Thermodynamic observables of mesoscopic systems can be expressed as integrated empirical currents. Their fluctuations are bound by thermodynamic uncertainty relations. We introduce the hyperaccurate current as the integrated empirical current with the least fluctuations in a given non-equilibrium system. For steady-state systems described by overdamped Langevin equations, we derive an equation for the hyperaccurate current by means of a variational principle. We show that the hyperaccurate current coincides with the entropy production if and only if the latter saturates the thermodynamic uncertainty relation, and it can be substantially more precise otherwise. The hyperaccurate current can be used to improve estimates of entropy production from experimental data.
Generalized empirical currents represent a vast class of thermodynamic observables of mesoscopic systems. Their fluctuations satisfy the thermodynamic uncertainty relations (TURs), as they can be bounded by the average entropy production. Here, we derive a general closed expression for the hyperaccurate current in discrete-state Markovian systems, i.e., the one with the least fluctuations, for both discrete- and continuous-time evolution. We show that its associated hyperaccurate bound is generally much tighter than the one given by the TURs, and might be crucial to providing a reliable estima…
Thermodynamic currents can fluctuate significantly at the nanoscale. But some currents fluctuate less than others. Hyperaccurate currents are those which fluctuate the least, in the sense that they maximize the signal-to-noise ratio (precision). In this letter we analytically determine the hyperaccurate current in the case of a quantum thermoelectric, modeled by the Landauer-Büttiker formalism.
In recent years, the unified theory of information and thermodynamics has been intensively discussed in the context of stochastic thermodynamics. The unified theory reveals that information theory would be useful to understand non-stationary dynamics of systems far from equilibrium. In this letter, we have found a new link between stochastic thermodynamics and information theory well known as information geometry. By applying this link, an information geometric inequality can be interpreted as a thermodynamic uncertainty relationship between speed and thermodynamic cost. We have numerically ap…
We present the stochastic thermodynamic analysis of a time-periodic single particle pump, including explicit results for flux, thermodynamic force, entropy production, work, heat and efficiency. These results are valid far from equilibrium. The deviations from the linear (Onsager) regime are discussed.
We develop a consistent stochastic thermodynamics for environments composed of thermodynamic reservoirs in an external conservative force field, that is environments described by the Generalized or Gibbs canonical ensemble. We demonstrate that small systems weakly coupled to such reservoirs exchange both heat and work by verifying a local detailed balance relation for the induced stochastic dynamics. Based on this analysis, we help to rationalize the observation that nonthermal reservoirs can increase the efficiency of thermodynamic heat engines.
Assuming time-scale separation, a simple and unified theory of thermodynamics and stochastic thermodynamics is constructed for small classical systems strongly interacting with its environment in a controllable fashion. The total Hamiltonian is decomposed into a bath part and a system part, the latter being the Hamiltonian of mean force. Both the conditional equilibrium of bath and the reduced equilibrium of the system are described by canonical ensemble theories with respect to their own Hamiltonians. The bath free energy is independent of the system variables and the control parameter. Furth…
The appeal of thermodynamics to problems outside physics is undeniable, as is the growing recognition of its apparent universality, yet in the absence of a rigorous formalism divorced from the peculiarities of molecular systems all attempts to generalize thermodynamics remain qualitative and heuristic at best. In this paper we formulate a probabilistic theory of thermodynamics and and set the basis for its application to generic stochastic populations.
