We consider the problem of constructing a thermodynamic theory of non-equilibrium steady states as a formal extension of the equilibrium theory. Specifically, studying a particular system, we attempt to construct a phenomenological theory describing the interplay between heat and mechanical work that takes place during operations through which the system undergoes transitions between non-equilibrium steady states. We find that, in contrast to the case of the equilibrium theory, apparently, there exists no systematic way within a phenomenological formulation to describe the work done by the system during such operations. With this observation, we conclude that the attempt to construct a thermodynamic theory of non-equilibrium steady states in analogy to the equilibrium theory has limited prospects for success and that the pursuit of such a theory should be directed elsewhere.
Following the proposal of steady state thermodynamics (SST) by Oono and Paniconi, we develop a phenomenological theory for steady nonequilibrium states in systems with heat conduction. We find that there is essentially a unique consistent thermodynamics, and make concrete predictions, i.e, the existence of a new osmotic pressure and a shift in the coexistence temperature. These predictions allow one to test for the quantitative validity of SST by comparing them with experiments.
The thermodynamic uncertainty relation offers a universal energetic constraint on the relative magnitude of current fluctuations in nonequilibrium steady states. However, it has only been derived for long observation times. Here, we prove a recently conjectured finite-time thermodynamic uncertainty relation for steady-state current fluctuations. Our proof is based on a quadratic bound to the large deviation rate function for currents in the limit of a large ensemble of many copies.
We study Langevin dynamics describing nonequilibirum steady states. Employing the phenomenological framework of steady state thermodynamics constructed by Oono and Paniconi [Prog. Theor. Phys. Suppl. {\bf130}, 29 (1998)], we find that the extended form of the second law which they proposed holds for transitions between steady states and that the Shannon entropy difference is related to the excess heat produced in an infinitely slow operation. A generalized version of the Jarzynski work relation plays an important role in our theory.
The seek for a new universal formulation for describing various non-equilibrium processes is a central task of modern non-equilibrium thermodynamics. In this paper, a novel steady-state thermodynamic formalism was established for general Markov processes described by the Chapman-Kolmogorov equation. Furthermore, corresponding formalisms of steady-state thermodynamics for master equation and Fokker-Planck equation could be rigorously derived in mathematics. To be concrete, we proved that: 1) in the limit of continuous time, the steady-state thermodynamic formalism for the Chapman-Kolmogorov equ…
The second law of ordinary thermodynamics and the second law of steady state thermodynamics, as proposed by Oono and Paniconi, are investigated from the microscopic point of view for the open quantum system. Based on the H-theorem of Lindblad, we show that both second laws are consistent with the quantum dynamics generated by the completely positive map. In addition, microscopic expressions of entropy production and ``housekeeping heat'' are obtained for some classes of dynamical quantum systems.
By considering general Markov stochastic dynamics and its coarse-graining, we study the framework of stochastic thermodynamics for the original and reduced descriptions corresponding to different scales. We are especially concerned with the case where the irreversible entropy production has a finite difference between the scales. We find that the sum of increment of nonequilibrium entropy and excess part of entropy production, which are key quantities in construction of steady state thermodynamics, is essentially kept invariant with respect to the change in the scales of description. This gene…
The accurate determination of transport coefficients in numerical simulations is becoming increasingly important in a wide range of applications. Here we consider the linear response in systems driven away from thermal equilibrium into a non-equilibrium steady state with non-vanishing entropy production rate. A simple derivation of a general response formula is presented. For Markov processes we explicitly construct the conjugate quantities and discuss their relation with known response formulas. Emphasis is put on the formal analogy with thermodynamic potentials and some consequences are disc…
We explore driven lattice gases for the existence of an intensive thermodynamic variable which could determine "equilibration" between two nonequilibrium steady-state systems kept in weak contact. In simulations, we find that these systems satisfy surprisingly simple thermodynamic laws, such as the zero-th law and the fluctuation-response relation between the particle-number fluctuation and the corresponding susceptibility remarkably well. However at higher densities, small but observable deviations from these laws occur due to nontrivial contact dynamics and the presence of long-range spatial…
Starting from microscopic mechanics, we derive thermodynamic relations for heat conducting nonequilibrium steady states. The extended Clausius relation enables one to experimentally determine nonequilibrium entropy to the second order in the heat current. The associated Shannon-like microscopic expression of the entropy is suggestive. When the heat current is fixed, the extended Gibbs relation provides a unified treatment of thermodynamic forces in linear nonequilibrium regime.
