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We discuss non-equilibrium thermodynamics of the mean-field Ising model from a geometric perspective, focusing on the thermodynamic limit. When the number of spins is finite, the Gibbs equilibria form a smooth Legendrian submanifold in the thermodynamic phase space whose points describe the stable macroscopic states of the system. We describe the convergence of these smooth Legendrian submanifolds, as the number of spins goes to infinity, to a singular Legendrian submanifold, admitting an analytic continuation that contains both the stable and metastable states. We also discuss the relaxation to a Gibbs equilibrium when the physical parameters are changed abruptly. The relaxation is defined via the gradient flow of the free energy with respect to the Wasserstein metric on microscopic states, that is, in the geometric language, via the gradient flow of the generating function of the equilibrium Legendrian with respect to the ghost variables. This leads to a discrete Fokker-Planck equation when the number of spins is finite. We show that in the thermodynamic limit this description is closely related to the seminal model of relaxation proposed by Glauber. Finally, we find a special range of parameters where such relaxation happens instantaneously, along the Reeb chords connecting the initial and the terminal Legendrian submanifolds. [ABSTRACT FROM AUTHOR]

We consider the classical measurement model for exploring nonequilibrium effects on the physical measurement and information estimation. With appropriate settings on the stochastic dynamics of the measurement and information estimation, we demonstrate the intrinsic statistical correspondence between the entropy production in the measurement and the best estimation accuracy—the Fisher information of the probability distribution of the observed object states. We uncover the general relationship between the moment generating functions of the entropy production and the acquired information from the measurement, which contains the fluctuation information of full orders compared to the Landauer principle on average level. With these relationships, we prove the limit of the best estimation accuracy, which is given by the entropy production in the measurement. This gives rise to a nonequilibrium constraint on the noise-signal-ratio of the estimation, which can be shown as an inequality that is analogous to the thermodynamics uncertainty relation. In applications, when the fluctuation in the observed data is large, the inequality relationships between the moment generating functions lead to a more suitable estimation of the minimum entropy production in the measurement, compared to the Landauer principle; and give rise to a more robust prediction of the best estimation accuracy, compared to the Fisher information. We demonstrate that the entropy production, acquired information and Fisher information are convex functions of the nonequilibrium strength of the measurement dynamics. Thus, the entropy production can be enlarged as the nonequilibrium strength increases in the measurement. As more entropy production is paid, more useful information can be extracted from the measurement, then more accuracy can be achieved in the related estimation. [ABSTRACT FROM AUTHOR]

We consider macroscopic systems in mild contact with boundary reservoirs and under the action of external fields. We present an explicit formula for the Hamiltonian of such systems, from which we deduce the equation of motions, the action functional, the hydrodynamic equation for the adjoint dynamics, and a formula for the quasi-potential. We examine the case in which the external forcing depends on time and drives the system from one nonequilibrium state to another. We extend the results presented in Bertini (J. Statist. Phys. 149: 773-802, 2012) on thermodynamic transformations for systems in strong contact with boundary reservoirs to the present situation. In particular, we propose a natural definition of renormalized work, and show that it satisfies a Clausius inequality, and that quasi-static transformations minimize the renormalized work. In addition, we connect the renormalized work to the quasi-potential describing the fluctuations in the stationary nonequilibrium ensemble. [ABSTRACT FROM AUTHOR]

We generalize an idea in the works of Landauer and Bennett on computations, and Hill’s in chemical kinetics, to emphasize the importance of kinetic cycles in mesoscopic nonequilibrium thermodynamics (NET). For continuous stochastic systems, a NET in phase space is formulated in terms of cycle affinity ∇ ∧ ( D - 1 b) and vorticity potential A (x) of the stationary flux J ∗ = ∇ × A . Each bivectorial cycle couples two transport processes represented by vectors and gives rise to Onsager’s notion of reciprocality; the scalar product of the two bivectors A · ∇ ∧ ( D - 1 b) is the rate of local entropy production in the nonequilibrium steady state. An Onsager operator that relates vorticity to cycle affinity is introduced. [ABSTRACT FROM AUTHOR]

