In this work, we investigate the relation between the concept of ``information rate'', an information geometric method for measuring the speed of the time evolution of the statistical states of a stochastic process, and stochastic thermodynamics quantities like entropy rate and entropy production. Then, we propose the application of entropy rate and entropy production to different practical applications such as abrupt event detection, correlation analysis, and control engineering. Specifically, by utilising the Fokker-Planck equation of multi-variable linear stochastic processes described by Langevin equations, we calculate the exact value for information rate, entropy rate, and entropy production and derive various inequalities among them. Inspired by classical correlation coefficients and control techniques, we create entropic-informed correlation coefficients as abrupt event detection methods and information geometric cost functions as optimal thermodynamic control policies, respectively. The methods are analysed via the numerical simulations of common prototypical systems.
In statistical physics, entropy is generally logarithm of probability. Therefore, if dynamics is decomposed by log, entropy production should be decomposed properly. In the present work, log-decomposition of dynamics is introduced. By which time evolution operator is logarithmically decomposed into a symmetric operator and an asymmetric factor. Path probability and path entropy production are also systematically and intuitively decomposed into symmetric and asymmetric parts. From symmetric operator, non-adiabatic entropy production is derived, whereas adiabatic entropy production is from asymm…
Based on the Fokker-Planck and the entropy balance equations we have studied the relaxation of a dissipative dynamical system driven by external Ornstein-Uhlenbeck noise processes in absence and presence of nonequilibrium constraint in terms of the thermodynamically inspired quantities like entropy flux and entropy production. The interplay of nonequilibrium constraint, dissipation and noise reveals some interesting extremal nature in the time dependence of entropy flux and entropy production.
*First-principles derivation of the entropy production in erectric static conduction. *The second-order (symmetric) density matrix contributes to the entropy production. *New schemes of steady states formulated using a relaxation-type von Neumann equation. *Stationary temperature is introduced to characterize steady states. *The mechanism of the entropy production in steady states is also clarified.
We propose an alternative method to compute the entropy production of a classical underdamped nonequilibrium system in a continuous phase space. This approach has the advantage that it is not necessary to distinguish between even and odd-parity variables. We show that the method leads to the same local entropy production as in previous studies while the differential entropy production along a stochastic trajectory turns out to be different. This demonstrates that the differential entropy production in continuous phase space systems is not uniquely defined.
The second law of thermodynamics, which asserts the non-negativity of the average total entropy production of a combined system and its environment, is a direct consequence of applying Jensen's inequality to a fluctuation relation. It is also possible, through this inequality, to determine an upper bound of the average total entropy production based on the entropies along the most extreme stochastic trajectories. In this work, we construct an upper bound inequality of the average of a convex function over a domain whose average is known. When applied to the various fluctuation relations, the u…
In the general process of eliminating dynamic variables in Markovian models, there exists a difference in the irreversible entropy production between the original and reduced dynamics. We call this difference the hidden entropy production, since it is an invisible quantity when only the reduced system's view is provided. We show that this hidden entropy production obeys a new integral fluctuation theorem for the generic case where all variables are time-reversal invariant, therefore supporting the intuition that entropy production should decrease by coarse graining. It is found, however, that …
We consider relaxation of an isolated system to the equilibrium using detailed balance condition and Onsager's fluctuation approximation. There is a small deviation from the equilibrium in two parameters. For this system, explicit expressions both for the dependence of trajectory entropy on random thermodynamic fluxes and for the dependence of entropy production on the most probable thermodynamic fluxes are obtained. Onsager's linear relations are obtained for the considered model using two methods (maximization of trajectory entropy and Ziegler's maximization of entropy production). Two exist…
From the previously obtained solutions of the Fokker - Planck equation for Rayleigh gas (small impurity of heavy particles in a thermostat of light particles) with sources and without them, the entropy production was calculated. In a system without source (isolated system) shown that it holds theorem of Prigogine, and in a system with sources (open system) implementation of the principle of Ziegler (MEPP) depends on the relaxation direction. In an open system entropy production is compensate by a negative production of entropy, i.e. by a negentropy production. The algebraic sum of entropy and …
The expressions for entropy production, free energy, and entropy extraction rates are derived for a Brownian particle that walks in an underdamped medium. Our analysis indicates that as long as the system is driven out of equilibrium, it constantly produces entropy at the same time it extracts entropy out of the system. At steady state, the rate of entropy production ${\dot e}_{p}$ balances the rate of entropy extraction ${\dot h}_{d}$. At equilibrium both entropy production and extraction rates become zero. The entropy production and entropy extraction rates are also sensitive to time. As tim…
In the present work, we study the entropy productions in a system controlled by double control parameters. By introducing a thermal fluctuation part, we solve the problem that the second law of the thermodynamics seems to be violated by the thermal fluctuation near equilibrium in the microscopic levels. Then we define the negative and the compensating entropy productions in the macroscopic levels.