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 …
We investigate the thermodynamics of simple (non-interacting) transport models beyond the scope of weak coupling. For a single fermionic or bosonic level -- tunnel-coupled to two reservoirs -- exact expressions for the stationary matter and energy current are derived from the solutions of the Heisenberg equations of motion. The positivity of the steady-state entropy production rate is demonstrated explicitly. Finally, for a configuration in which particles are pumped upwards in chemical potential by a downward temperature gradient, we demonstrate that the thermodynamic efficiency of this proce…
We numerically determine the entropy for heat-conducting states, which is connected to the so-called excess heat considered as a basic quantity for steady-state thermodynamics in nonequilibrium. We adopt an efficient method to estimate the entropy from the bare heat current and find that the obtained entropy agrees with the familiar local equilibrium hypothesis well. Our method possesses a wider applicability than local equilibrium and opens a possibility to compare thermodynamic properties of complex systems with those in the local equilibrium. We further investigate the entropy for heat-cond…
We extend Jarzynski's work relation and the second law of thermodynamics to a heat conducting system which is operated by an external agent. These extensions contain a new non equilibrium contribution expressed as the violation of the (linear) response relation caused by the operation. We find that a natural extension of the minimum work principle involves information about the time-reversed operation, and is far from straightforward. Our work relation may be tested experimentally especially when the temperature gradient is small.
We discuss fluctuation relations in simple cases of non-equilibrium Langevin dynamics. In particular, we show that close to non-equilibrium steady states with non-vanishing probability currents some of these relations reduce to a modified version of the fluctuation-dissipation theorem. The latter may be interpreted as the equilibrium-like relation in the reference frame moving with the mean local velocity determined by the probability current.
Thermodynamical arguments are known to be useful in the construction of physically motivated Lyapunov functionals for nonlinear stability analysis of spatially homogeneous equilibrium steady states in thermodynamically isolated systems. Unfortunately, the limitation to thermodynamically isolated systems is essential, and standard arguments are not applicable even for some very simple thermodynamically open systems. On the other hand, the nonlinear stability of thermodynamically open systems is usually investigated using the so-called energy method. Unfortunately, the designation "energy method…
We show that the system entropy change for the transitions between non-equilibrium steady states arbitrarily far from equilibrium for any constituting process is given by the relative entropy of the distributions of these steady states. This expression is then shown to relate to the dissipation relations of both Vaikuntanathan and Jarzynski [EPL 87, 60005 (2009)] and Kawai, Parrondo and Van den Broeck [Phys. Rev. Lett. 98, 080602 (2007)] in the case of energy-conserving driving.