The moment system associated with the Boltzmann equation is largely used in many applications and is the main ingredient of Rational Extended Thermodynamics (RET). The choice of truncation of moments is arbitrary and many possibilities are present in the literature depending in particular on many contracted indexes in each moment tensor considered. Moreover, in a polyatomic gas, we have two hierarchies of moments with different indexes of truncation. As any classical theory can be considered as a limiting case of a relativistic one, we study in this paper the moments associated with the relativistic Boltzmann–Chernikov equation until the index of truncation N. We take the classical limit for both polyatomic and monatomic rarefied gases and prove that there exists only unique possible choice of the moments in the classical case for a given N. [ABSTRACT FROM AUTHOR]

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 Δ S tot equivalently, reaching its lower bound. Properties of the optimal thermodynamic machines are discussed, with explicit expressions of upper bounds of work output W, power P, and energy efficiency η are presented. To illustrate the results obtained, typical examples with optimal protocols (external potentials) are also presented. [ABSTRACT FROM AUTHOR]

We propose the concept of global temperature for spatially non-uniform heat conduction systems. With this novel quantity, we present an extended framework of thermodynamics for the whole system such that the fundamental relation of thermodynamics holds, which we call "global thermodynamics" for heat conduction systems. Associated with this global thermodynamics, we formulate a variational principle for determining thermodynamic properties of the liquid-gas phase coexistence in heat conduction, which corresponds to the natural extension of the Maxwell construction for equilibrium systems. We quantitatively predict that the temperature of the liquid–gas interface deviates from the equilibrium transition temperature. This result indicates that a super-cooled gas stably appears near the interface. [ABSTRACT FROM AUTHOR]

Height fluctuations are studied in the one-dimensional totally asymmetric simple exclusion process with periodic boundaries, with a focus on how late time relaxation towards the non-equilibrium steady state depends on the initial condition. Using a reformulation of the matrix product representation for the dominant eigenstate, the statistics of the height at large scales is expressed, for arbitrary initial conditions, in terms of extremal values of independent standard Brownian bridges. Comparison with earlier exact Bethe ansatz asymptotics leads to explicit conjectures for some conditional probabilities of non-intersecting Brownian bridges with exponentially distributed distances between the endpoints. [ABSTRACT FROM AUTHOR]

We continue the investigation of kinetic models of a system in contact via stochastic interactions with several spatially homogeneous thermal reservoirs at different temperatures. Considering models different from those investigated in Carlen et al. (Braz J Probab Stat 29:372-386, 2015), we explicitly compute the unique spatially uniform non-equilibrium steady state (NESS) and prove that it is approached exponentially fast from any uniform initial state. This leaves open the question of whether there exist NESS that are not spatially uniform. Making a further simplification of our models, we then prove non-existence of such NESS and exponential approach to the unique spatially uniform NESS (with a computably boundable rate). The method of proof relies on refined Doeblin estimates and other probabilistic techniques, and is quite different form the analysis in Carlen et al. (Braz J Probab Stat 29:372-386, 2015) that was based on contraction mapping methods. [ABSTRACT FROM AUTHOR]

Recently, remarkably simple exact results were presented about the dynamics of heat transport in the local Luttinger model for nonequilibrium initial states defined by position-dependent temperature profiles. We present mathematical details on how these results were obtained. We also give an alternative derivation using only algebraic relations involving the energy-momentum tensor which hold true in any unitary conformal field theory (CFT). This establishes a simple universal correspondence between initial temperature profiles and the resulting heat-wave propagation in CFT. We extend these results to larger classes of nonequilibrium states. It is proposed that such universal CFT relations provide benchmarks to identify nonuniversal properties of nonequilibrium dynamics in other models. [ABSTRACT FROM AUTHOR]