The nonequilibrium thermodynamics feature of a Brownian motor operating between two different heat baths is explored as a function of time $t$. Using the Gibbs entropy and Schnakenberg microscopic stochastic approach, we find exact closed form expressions for the free energy, the rate of entropy production and the rate of entropy flow from the system to the outside. We show that when the system is out of equilibrium, it constantly produces entropy and at the same time extract entropy out of the system. Its entropy production and extraction rates decrease in time and saturate to constant value.…
We show a relationship between the entropy production in stochastic thermodynamics and the stochastic interaction in the information integrated theory. To clarify this relationship, we newly introduce an information geometric interpretation of the entropy production for a total system and the partial entropy productions for subsystems. We show that the violation of the additivity of the entropy productions is related to the stochastic interaction. This framework is a thermodynamic foundation of the integrated information theory. We also show that our information geometric formalism leads to a …
Using fluorescence spectroscopy we directly measure entropy production of a single two-level system realized experimentally as an optically driven defect center in diamond. We exploit a recent suggestion to define entropy on the level of a single stochastic trajectory (Seifert, Phys. Rev. Lett. {\bf 95}, 040602 (2005)). Entropy production can then be split into one of the system itself and one of the surrounding medium. We demonstrate that the total entropy production obeys various exact relations for finite time trajectories.
A rigorous derivation of nonequilibrium entropy production via the path-integral formalism is presented. Entropy production is defined as the entropy change piled in a heat reservoir as a result of a nonequilibrium thermodynamic process. It is a central quantity by which various forms of the fluctuation theorem are obtained. The two kinds of the stochastic dynamics are investigated: the Langevin dynamics for an even-parity state and the Brownian motion of a single particle. Mathematical ambiguities in deriving the functional form of the entropy production, which depends on path in state space,…
We consider conservative quantum evolutions possibly interrupted by macroscopic measurements. When started in a nonequilibrium state, the resulting path-space measure is not time-reversal invariant and the weight of time-reversal breaking equals the exponential of the entropy production. The mean entropy production can then be expressed via a relative entropy on the level of histories. This gives a partial extension of the result for classical systems, that the entropy production is given by the source term of time-reversal breaking in the path-space measure.
We derive general expressions for the free energy, entropy production and entropy extraction rates for a Brownian particle that walks in a viscous medium where the dynamics of its motion is governed by the Langevin equation. It is shown that when the system is out of equilibrium, it constantly produces entropy and at the same time extracts entropy out of the system. Its entropy production and extraction rates decrease in time and saturate to a constant value. In long time limit, the rate of entropy production balances the rate of entropy extraction and at equilibrium both entropy production an…
Linear irreversible thermodynamics asserts that the instantaneous local spontaneous entropy production is always nonnegative. However for a viscoelastic fluid this is not always the case. Given the fundamental status of the Second Law, this presents a problem. We provide a new derivation of the Second Law, from first principles, which is valid for the appropriately time averaged entropy production allowing the instantaneous entropy production to be negative for short intervals of time. We show that time averages (rather than instantaneous values) of the entropy production are nonnegative. We i…
The total entropy production fluctuations are studied in some exactly solvable models. For these systems, the detailed fluctuation theorem holds even in the transient state, provided initially the system is prepared in thermal equilibrium. The nature of entropy production during the relaxation of a system to equilibrium is analyzed. The averaged entropy production over a finite time interval gives a better bound for the average work performed on the system than that obtained from the well known Jarzynski equality. Moreover, the average entropy production as a quantifier for information theoret…
Entropy production is one of the most important characteristics of non-equilibrium steady states. We study here the steady-state entropy production, both at short times as well as in the long-time limit, of two important classes of non-equilibrium systems: transport systems and reaction-diffusion systems. The usefulness of the mean entropy production rate and of the large deviation function of the entropy production for characterizing non-equilibrium steady states of interacting many-body systems is discussed. We show that the large deviation function displays a kink-like feature at zero entro…
Systems driven away from thermal equilibrium constantly deliver entropy to their environment. Determining this entropy production requires detailed information about the system's internal states and dynamics. However, in most practical scenarios, only a part of a complex experimental system is accessible to an external observer. In order to address this challenge, two notions of partial entropy production have been introduced in the literature as a way to assign an entropy production to an observed subsystem: one due to Shiraishi and Sagawa [Phys. Rev. E 91, 012130 (2015)] and another due to P…
The entropy production rate of nonequilibrium systems is studied via the Fokker-Planck equation. This approach, based on the entropy production rate equation given by Schnakenberg from a master equation, requires information of the transition rate of the system under study. We obtain the transition rate from the conditional probability extracted from the Fokker-Planck equation and then derive a new and more operable expression for the entropy production rate. Examples are presented as applications of our approach.