We use the work done on and the heat removed from a system to maintain it in a nonequilibrium steady state for a thermodynamic-like description of such a system as well as of its fluctuations. Based on a generalized Onsager-Machlup theory for nonequilibrium steady states we indicate two ambiguities, not present in an equilibrium state, in defining such work and heat: one due to a non-uniqueness of time-reversal procedures and another due to multiple possibilities to separate heat into work and an energy difference in nonequilibrium steady states. As a consequence, for such systems, the work an…
The present paper reports our attempt to search for a new universal framework in nonequilibrium physics. We propose a thermodynamic formalism that is expected to apply to a large class of nonequilibrium steady states including a heat conducting fluid, a sheared fluid, and an electrically conducting fluid. We call our theory steady state thermodynamics (SST) after Oono and Paniconi's original proposal. The construction of SST is based on a careful examination of how the basic notions in thermodynamics should be modified in nonequilibrium steady states. We define all thermodynamic quantities thr…
Kinetics of steady-state copolymerization has been investigated since 1940s. Irreversible terminal and penultimate models were successfully applied to a number of comonomer systems, but failed for systems where depropagation is significant. Although a general mathematical treatment of the terminal model with depropagation was established in 1980s, penultimate model and higher-order terminal models with depropagation have not been systematically studied, since depropagation leads to hierarchically-coupled and unclosed kinetic equations which are hard to be solved analytically. In this work, we …
These lecture notes give a short review of methods such as the matrix ansatz, the additivity principle or the macroscopic fluctuation theory, developed recently in the theory of non-equilibrium phenomena. They show how these methods allow to calculate the fluctuations and large deviations of the density and of the current in non-equilibrium steady states of systems like exclusion processes. The properties of these fluctuations and large deviation functions in non-equilibrium steady states (for example non-Gaussian fluctuations of density or non-convexity of the large deviation function which g…
We ask what happens when two systems having a nonequilibrium steady state are kept in contact and allowed to exchange a quantity, say mass, which is conserved in the combined system. Will the systems eventually evolve to a new stationary state where certain intensive thermodynamic variable, like equilibrium chemical potential, equalizes following zeroth law of thermodynamics and, if so, under what conditions is it possible? We argue that the zeroth law would hold, provided both systems have short-ranged spatial correlations and they {\it interact weakly} to exchange mass with rates satisfying …
To be useful, steady state thermodynamics (SST) must be self-consistent and have predictive value. Although consistency of SST was recently verified for driven lattice gases under global weak exchange, I show here that it does not predict the coexisting densities in athermal stochastic lattice gases under a nonuniform drive. I consider the lattice gas with nearest-neighbor exclusion on the square lattice, with nearest-neighbor hopping (NNE dynamics), and with hopping to both nearest and next-nearest neighbors (NNE2 dynamics). Part of the system is subject to a drive $D$ that favors hopping alo…
When modelling driven steady states of matter, it is common practice either to choose transition rates arbitrarily, or to assume that the principle of detailed balance remains valid away from equilibrium. Neither of those practices is theoretically well founded. Hypothesising ergodicity constrains the transition rates in driven steady states to respect relations analogous to, but different from the equilibrium principle of detailed balance. The constraints arise from demanding that the design of any model system contains no information extraneous to the microscopic laws of motion and the macro…
Starting from the pioneering work of G. S. Agarwal [Zeitschrift für Physik 252, 25 (1972)], we present a unified derivation of a number of modified fluctuation-dissipation relations (MFDR) that relate response to small perturbations around non-equilibrium steady states to steady-state correlations. Using this formalism we show the equivalence of velocity forms of MFDR derived using continuum Langevin and discrete master equation dynamics. The resulting additive correction to the Einstein relation is exemplified using a flashing ratchet model of molecular motors.
An equivalence between non equilibrium steady states (NESS) driven by a time-independent force and stochastic pumps (SP) stirred by a time-varying conservative force is studied for general many-body diffusive systems. When the particle density and current of NESS are imitated by SP time-averaged counterparts, we prove that the entropy production rate in NESS is always greater than that of SP, provided that the conductivity of the particle current is concave as a function of the particle density. Searching for a SP protocol that saturates the entropy production bound reveals an unexpected conne…
When a Brownian object interacts with non-interacting gas particles under non-equilibrium conditions, the energy dissipation associated to the Brownian motion causes an additional force on the object as a `momentum transfer deficit'. This principle is demonstrated first by a new NESS model and then applied to several known models such as adiabatic piston for which simple explanation has been lacking.
We investigate nonequilibrium chemical reaction systems from the view point of steady state thermodynamics proposed by Oono and Paniconi [Prog. Theor. Phys. Suppl. 130, 29 (1998)]. The concentrations of some compounds are operated by an external system, so that a transition from a steady state to other steady state takes place. We show that an analogue of Clausius inequality holds macroscopically for the operation processes. This implies that the second law of thermodynamics can be generalized, including nonequilibrium steady states.
