We study the problem of optimizing released heat or dissipated work in stochastic thermodynamics. In the overdamped limit these functionals have singular solutions, previously interpreted as protocol jumps. We show that a regularization, penalizing a properly defined acceleration, changes the jumps into boundary layers of finite width. We show that in the limit of vanishing boundary layer width no heat is dissipated in the boundary layer, while work can be done. We further give a new interpretation of the fact that the optimal protocols in the overdamped limit are given by optimal deterministic transport (Burgers equation).
Fluctuation relations are identities, holding in non-equilibrium systems, that have attracted a lot of interest in the last 20 years. This is a series of 4 lectures discussing various aspects of such relations for stochastic equations modeling non-equilibrium processes.
We discuss the relevance of geometric concepts in the theory of stochastic differential equations for applications to the theory of non-equilibrium thermodynamics of small systems. In particular, we show how the Eells-Elworthy-Malliavin covariant construction of the Wiener process on a Riemann manifold provides a physically transparent formulation of optimal control problems of finite-time thermodynamic transitions. Based on this formulation, we turn to an evaluative discussion of recent results on optimal thermodynamic control and their interpretation.
Stochastic Thermodynamics (ST) extends the notions of classical thermodynamics to trajectories taken from a nonequilibrium ensemble. This extension yields a simple approach to fluctuation relations in small systems. Multiple time- and length scales play an important role for measurements but also for the foundations of nonequilibrium statistical mechanics. Here, under the assumptions of local equilibrium we derive the trajectory functionals of ST in the context of reversible deterministic thermostats. Further, the connection to previous work is made.
We investigate the thermodynamic limit of the su(n)-invariant spin chain models with unparallel boundary fields. It is found that the contribution of the inhomogeneous term in the associated T-Q relation to the ground state energy does vanish in the thermodynamic limit. This fact allows us to calculate the boundary energy of the system. Taking the su(2) (or the XXX) spin chain and the su(3) spin chain as concrete examples, we have studied the corresponding boundary energies of the models. The method used in this paper can be generalized to study the thermodynamic properties and boundary energy…
The thermodynamic uncertainty relation is a prominent result in stochastic thermodynamics that provides a bound on the fluctuations of any thermodynamic flux, also known as current, in terms of the average rate of entropy production. Such fluctuations are quantified by the second moment of the probability distribution of the current. The role of higher order standardized moments such as skewness and kurtosis remains largely unexplored. We analyze the skewness and kurtosis associated with the first passage time of thermodynamic currents within the framework of stochastic thermodynamics. We deve…
The applicability of stochastic differential equations to thermodynamics is considered and a new form, different from the classical Ito and Stratonovich forms, is introduced. It is shown that the new presentation is more appropriate for the description of thermodynamic fluctuations. The range of validity of the Boltzmann-Einstein principle is also discussed and a generalized alternative is proposed. Both expressions coincide in the small fluctuation limit, providing a normal distribution density.
We introduce a model of the quantum Brownian motion coupled to a classical neat bath by using the operator differential proposed in the quantum analysis. We then define the heat operator by adapting the idea of the stochastic energetics. The introduced operator satisfies the relations which are analogous to the first and second laws of thermodynamics.
Stochastic Thermodynamics on Time-Evolving Curved Spaces
The University of Tokyo（东京大学）；RIKEN Center for Advanced Intelligence Project (AIP)（理化学研究所先进智能项目中心）；Inamori Research Institute for Science (InaRIS)（稻盛科学技术研究所）
In the present paper, we study the power output and efficiency of overdamped stochastic thermodynamic engines that are in contact with a heat bath having a temperature that varies periodically with time. This is in contrast to most of the existing literature that considers the Carnot paradigm of alternating contact with heat baths having different fixed temperatures, hot and cold. Specifically, we consider a periodic and bounded but otherwise arbitrary temperature profile and derive explicit bounds on the power and efficiency achievable by a suitable controlling potential that couples the ther…
Seifert derived an exact fluctuation relation for diffusion processes using the concept of "stochastic system entropy". In this note we extend his formalism to entropic transport. We introduce the notion of relative stochastic entropy, or "relative surprisal", and use it to generalize Seifert's system/medium decomposition of the total entropy. This result allows to apply the concepts of stochastic thermodynamics to diffusion processes in confined geometries, such as ion channels, cellular pores or nanoporous materials. It can be seen as the equivalent for diffusion processes of Esposito and Sc…
Thermodynamics of small systems has become an important field of statistical physics. They are driven out of equilibrium by a control, and the question is naturally posed how such a control can be optimized. We show that optimization problems in small system thermodynamics are solved by (deterministic) optimal transport, for which very efficient numerical methods have been developed, and of which there are applications in Cosmology, fluid mechanics, logistics, and many other fields. We show, in particular, that minimizing expected heat released or work done during a non-equilibrium transition …
The meaning of thermodynamic descriptions is found in large-deviations scaling of the fluctuations probabilities. The primary large-deviations rate function is the entropy, which is the basis for both fluctuation theorems and for characterizing the thermodynamic interactions of systems. Freidlin-Wentzell theory provides a general formulation of large-deviations scaling for non-equilibrium stochastic processes, through a representation in terms of a Hamiltonian dynamical system. A number of related methods now exist to construct the Freidlin-Wentzell Hamiltonian for many kinds of stochastic pro…
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 are describing relations between Schrödinger's variational problem and Onsager's approach to nonequilibrium statistical mechanics. Although the second work on reciprocal relations and detailed balance has been published in the same year (1931) as the first one, the impact of Schrödinger's idea has not yet been considered in the classical context of Onsager.