We develop the stochastic approach to thermodynamics based on the stochastic dynamics, which can be discrete (master equation) continuous (Fokker-Planck equation), and on two assumptions concerning entropy. The first is the definition of entropy itself and the second, the definition of entropy production rate which is nonnegative and vanishes in thermodynamic equilibrium. Based on these assumptions we study interacting systems with many degrees of freedom in equilibrium or out of thermodynamic equilibrium, and how the macroscopic laws are derived from the stochastic dynamics. These studies inc…
The study of stochastic thermodynamic machines is one of the main topics in nonequilibrium thermodynamics. In this study, within the framework of Fokker-Planck equation, and using the method of characteristics of partial differential equation as well as the variational method, performance of stochastic thermodynamic machines is optimized according to the external potential, with the irreversible work $W_{irr}$, or the total entropy production $ΔS_{\rm tot}$ equivalently, reaching its lower bound. Properties of the optimal thermodynamic machines are discussed, with explicit expressions of upper…
In stochastic thermodynamics standard concepts from macroscopic thermodynamics, such as heat, work, and entropy production, are generalized to small fluctuating systems by defining them on a trajectory-wise level. In Langevin systems with continuous state-space such definitions involve stochastic integrals along system trajectories, whose specific values depend on the discretization rule used to evaluate them (i.e. the "interpretation" of the noise terms in the integral). Via a systematic mathematical investigation of this apparent dilemma, we corroborate the widely used standard interpretatio…
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…
Stochastic thermodynamics is formulated for variables that are odd under time reversal. The invariance under spatial rotation of the collision rates due to the isotropy of the heat bath is shown to be a crucial ingredient. An alternative detailed fluctuation theorem is derived, expressed solely in terms of forward statistics. It is illustrated for a linear kinetic equation with kangaroo rates.
We prove the Jarzynski relation for general stochastic processes including non-Markovian systems with memory. The only requirement for our proof is the existence of a stationary state, therefore excluding non-ergodic systems. We then show how the concepts of stochastic thermodynamics can be used to prove further exact non-equilibrium relations like the Crooks relation and the fluctuation theorem on entropy production for non-Markovian dynamics.
At very small scales, thermodynamic energy exchanges like work and heat become comparable to thermal energy of the system, which leads to unusual phenomena like the transient violations of Second Law. We explore the generic characters of such systems using the framework of Stochastic Thermodynamics and provide a preliminary overview of the basic concepts. Here we have attempted to put into simple terms some actively pursued topics like the arrow of time, effect of information gain on Second Law, explanation of origin of life using Crooks theorem and the thermodynamic uncertainty relations.
From a viewpoint of stochastic thermodynamics, we derive equations that describe the collective dynamics near the order-disorder transition in the globally coupled XY model and near the synchronization-desynchronization transition in the Kuramoto model. A new way of thinking is to interpret the deterministic time evolution of a macroscopic variable as an external operation to a thermodynamic system. We then find that the irreversible work determines the equation for the collective dynamics. When analyzing the Kuramoto model, we employ a generalized concept of irreversible work which originates…
Building on our earlier work (Proesmans et. al., Phys.~Rev.~X \textbf{6} (2016), 041010), we introduce the underdamped Brownian duet as a prototype model of a dissipative system or of a work-to-work engine. Several recent advances from the theory of stochastic thermodynamics are illustrated with explicit analytic calculations and corresponding Langevin simulations. In particular, we discuss Onsager-Casimir symmetry, the trade-off relations between power, efficiency and dissipation, and stochastic efficiency.
Recently, there has been a considerable progress on the issue of the thermodynamic second law, which is known as the law of entropy increase or irreversibility. In particular, a novel symmetry known as the Gallavotti-Cohen symmetry is found in nonequilibrium (NEQ) fluctuations, which leads to so-called fluctuation theorems. The thermodynamic second law is a simple corollary of fluctuation theorems, from which one can predict quantitatively how often NEQ processes violate the law of entropy increase. Violations disappear in the thermodynamic limit, but can be observed reasonably well in small s…
We propose a way to incorporate the effect of a specific class of feedback processes into stochastic thermodynamics. These "Maxwell demon" feedbacks do not affect the system energetics but only the energy barriers between the system states (in a way which depends on the system states). They are thus of a purely informational nature. We show that the resulting formalism can be applied to study the thermodynamic effect of a feedback process acting on electron transfers through a junction.