Electrical circuits with transient elements can be good examples of systems where non--steady irreversible processes occur, so in the same way as a steady state energy converter, we use the formal construction of the first order irreversible thermodynamic (FOIT) to describe the energetics of these circuits. In this case, we propose an isothermic model of two meshes with transient and passive elements, besides containing two voltage sources (which can be functions of time); this is a non--steady energy converter model. Through the Kirchhoff equations, we can write the circuit phenomenological e…
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…
There is a long-standing question of whether it is possible to extend the formalism of equilibrium thermodynamics to the case of non-equilibrium systems in steady states. We have made such an extension for an ideal gas in a heat flow [Hołyst \emph{et al.}, J. Chem. Phys. 157, 194108 (2022)]. Here we investigate whether such a description exists for the system with interactions: the Van der Waals gas in a heat flow. We introduce the parameters of state, each associated with a single way of changing energy. The first law of non-equilibrium thermodynamics follows from these parameters. The intern…
Long-ranged correlations generically exist in non-equilibrium fluid systems. In the case of a non-equilibrium steady state caused by a temperature gradient the correlations are especially long-ranged and strong. The anomalous light scattering predicted to exist in these systems is well-confirmed by numerous experiments. Recently the Casimir force or pressure due to these fluctuations or correlations have been discussed in great detail. In this paper the notion of a Casimir work is introduced and a novel way to measure the non-equilibrium Casimir force is suggested. In particular, the non-equil…
By considering a thermodynamic force as gauge field, we extend constitutive equations of Onsager's non-equilibrium thermodynamics to non-linear equations. In Onsager's non-equilibrium thermodynamics, the thermodynamic force corresponds to a pure gauge, for which the constitutive equations are obtained by gauge fixing. If we extend the thermodynamic force from pure gauge to physical one, we obtain the non-linear constitutive equations of non-equilibrium thermodynamics.
Motivated by fundamental issues in non-equilibrium statistical mechanics (NESM), we study the venerable susceptible-infected (SIS) model of disease spreading in an idealized, simple setting. Using Monte Carlo and analytic techniques, we consider a fully connected, uni-directional network of odd number of nodes, each having an equal number of in- and out-degrees. With the standard SIS dynamics at high infection rates, this system settles into an active non-equilibrium steady state. We find the exact probability distribution and explore its implications for NESM, such as the presence of persiste…
Stochastic processes that are randomly reset to an initial condition serve as a showcase to investigate non-equilibrium steady states. However, all existing results have been restricted to the special case of memoryless resetting protocols. Here, we obtain the general solution for the distribution of processes in which waiting times between reset events are drawn from an arbitrary distribution. This allows for the investigation of a broader class of much more realistic processes. As an example, our results are applied to the analysis of the efficiency of constrained random search processes.
We consider effect of stochastic sources upon self-organization process being initiated with creation of the limit cycle. General expressions obtained are applied to the stochastic Lorenz system to show that departure from equilibrium steady state can destroy the limit cycle at certain relation between characteristic scales of temporal variation of principle variables. Noise induced resonance related to the limit cycle is found to appear if the fastest variations displays a principle variable, which is coupled with two different degrees of freedom or more.
A century ago, the foundations of equilibrium statistical mechanics were laid. For a system in equilibrium with a thermal bath, much is understood through the Boltzmann factor, exp{-H[C]/kT}, for the probability of finding the system in any microscopic configuration C. In contrast, apart from some special cases, little is known about the corresponding probabilities, if the same system is in contact with more than one reservoir of energy, so that, even in stationary states, there is a constant energy flux through our system. These non-equilibrium steady states display many surprising properties…
The various versions of cond-mat/0312353 discuss results obtained by me&coworkers in the last decade. I have received requests to comment on the paper and the comments are collected here, including some that I tried to point in the course of discussions that can be found in the web page http://ipparco.roma1.infn.it
We present a theoretical framework to understand a modified fluctuation-dissipation theorem valid for systems close to non-equilibrium steady-states and obeying markovian dynamics. We discuss the interpretation of this result in terms of trajectory entropy excess. The framework is illustrated on a simple pedagogical example of a molecular motor. We also derive in this context generalized Green-Kubo relations similar to the ones derived recently by Seifert., Phys. Rev. Lett., 104, 138101 (2010) for more general networks of biomolecular states.
