for this new model and compare the results to the original one. Our analysis shows that for large pulling forces the predictions of both models can differ strongly even if the (un)binding rates are large in comparison to the diffusion time-scale but still finite. Furthermore, implications of the thermodynamic uncertainty relation on the efficiency of Brownian ratchets are discussed.

Recently, the fluctuations-dissipation theorem (valid close to equilibrium) has been transformed into an inequality constraining far-from-equilibrium systems. This result is referred to as ``thermodynamic uncertainty relations'' that holds for various levels of description, from local currents to global currents. Building on these recent results for stationary out of equilibrium systems, we have introduced a non-equilibrium conductance matrix connecting fundamental currents to thermodynamic affinities (or forces), offering the highest level of description of an irreversible system. In this oral presention, I will derive an expression for this conductance matrix and I will show that it has all the properties of an Onsager response Matrix, excepted that it is now a function of the affinities. Using the Loewner partial order for matrices, I will prove a thermodynamic uncertainty relation at the level of conductance and covariance matrices and show that this inequality saturates in the close to equilibrium limit leading back to the fluctuations-dissipation theorem. The non-equilibrium conductance matrix is a central object in the study of stochastic thermodynamic machines since it allows to define the notion of degree of coupling far beyond equilibrium. The poster of Hadrien Vroylandt will exhibit the constraints for thermodynamic machines (maximum efficiency and power efficiency trade-off) that proceed from our framework extending the seminal work of Kedem and Kaplan.

Dynamical ensembles have been introduced to study constrained stochastic processes. In the microcanonical ensemble, the value of a dynamical observable is constrained to a given value. In the canonical ensemble a bias is introduced in the process to move the mean value of this observable. The equivalence between the two ensembles means that calculations in one or the other ensemble lead to the same result. In this paper, we study the physical conditions associated with ensemble equivalence and the consequences of non-equivalence. For continuous time Markov jump processes, we show that ergodicity guarantees ensemble equivalence. For non-ergodic systems or systems with emergent ergodicity breaking, we adapt a method developed for equilibrium ensembles to compute asymptotic probabilities while caring about the initial condition. We illustrate our results on the infinite range Ising model by characterizing the fluctuations of magnetization and activity. We discuss the emergence of non ergodicity by showing that the initial condition can only be forgotten after a time that scales exponentially with the number of spins.