We present a general formalism for the construction of thermodynamically consistent stochastic models of non-linear electronic circuits. The devices constituting the circuit can have arbitrary I-V curves and may include tunnel junctions, diodes, and MOS transistors in subthreshold operation, among others. We provide a full analysis of the stochastic non-equilibrium thermodynamics of these models, identifying the relevant thermodynamic potentials, characterizing the different contributions to the irreversible entropy production, and obtaining different fluctuation theorems. Our work provides a …
We study a relationship between optimal transport theory and stochastic thermodynamics for the Fokker-Planck equation. We show that the lower bound on the entropy production is the action measured by the path length of the $L^2$-Wasserstein distance. Because the $L^2$-Wasserstein distance is a geometric measure of optimal transport theory, our result implies a geometric interpretation of the entropy production. Based on this interpretation, we obtain a thermodynamic trade-off relation between transition time and the entropy production. This thermodynamic trade-off relation is regarded as a the…
We apply the stochastic thermodynamics formalism to describe the dynamics of systems of complex Langevin and Fokker-Planck equations. We provide in particular a simple and general recipe to calculate thermodynamical currents, dissipated and propagating heat for networks of nonlinear oscillators. By using the Hodge decomposition of thermodynamical forces and fluxes, we derive a formula for entropy production that generalises the notion of non-potential forces and makes trans- parent the breaking of detailed balance and of time reversal symmetry for states arbitrarily far from equilibrium. Our f…
A general formulation of stochastic thermodynamics is presented for open systems exchanging energy and particles with multiple reservoirs. By introducing a partition in terms of "macrostates" (e.g. sets of "microstates"), the consequence on the thermodynamic description of the system is studied in detail. When microstates within macrostates rapidly thermalize, the entire structure of the microscopic theory is recovered at the macrostate level. This is not the case when these microstates remain out of equilibrium leading to additional contributions to the entropy balance. Some of our results ar…
The stochastic thermodynamics provides a framework for the description of systems that are out of thermodynamic equilibrium. It is based on the assumption that the elementary constituents are acted by random forces that generate a stochastic dynamics, which is here represented by a Fokker-Planck-Kramers equation. We emphasize the role of the irreversible probability current, the vanishing of which characterizes the thermodynamic equilibrium and yields a special relation between fluctuation and dissipation. The connection to thermodynamics is obtained by the definition of the energy function an…
Stochastic thermodynamics extends the notions and relations of classical thermodynamics to small systems that experience strong fluctuations. The definitions of work and heat and the microscopically reversible condition are two key concepts in the current framework of stochastic thermodynamics. Herein, we apply stochastic thermodynamics to small systems with odd controlling parameters and find that the definition of heat and the microscopically reversible condition are incompatible. Such a contradiction also leads to a revision to the fluctuation theorems and nonequilibrium work relations. By …
Strong-coupling statistical thermodynamics is formulated as Hamiltonian dynamics of an observed system interacting with another unobserved system (a bath). It is shown that the entropy production functional of stochastic thermodynamics, defined as the log-ratio of forward and backward system path probabilities, is in one-to-one relation with the log-ratios of joint initial conditions of the system and the bath. A version of strong-coupling statistical thermodynamics where the system-bath interaction vanishes at the beginning and the end of a process is, as is also weak-coupling stochastic ther…
We study the thermodynamic properties of a microscopic model of coupled oscillators that exhibits a dynamical phase transition from a desynchronized to a synchronized phase. We consider two different configurations for the thermodynamic forces applied on the oscillators, one resembling the macroscopic power grids, and one resembling autonomous molecular motors. We characterize the input and the output power as well as the efficiency at maximum power, providing analytic expressions for such quantities near the critical coupling strength. We discuss the role of the quenched disorder in the therm…
The total entropy production of stochastic systems can be divided into three quantities. The first corresponds to the excess heat, whilst the second two comprise the house-keeping heat. We denote these two components the transient and generalised house-keeping heat and we obtain an integral fluctuation theorem for the latter, valid for all Markovian stochastic dynamics. A previously reported formalism is obtained when the stationary probability distribution is symmetric for all variables that are odd under time reversal which restricts consideration of directional variables such as velocity.