Acceleration of relaxation toward a fixed stationary distribution via violation of detailed balance was reported in the context of a Markov chain Monte Carlo method recently. Inspired by this result, systematic methods to violate detailed balance in Langevin dynamics were formulated by using exponential and rotational nonconservative forces. In the present paper, we accentuate that such specific nonconservative forces relate to the large deviation of total heat in an equilibrium state. The response to these nonconservative forces can be described by the intrinsic fluctuation of the total heat …
We analyse the steady state regime of a one dimensional Ising model under a tapping dynamics recently introduced by analogy with the dynamics of mechanically perturbed granular media. The idea that the steady state regime may be described by a flat measure over metastable states of fixed energy is tested by comparing various steady state time averaged quantities in extensive numerical simulations with the corresponding ensemble averages computed analytically with this flat measure. The agreement between the two averages is excellent in all the cases examined, showing that a static approach is …
Development of steady state thermodynamics and statistical mechanics depends crucially on our ability to develop notions of equilibrium thermodynamics for nonequilibrium steady states (NESS). The present paper considers the development of heat capacity. A modified definition is proposed which continues to maintain the same relation to steady state Shannon entropy as in equilibrium, thus providing a thermodynamically consistent treatment of NESS heat capacity.
It is known that equilibrium thermodynamics can be deduced from a constrained Fisher information extemizing process. We show here that, more generally, both non-equilibrium and equilibrium thermodynamics can be obtained from such a Fisher treatment. Equilibrium thermodynamics corresponds to the ground state solution, and non-equilibrium thermodynamics corresponds to excited state solutions, of a Schroedinger wave equation (SWE). That equation appears as an output of the constrained variational process that extremizes Fisher information. Both equilibrium- and non-equilibrium situations can ther…
We develop a unified treatment of pumping and nonequilibrium thermodynamics. We show that the pumping current generated through an adiabatic mechanical operation in equilibrium can be expressed in terms of the stationary distribution of the corresponding driven nonequilibrium system. We also show that the total transfer in pumping can be evaluated from the work imported to the driven counterpart. These findings lead us to a unified viewpoint for pumping and nonequilibrium thermodynamics.
We study the steady state of a finite XX chain coupled at its boundaries to quantum reservoirs made of free spins that interact one after the other with the chain. The two-point correlations are calculated exactly and it is shown that the steady state is completely characterized by the magnetization profile and the associated current. Except at the boundary sites, the magnetization is given by the average of the reservoirs' magnetizations. The steady state current, proportional to the difference in the reservoirs' magnetizations, shows a non-monotonous behavior with respect to the system-reser…
A modified fluctuation-dissipation-theorem (MFDT) for a non-equilibrium steady state (NESS) is experimentally checked by studying the position fluctuations of a colloidal particle whose motion is confined in a toroidal optical trap. The NESS is generated by means of a rotating laser beam which exerts on the particle a sinusoidal conservative force plus a constant non-conservative one. The MFDT is shown to be perfectly verified by the experimental data. It can be interpreted as an equilibrium-like fluctuation-dissipation relation in the Lagrangian frame of the mean local velocity of the particl…
The connection between dissipation and symmetry breaking is a long-standing enigma in statistical physics. It is intimately connected to the quest of a non-equilibrium functional whose minimization gives the non-equilibrium steady state (NESS). Writing down such a functional, we show that, in the presence of additive noise, any NESS is characterized by the minimum entropy production compatible with the maximum dissipation along cycles in the trajectory space. This result sheds light on the excess entropy production principle and the onset of chiral symmetry breaking out-of-equilibrium, indicat…
Using a generalisation of the detailed balance for systems maintained out of equilibrium by contact with 2 reservoirs at unequal temperatures or at unequal densities, we recover the fluctuation theorem for the large deviation funtion of the current. For large diffusive systems, we show how the large deviation funtion of the current can be computed using a simple additivity principle. The validity of this additivity principle and the occurence of phase transitions are discussed in the framework of the macroscopic fluctuation theory.