It is shown that Onsager's principle of the least dissipation of energy is equivalent to the maximum entropy production principle. It is known that solutions of the linearized Boltzmann equation make extrema of entropy production. It is argued, in the case of stationary processes, that this extremum is a maximum rather than a minimum.
The probability distribution of the entropy production for the effusion of a relativistic ideal gas is calculated explicitly. This result is then extended to include particle and anti-particle pair production and annihilation. In both cases, the fluctuation theorem is verified.
We investigate the total entropy production of a Brownian particle in a driven bistable system. This system exhibits the phenomenon of stochastic resonance. We show that in the time-periodic steady state, the probability density function for the total entropy production satisfies Seifert's integral and detailed fluctuation theorems over finite time trajectories.
We analyzed the entropy production in the majority-vote model by using a mean-field approximation and Monte Carlo simulations. The dynamical rules of the model do not obey detailed balance so that entropy is continuously being produced. This nonequilibrium stochastic model is known to have a critical behavior belonging to the universality class of the equilibrium Ising model. We show that the entropy production also exhibits a singularity at the critical point similar to the one occurring in the entropy, or the energy, of the equilibrium Ising model.
We study the entropy production rate in systems described by linear Langevin equations, containing mixed even and odd variables under time reversal. Exact formulas are derived for several important quantities in terms only of the means and covariances of the random variables in question. These include the total rate of change of the entropy, the entropy production rate, the entropy flux rate and the three components of the entropy production. All equations are cast in a way suitable for large-scale analysis of linear Langevin systems. Our results are also applied to different types of electric…
We will discuss the maximum entropy production (MaxEP) principle based on Jaynes' information theoretical arguments, as was done by Dewar (2003, 2005). With the help of a simple mathematical model of a non-equilibrium system, we will show how to derive minimum and maximum entropy production. Furthermore, the model will help us to clarify some confusing points and to see differences between some MaxEP studies in the literature.
We study by Montecarlo simulation the non-equilibrium stationary behavior of a three-dimensional stochastic lattice gas with reversible kinetic constraints and in diffusive contact with two particle reservoirs at different chemical potential. When one of the boundaries is placed in the glassy phase, the statistics of entropy production fluctuations driven by the chemical potential gradient satisfy the Gallavotti-Cohen fluctuation theorem provided that an effective entropy production is introduced.
We propose to construct Markov dynamics with specified characteristics using a cycle decomposition of the space of Markov chains introduced by Cohen and Alpern. As an application of this approach, we derive the minimal entropy production required to generate prescribed thermodynamic currents.
We study the entropy production of a microscopic model for nonequilibrium wetting. We show that, in contrast to the equilibrium case, a bound interface in a nonequilibrium steady state produces entropy. Interestingly, in some regions of the phase diagram a bound interface produces more entropy than a free interface. Moreover, by solving exactly a four-site system, we find that the first derivative of the entropy production with respect to the control parameter displays a discontinuity at the critical point of the wetting transition.
Recently, the fundamental laws of thermodynamics have been reconsidered for small systems. The discovery of the fluctuation relations has spurred theoretical and experimental studies on thermodynamics of systems with few degrees of freedom. The concept of entropy production has been extended to the microscopic level by considering stochastic trajectories of a system coupled to a heat bath. However, the experimental observation of the microscopic entropy production remains elusive. We measure distributions of the microscopic entropy production in a single-electron box consisting of two islands …
Allen-Cahn (Ginzburg-Landau) dynamics for scalar fields with heat conduction is treated in rigid bodies using a non-equilibrium thermodynamic framework with weakly nonlocal internal variables. The entropy production and entropy flux is calculated with the classical method of irreversible thermodynamics by separating full divergences.