The nonequilibrium dynamics of anharmonic chains is studied by imposing an initial domain-wall state, in which the two half lattices are prepared in equilibrium with distinct parameters. We analyse the Riemann problem for the corresponding Euler equations and, in specific cases, compare with molecular dynamics. Additionally, the fluctuations of time-integrated currents are investigated. In analogy with the KPZ equation, their typical fluctuations should be of size $$t^{1/3}$$ and have a Tracy-Widom GUE distributed amplitude. The proper extension to anharmonic chains is explained and tested through molecular dynamics. Our results are calibrated against the stochastic LeRoux lattice gas. [ABSTRACT FROM AUTHOR]

Experience collected in mesoscopic dynamic modeling of externally driven systems indicates absence of potentials that could play role of equilibrium or nonequilibrium thermodynamic potentials yet their thermodynamics-like modeling is often found to provide a good description, good understanding, and predictions that agree with results of experimental observations. This apparent contradiction is explained by noting that the dynamic and the thermodynamics-like investigations on a given mesoscopic level of description are not directly related. Their relation is indirect. They both represent two aspects of dynamic modeling on a more microscopic level of description. The thermodynamic analysis arises in the investigation of the way the more microscopic dynamics reduces to the mesoscopic dynamics (reducing dynamics) and the mesoscopic dynamic analysis in the investigation of the result of the reduction (reduced dynamics). [ABSTRACT FROM AUTHOR]

This paper studies a mathematical formalism of nonequilibrium thermodynamics for chemical reaction models with N species, M reactions, and general rate law. We establish a mathematical basis for J. W. Gibbs' macroscopic chemical thermodynamics under G. N. Lewis' kinetic law of entire equilibrium (detailed balance in nonlinear chemical kinetics). In doing so, the equilibrium thermodynamics is then naturally generalized to nonequilibrium settings without detailed balance. The kinetic models are represented by a Markovian jumping process. A generalized macroscopic chemical free energy function and its associated balance equation with nonnegative source and sink are the major discoveries. The proof is based on the large deviation principle of this type of Markov processes. A general fluctuation dissipation theorem for stochastic reaction kinetics is also proved. The mathematical theory illustrates how a novel macroscopic dynamic law can emerges from the mesoscopic kinetics in a multi-scale system. [ABSTRACT FROM AUTHOR]

We extend the duality of Kipnis et al. (J Stat Phys 27:65-74, ) to inhomogeneous lattice gas systems where either the components have different degrees of freedom or the rate of interaction depends on the spatial location. Then the dual process is applied to prove local equilibrium in the hydrodynamic limit for some inhomogeneous high dimensional systems and in the nonequilibrium steady state for one dimensional systems with arbitrary inhomogeneity. [ABSTRACT FROM AUTHOR]

The kinetics and thermodynamics of free living copolymerization are studied for processes with rates depending on k monomeric units of the macromolecular chain behind the unit that is attached or detached. In this case, the sequence of monomeric units in the growing copolymer is a kth-order Markov chain. In the regime of steady growth, the statistical properties of the sequence are determined analytically in terms of the attachment and detachment rates. In this way, the mean growth velocity as well as the thermodynamic entropy production and the sequence disorder can be calculated systematically. These different properties are also investigated in the regime of depolymerization where the macromolecular chain is dissolved by the surrounding solution. In this regime, the entropy production is shown to satisfy Landauer's principle. [ABSTRACT FROM AUTHOR]

We study thermodynamic operations which bring a nonequilibrium steady state (NESS) to another NESS in physical systems under nonequilibrium conditions. We model the system by a suitable Markov jump process, and treat thermodynamic operations as protocols according to which the external agent varies parameters of the Markov process. Then we prove, among other relations, a NESS version of the Jarzynski equality and the extended Clausius relation. The latter can be a starting point of thermodynamics for NESS. We also find that the corresponding nonequilibrium entropy has a microscopic representation in terms of symmetrized Shannon entropy in systems where the microscopic description of states involves 'momenta'. All the results in the present paper are mathematically rigorous. [ABSTRACT FROM AUTHOR]