The Mpemba effect denotes an anomalous relaxation phenomenon where a system initially at a hot temperature cools faster than a system that starts at a less elevated temperature. We introduce an isothermal analog of this effect for a system prepared in a non-equilibrium steady state that then relaxes towards equilibrium. Here, the driving strength, which determines the initial non-equilibrium steady state, takes the role of the temperature in the original version. As a paradigm, we consider a particle initially driven by a non-conservative force along a one-dimensional periodic potential. We sh…
The fluctuation relation of the Gallavotti-Cohen Fluctuation Theorem (GCFT) concerns fluctuations in the phase space compression rate of dissipative, reversible dynamical systems. It has been proven for Anosov systems, but it is expected to apply more generally. This raises the question of which non-Anosov systems satisfy the fluctuation relation. We analyze time dependent fluctuations in the phase space compression rate of a class of N-particle systems that are at equilibrium or in near equilibrium steady states. This class does not include Anosov systems or isoenergetic systems, however, it …
We present a unified and simple method for deriving work theorems for classical and quantum Hamiltonian systems, both under equilibrium conditions and in a steady state. Throughout the paper, we adopt the partitioning of the total Hamiltonian into the system part, the bath part, and their coupling. We rederive many equalities which are available in the literature and obtain a number of new equalities for nonequilibrium classical and quantum systems. Our results can be useful for determining partition functions and (generalized) free energies through simulations and/or measurements performed on…
In non-equilibrium steady states (NESS) far from equilibrium, it is known that the Einstein relation is violated. Then, the ratio of the diffusion coefficient to the mobility is called an effective temperature, and the physical relevance of this effective temperature has been studied in several works. Although the physical relevance is not yet completely clear, it has been found that the role of an effective temperature in NESS is indeed analogous to that of the temperature in equilibrium systems in a number of respects. In this paper, we find further evidence establishing this analogy. We emp…
We develop an exact quantum thermodynamic description for a noninteracting nanoscale steady state that couples strongly with multiple reservoirs. It is demonstrated that there exists a steady-state extension of the thermodynamic function that correctly accounts for the multiterminal Landauer-Büttiker formula of quantum transport of charge, energy or heat, via the nonequilibrium thermodynamic relations. Its explicit form is obtained for a single bosonic or fermionic level in the wide-band limit, and corresponding thermodynamic forces (affinities) are identified. Nonlinear generalization of the …
We examine how systems in non-equilibrium steady states close to a continuous phase transition can still be described by a Landau potential if one forgoes the assumption of analyticity. In a system simultaneously coupled to several baths at different temperatures, the non-analytic potential arises from the different density of states of the baths. In periodically driven-dissipative systems, the role of multiple baths is played by a single bath transferring energy at different harmonics of the driving frequency. The mean-field critical exponents become dependent on the low-energy features of th…
We explore thermodynamic relations in non-equilibrium steady states with numerical experiments on a driven lattice gas. After operationally defining the pressure and chemical potential in the driven lattice gas, we confirm numerically the validity of the integrability condition (the Maxwell relation) for the two quantities whose values differ from those for an equilibrium system. This implies that a free energy function can be constructed for the non-equilibrium steady state that we consider. We also investigate a fluctuation relation associated with this free energy function. Our result sugge…
Macroscopic thermodynamics of equilibrium is constructed for systems obeying power-law canonical distributions. With this, the connection between macroscopic thermodynamics and microscopic statistical thermodynamics is generalized. This is complementary to the Gibbs theorem for the celebrated exponential canonical distributions of systems in contact with a heat bath. Thereby, a thermodynamic basis is provided for power-law phenomena ubiquitous in nature.