The entropy production of a nonequilibrium system with broken detailed balance is a random variable whose mean value is nonnegative. Among the total entropy production, the housekeeping entropy production is associated with the heat dissipation in maintaining a nonequilibrium steady state. We derive a Langevin-type stochastic differential equation for the housekeeping entropy production. The equation allows us to define a housekeeping entropic time $τ$. Remarkably, it turns out that the probability distribution of the housekeeping entropy production at a fixed value of $τ$ is given by the Gaus…
In contrast to the standard derivation of Kirchhoff's loop law, which invokes electric potential, we show, for the linear planar electric network in a stationary state at the fixed temperature,that loop law can be derived from the maximum entropy production principle. This means that the currents in network branches are distributed in such a way as to achieve the state of maximum entropy production.
In this paper we discuss the meaning of the Schnakenberg formula for entropy production in non-equilibrium systems. To this end we consider a non-equilibrium system as part of a larger isolated system which includes the environment. We prove that the Schnakenberg formula provides only a lower bound to the actual entropy production in the environment. This is also demonstrated in the simplest example of a three-state clock model.
We derive a geometrical expression of the excess entropy production for quasi-static transitions between nonequilibrium steady states of Markovian jump processes, which can be exactly applied to nonlinear and nonequilibrium situations. The obtained expression is geometrical; the excess entropy production depends only on a trajectory in the parameter space, analogous to the Berry phase in quantum mechanics. Our results imply that vector potentials are needed to construct thermodynamics of nonequilibrium steady states.
The Tsallis entropy, which is a generalization of the Boltzmann-Gibbs entropy, plays a central role in nonextensive statistical mechanics of complex systems. A lot of efforts have recently been made on establishing a dynamical foundation for the Tsallis entropy. They are primarily concerned with nonlinear dynamical systems at the edge of chaos. Here, it is shown by generalizing a formulation of thermostatistics based on time averages recently proposed by Carati [A. Carati, Physica A 348, 110 (2005)] that, whenever relevant, the Tsallis entropy indexed by $q$ is temporally extensive: linear gro…
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 discuss the validity of close-to-equilibrium entropy production principles in the context of linear electrical circuits. Both the minimum and the maximum entropy production principle are understood within dynamical fluctuation theory. The starting point are Langevin equations obtained by combining Kirchoff's laws with a Johnson-Nyquist noise at each dissipative element in the circuit. The main observation is that the fluctuation functional for time averages, that can be read off from the path-space action, is in first order around equilibrium given by an entropy production rate. That allows…
We define the entropy operator as the negative of the logarithm of the density matrix, give a prescription for extracting its thermodynamically measurable part, and discuss its dynamics. For an isolated system we derive the first, second and third laws of thermodynamics. For weakly-coupled subsystems of an isolated system, an expression for the long time limit of the expectation value of the rate of change of the thermodynamically measurable part of the entropy operator is derived and interpreted in terms of entropy production and entropy transport terms. The interpretation is justified by com…
We study a spatial diffusion process generated by velocity fluctuations of intermittent nature. We note that intermittence reduces the entropy production rate while enhancing the diffusion strength. We study a case of space-dependent intermittence and prove it to result in a deviation from uniform distribution. This macroscopic effect can be used to measure the relative value of the trajectory entropy.