Topological protection has emerged as an organizing principle for understanding and engineering robust collective behavior in electronic and material systems. Recent work suggests that topology may also play a role in organizing stochastic processes relevant to biology and self-assembly. Here, we show that topological protection in chemical networks can be understood entirely in terms of nonequilibrium thermodynamics. We illustrate these ideas using simple examples inspired by the literature.
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.
The stochastic thermodynamics of a dilute, well-stirred mixture of chemically-reacting species is built on the stochastic trajectories of reaction events obtained from the Chemical Master Equation. However, when the molecular populations are large, the discrete Chemical Master Equation can be approximated with a continuous diffusion process, like the Chemical Langevin Equation or Low Noise Approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the Chemical Master Equation. We find that a stochastic-thermodynamic descr…
A stochastic deformation of a thermodynamic symplectic structure is studied. The stochastic deformation procedure is analogous to the deformation of an algebra of observables like deformation quantization, but for an imaginary deformation parameter (the Planck constant). Gauge symmetries of thermodynamics and corresponding stochastic mechanics, which describes fluctuations of a thermodynamic system, are revealed and gauge fields are introduced. A physical interpretation to the gauge transformations and gauge fields is given. An application of the formalism to a description of systems with dist…
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…
Stochastic thermodynamics provides a framework for describing small systems like colloids or biomolecules driven out of equilibrium but still in contact with a heat bath. Both, a first-law like energy balance involving exchanged heat and entropy production entering refinements of the second law can consistently be defined along single stochastic trajectories. Various exact relations involving the distribution of such quantities like integral and detailed fluctuation theorems for total entropy production and the Jarzynski relation follow from such an approach based on Langevin dynamics. Analogu…
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…
The consistency across scales of a recently developed mathematical thermodynamic structure, between a continuous stochastic nonlinear dynamical system (diffusion process with Langevin or Fokker-Planck equations) and its emergent discrete, inter-attractoral Markov jump process, is investigated. We analyze how the system's thermodynamic state functions, e.g. free energy $F$, entropy $S$, entropy production $e_p$, and free energy dissipation $\dot{F}$, etc., are related when the continuous system is describe with a coarse-grained discrete variable. We show that the thermodynamics derived from the…
We consider stochastic thermodynamics as a theory of statistical inference for experimentally observed fluctuating time-series. To that end, we introduce a general framework for quantifying the knowledge about the dynamical state of the system on two scales: a fine-grained or microscopic, deterministic and a coarse-grained or mesoscopic, stochastic level of description. For a generic model dynamics, we show how the mathematical expressions for fluctuating entropy changes used in Markovian stochastic thermodynamics emerge naturally. Our ideas are conceptional approaches towards (i) connecting e…
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 propose a piecewise deterministic Markovian jump process in Hilbert space such that the covariance matrix of this stochastic process solves the thermodynamic quantum master equation. The proposed stochastic process is particularly simple because the normalization of the vectors in Hilbert space is preserved only on average. As a consequence of the nonlinearity of the thermodynamic master equation, the construction of stochastic trajectories involves the density matrix as a running ensemble average. We identify a principle of detailed balance and a fluctuation-dissipation relation for our Ma…
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.
The thermodynamic framework of repeated interactions is generalized to an arbitrary open quantum system in contact with a heat bath. Based on these findings the theory is then extended to arbitrary measurements performed on the system. This constitutes a direct experimentally testable framework in strong coupling quantum thermodynamics. By construction, it provides many quantum stochastic processes and quantum causal models with a consistent thermodynamic interpretation. The setting can be further used, for instance, to rigorously investigate the interplay between non-Markovianity and nonequil…
Virtually every organism gathers information about its noisy environment and builds models from that data, mostly using neural networks. Here, we use stochastic thermodynamics to analyse the learning of a classification rule by a neural network. We show that the information acquired by the network is bounded by the thermodynamic cost of learning and introduce a learning efficiency $η\le1$. We discuss the conditions for optimal learning and analyse Hebbian learning in the thermodynamic limit.
This article sets up a formalism to describe stochastic thermodynamics for driven out-of-equilibrium open quantum systems. A stochastic Schrödinger equation allows to construct quantum trajectories describing the dynamics of the system state vector in presence of an eventually monitored environment. Thermodynamic quantities are defined at the single quantum trajectory level, independently of any energy measurement, at any time of the protocol. We thereby identify coherent contributions, without classical counterparts, leading to quantum fluctuations of thermodynamic quantities. This formalism …
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…
We propose a mathematical formulation of the zeroth law of thermodynamics and develop a stochastic dynamical theory, with a consistent irreversible thermodynamics, for systems possessing sustained conservative stationary current in phase space while in equilibrium with a heat bath. The theory generalizes underdamped mechanical equilibrium: $dx=gdt+\{-D\nablaϕdt+\sqrt{2D}dB(t)\}$, with $\nabla\cdot g=0$ and $\{\cdots\}$ respectively representing phase-volume preserving dynamics and stochastic damping. The zeroth law implies stationary distribution $u^{ss}(x)=e^{-ϕ(x)}$. We find an orthogonality…