We extend stochastic thermodynamics by relaxing the two assumptions that the Markovian dynamics must be linear and that the equilibrium distribution must be a Boltzmann distribution. We show that if we require the second law to hold when those assumptions are relaxed, then it cannot be formulated in terms of Shannon entropy. However, thermodynamic consistency is salvaged if we reformulate the second law in terms of generalized entropy; our first result is an equation relating the precise form of the non-linear master equation to the precise associated generalized entropy which results in therm…
We experimentally study a piezoelectric energy harvester driven by broadband random vibrations. We show that a linear model, consisting of an underdamped Langevin equation for the dynamics of the tip mass, electromechanically coupled with a capacitor and a load resistor, can accurately describe the experimental data. In particular, the theoretical model allows us to define fluctuating currents and to study the stochastic thermodynamics of the system, with focus on the distribution of the extracted work over different time intervals. Our analytical and numerical analysis of the linear model is …
Stochastic dynamics with random resetting leads to a non-equilibrium steady state. Here, we consider the thermodynamics of resetting by deriving the first and second law for reset processes far from equilibrium. We identify the contributions to the entropy production of the system which arise due to resetting and show that they correspond to the rate with which information is either erased or created. Using Landauer's principle, we derive a bound on the amount of work that is required to maintain a resetting process. We discuss different regimes of resetting, including a Maxwell's demon scenar…
The piston system (particles in a box) is the simplest and 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 stochastic thermodynamics 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 veri…
We provide a stochastic thermodynamic description across scales for $N$ identical units with all-to-all interactions that are driven away from equilibrium by different reservoirs and external forces. We start at the microscopic level with Poisson rates describing transitions between many-body states. We then identify an exact coarse graining leading to a mesoscopic description in terms of Poisson transitions between system occupations. We proceed studying macroscopic fluctuations using the Martin-Siggia-Rose formalism and large deviation theory. In the macroscopic limit ($N \to \infty$), we de…
So far, feedback-driven systems have been discussed using (i) measurement and control, (ii) a tape interacting with a system or (iii) by identifying an implicit Maxwell demon in steady state transport. We derive the corresponding second laws from one master fluctuation theorem and discuss their relationship. In particular, we show that both the entropy production involving mutual information between system and controller and the one involving a Shannon entropy difference of an information reservoir like a tape carry an extra term different from the usual current times affinity. We thus general…
We generalize stochastic thermodynamics to include information reservoirs. Such information reservoirs, which can be modeled as a sequence of bits, modify the second law. For example, work extraction from a system in contact with a single heat bath becomes possible if the system also interacts with an information reservoir. We obtain an inequality, and the corresponding fluctuation theorem, generalizing the standard entropy production of stochastic thermodynamics. From this inequality we can derive an information processing entropy production, which gives the second law in the presence of info…
Phenomenological nonequilibrium thermodynamics describes how fluxes of conserved quantities such as matter, energy and charge flow from outer reservoirs across a system, and how they irreversibly degrade from one form to another. Stochastic thermodynamics is formulated in terms of probability fluxes circulating in the system's configuration space. The consistency of the two frameworks is granted by the condition of local detailed balance, which specifies the amount of physical quantities exchanged with the reservoirs during single transitions between configurations. We demonstrate that the top…
We explore the idea that non-equilibrium steady states breaking detailed balance are obtained by deforming trajectories (lines in space-time) that have been sampled in a reference system with stochastic dynamics obeying detailed balance, and we ask for the work required to perform this task. These geometric deformations are not arbitrary but arise through interactions with the environment, either the manipulation of conserved quantities by an external agent, or by their exchange with a work reservoir. This view allows to consistently model the breaking of detailed balance and the accompanying …
The Boltzmann kinetic equation is obtained from an integro-differential master equation that describes a stochastic dynamics in phase space of an isolated thermodynamic system. The stochastic evolution yields a generation of entropy, leading to an increase of Gibbs entropy, in contrast to a Hamiltonian dynamics, described by the Liouville equation, for which the entropy is constant in time. By considering transition rates corresponding to collisions of two particles, the Boltzmann equation is attained. When the angle of the scattering produced by collisions is small, the master equation is sho…