We investigate nonequilibrium underdamped Langevin dynamics of Brownian particles that interact through a harmonic potential with coupling constant $K$ and are in thermal contact with two heat baths at different temperatures. The system is characterized by a net heat flow and an entropy production in the steady state. We compare the entropy production of the harmonic system with that of Brownian particles linked with a rigid rod. The harmonic system may be expected to reduce to the rigid rod system in the infinite $K$ limit. However, we find that the harmonic system in the $K\to\infty$ limit p…
We show that under local detailed balance the expected entropy production rate is always bounded in terms of the dynamical activity. The activity refers to the time-symmetric contribution in the action functional for path-space probabilities and relates to escape rates and unoriented traffic. Under global detailed balance we get a lower bound on the decrease of free energy which is known from gradient flow analysis. For stationary driven systems we recover some of the recently studied "uncertainty" relations for the entropy production, appearing in studies about the effectiveness of mesoscopic…
We consider a system of two Brownian particles (say A and B), coupled to each other via harmonic potential of stiffness constant $k$. Particle-A is connected to two heat baths of constant temperatures $T_1$ and $T_2$, and particle-B is connected to a single heat bath of a constant temperature $T_3$. In the steady state, the total entropy production for both particles obeys the fluctuation theorem. We compute the total entropy production due to one of the particles called as partial or apparent entropy production, in the steady state for a time segment $τ$. When both particles are weakly intera…
Heat conduction is investigated on three levels: equilibrium, Fourier, and Cattaneo. The Fourier level is either the point of departure for investigating the approach to equilibrium or the final stage in the investigation of the approach from the Cattaneo level. Both investigations bring to the Fourier level an entropy and a thermodynamics. In the absence of external and internal influences preventing the approach to equilibrium the entropy that arises in the latter investigation is the production of the classical entropy that arises in the former investigation. If the approach to equilibrium …
We analyse a non-equilibrium exclusion process in which particles are created and annihilated in pairs and hop to the the right or to the left with different transition rates, $p$ and $q$, respectively. We have studied the dynamics of a single particle, and exactly determined the entropy, entropy production rate and entropy flux as functions of time. In the system of many particles, we have characterised the system by its probability distribution, as well as the entropy production rate in close forms, provided that $p+q$ equals the sum of dimers creation and annihilation rates. The general cas…
We challenge claims that the principle of maximum entropy production produces physical phenomenological relations between conjugate currents and forces, even beyond the linear regime, and that currents in networks arrange themselves to maximize entropy production as the system approaches the steady state. In particular: (1) we show that Ziegler's principle of thermodynamic orthogonality leads to stringent reciprocal relations for higher order response coefficients, and in the framework of stochastic thermodynamics, we exhibit a simple explicit model that does not satisfy them; (2) on a network…
For a colloidal particle driven by a constant force across a periodic potential, we investigate the distribution of entropy production both experimentally and theoretically. For short trajectories, the fluctuation theorem holds experimentally. The mean entropy production rate shows two regimes as a function of the applied force. Theoretically, both mean and variance of the pronounced non-Gaussian distribution can be obtained from a differential equation in good agreement with the experimental data.
We study irreversible processes for nonlinear oscillators networks described by complex-valued Langevin equations that account for coupling to different thermo-chemical baths. Dissipation is introduced via non-Hermitian terms in the Hamiltonian of the model. We apply the stochastic thermodynamics formalism to compute explicit expressions for the entropy production rates. We discuss in particular the non-equilibrium steady states of the network characterised by a constant production rate of entropy and flows of energy and particle currents. For two specific examples, a one-dimensional chain and…
The large deviation function for entropy production is calculated for a particle driven along a periodic potential by solving a time-independent eigenvalue problem. In an intermediate force regime, the large deviation function shows pronounced deviations from a Gaussian behavior with a characteristic ``kink'' at zero entropy production. Such a feature can also be extracted from the analytical solution of the asymmetric random walk to which the driven particle can be mapped in a certain parameter range.
We study the time evolution of a quantum system without classical counterpart, undergoing a process of entropy increase due to the environment influence. We show that if the environment-induced decoherence is interpreted in terms of wave-function collapses, a symbolic sequence can be generated. We prove that the Kolmogorov-Sinai entropy of this sequence coincides with rate of von Neumann entropy increase.
With nontrivial entropy production, first passage process is one of the most common nonequilibrium process in stochastic thermodynamics. Using one dimensional birth and death precess as a model framework, approximated expressions of mean first passage time (FPT), mean total number of jumps (TNJ), and their coefficients of variation (CV), are obtained for the case far from equilibrium. Consequently, uncertainty relations for FPT and TNJ are presented. Generally, mean FPT decreases exponentially with entropy production, while mean TNJ decreases exponentially first and then tends to a starting si…
We study the large deviation function for the entropy production rate in two driven one-dimensional systems: the asymmetric random walk on a discrete lattice and Brownian motion in a continuous periodic potential. We compare two approaches: the Donsker-Varadhan theory and the Freidlin-Wentzell theory. We show that the wings of the large deviation function are dominated by a single optimal trajectory: either in forward (positive rate) or in backward direction (negative rate). The joining of both branches at zero entropy production implies a non-differentiability and thus the appearance of a "ki…