We establish the fluctuation theorem in the presence of information exchange between a nonequilibrium system and other degrees of freedom such as an observer and a feedback controller, where the amount of information exchange is added to the entropy production. The resulting generalized second law sets the fundamental limit of energy dissipation and energy cost during the information exchange. Our results apply not only to feedback-controlled processes but also to a much broader class of information exchanges, and provides a unified framework of nonequilibrium thermodynamics of measurement and…
The formalism of state estimation and hidden Markov models (HMMs) can simplify and clarify the discussion of stochastic thermodynamics in the presence of feedback and measurement errors. After reviewing the basic formalism, we use it to shed light on a recent discussion of phase transitions in the optimized response of an information engine, for which measurement noise serves as a control parameter. The HMM formalism also shows that the value of additional information shows a maximum at intermediate signal-to-noise ratios. Finally, we discuss how systems open to information flow can apparently…
A set of core features is set forth as the essence of a thermodynamic description, which derive from large-deviation properties in systems with hierarchies of timescales, but which are \emph{not} dependent upon conservation laws or microscopic reversibility in the substrate hosting the process. The most fundamental elements are the concept of a macrostate in relation to the large-deviation entropy, and the decomposition of contributions to irreversibility among interacting subsystems, which is the origin of the dependence on a concept of heat in both classical and stochastic thermodynamics. A …
Nonequilibrium thermodynamics (NET) investigates processes in systems out of global equilibrium. On a mesoscopic level, it provides a statistical dynamic description of various complex phenomena such as chemical reactions, ion transport, diffusion, thermochemical, thermomechanical and mechanochemical fluxes. In the present review, we introduce a mesoscopic stochastic formulation of NET by analyzing entropy production in several simple examples. The fundamental role of nonequilibrium steady-state cycle kinetics is emphasized. The statistical mechanics of Onsager's reciprocal relations in this c…
Physical scenarios that require a relativistic treatment are ubiquitous in nature, ranging from cosmological objects to charge carriers in Dirac materials. Interestingly all of these situations have in common that the systems typically operate very far from thermal equilibrium. Therefore, if and how the framework of stochastic thermodynamics applies at relativistic energies is a salient question. In the present work we generalize the notions of stochastic heat and work for the relativistic Langevin equation and derive the fluctuation theorems without and with feedback. For processes with feedb…
Thermodynamic Length in Stochastic Thermodynamics of Far-From-Equilibrium Systems: Unification of Fluctuation Relation and Thermodynamic Uncertainty Relation
The theoretical understanding of active matter, which is driven out of equilibrium by directed motion, is still fragmental and model oriented. Stochastic thermodynamics, on the other hand, is a comprehensive theoretical framework for driven systems that allows to define fluctuating work and heat. We apply these definitions to active matter, assuming that dissipation can be modelled by effective non-conservative forces. We show that, through the work, conjugate extensive and intensive observables can be defined even in non-equilibrium steady states lacking a free energy. As an illustration, we …
Stochastic thermodynamics provides an important framework to explore small physical systems where thermal fluctuations are inevitable. In the studies of stochastic thermodynamics, some thermodynamic quantities, such as the trajectory work, associated with the complete Langevin equation (the Kramers equation) are often assumed to converge to those associated with the overdamped Langevin equation (the Smoluchowski equation) in the overdamped limit under the overdamped approximation. Nevertheless, a rigorous mathematical proof of the convergence of the work distributions to our knowledge has not …
We verify the finite time fluctuation theorem for a linear Ising chain at its ends in contact with heat reservoirs. Analytic results are derived for a chain consisting of only two spins. The system can be mapped onto a model for particle transport, namely the symmetric exclusion process, in contact with thermal and particle reservoirs. We modify the symmetric exclusion process to represent a thermal engine and reproduce universal features of the efficiency at maximum power.
It is nearly 150 years since Maxwell challenged the validity of the second law of thermodynamics by imagining a tiny creature who could sort the molecules of a gas in such a way that would decrease entropy without exerting any work. The demon has been discussed largely using thought experiments, but it has recently become possible to exert control over nanoscale systems, just as Maxwell imagined, and the status of the second law has become a more practical matter, raising the issue of how measurements manage our ignorance in a way that can be exploited. The framework of stochastic thermodynami…
We describe a single-level quantum dot in contact with two leads as a nanoscale finite-time thermodynamic machine. The dot is driven by an external stochastic force that switches its energy between two values. In the isothermal regime, it can operate as a rechargeable battery by generating an electric current against the applied bias in response to the stochastic driving, and re-delivering work in the reverse cycle. This behavior is reminiscent of the Parrondo paradox. If there is a thermal gradient the device can function as a work-generating thermal engine, or as a refrigerator that extracts…