The Sun's radiation is a source of origin and maintenance of life on Earth. The Sun-Earth system is a thermodynamic machine transforming radiation into useful work of living organisms. Despite the importance of efficiency for such a thermodynamic machine, the analysis of its efficiency coefficient (EC) available in the literature has considerable shortcomings: As is noted by the author of the classical study on this subject (Oxenius in J Quant Spectrosc Radiat Transf 6:65-91, ), the second law of thermodynamics is violated for the radiation beam (without direction integration). The typical thermodynamic analysis of the interaction between radiation and matter is performed assuming an equilibrium of the chemical composition thereof as opposed to the radiation work in the biosphere (photosynthesis), which usually occurs under the conditions of a significant deviation of the active substance's composition from its equilibrium values. The 'black box' model (Aoki in J Phys Soc Jpn 52:1075-1078, ) is traditionally used to analyze the work efficiency of the Sun-Earth thermodynamic machine. It fails to explain the influence of many internal characteristics of the radiation-matter interaction on the process's EC. The present paper overcomes the above shortcomings using a relatively simple model of interaction between anisotropic radiation and two-level molecules of a rarefied component in a buffer substance. [ABSTRACT FROM AUTHOR]

Glansdorff and Prigogine (1970) proposed a decomposition of the entropy production rate, which today is mostly known for Markov processes as the Hatano-Sasa approach. Their context was irreversible thermodynamics which, while ignoring fluctuations, still allows a somewhat broader treatment than the one based on the Master or Fokker-Planck equation. Glansdorff and Prigogine were the first to introduce a notion of excess entropy production rate $$\delta ^2$$ EP and they suggested as sufficient stability criterion for a nonequilibrium macroscopic condition that $$\delta ^2$$ EP be positive. We find for nonlinear diffusions that their excess entropy production rate is itself the time-derivative of a local free energy which is the close-to-equilibrium functional governing macroscopic fluctuations. The positivity of the excess $$\delta ^2$$ EP, for which we state a simple sufficient condition, is therefore equivalent with the monotonicity in time of that functional in the relaxation to steady nonequilibrium. There also appears a relation with recent extensions of the Clausius heat theorem close-to-equilibrium. The positivity of $$\delta ^2$$ EP immediately implies a Clausius (in)equality for the excess heat. A final and related question concerns the operational meaning of fluctuation functionals, nonequilibrium free energies, and how they make their entrée in irreversible thermodynamics. [ABSTRACT FROM AUTHOR]

We consider the non-equilibrium dynamics for the Widom-Rowlinson model (without hard-core) in the continuum. The Lebowitz-Penrose-type scaling of the dynamics is studied and the system of the corresponding kinetic equations is derived. In the space-homogeneous case, the equilibrium points of this system are described. Their structure corresponds to the dynamical phase transition in the model. The bifurcation of the system is shown. [ABSTRACT FROM AUTHOR]

Baths produce friction and random forcing on particles suspended in them. The relation between noise and friction in (generalized) Langevin equations is usually referred to as the second fluctuation-dissipation theorem. We show what is the proper nonequilibrium extension, to be applied when the environment is itself active and driven. In particular we determine the effective Langevin dynamics of a probe from integrating out a steady nonequilibrium environment. The friction kernel picks up a frenetic contribution, i.e., involving the environment's dynamical activity, responsible for the breaking of the standard Einstein relation. [ABSTRACT FROM AUTHOR]

We generalize the Clausius (in)equality to overdamped mesoscopic and macroscopic diffusions in the presence of nonconservative forces. In contrast to previous frameworks, we use a decomposition scheme for heat which is based on an exact variant of the Minimum Entropy Production Principle as obtained from dynamical fluctuation theory. This new extended heat theorem holds true for arbitrary driving and does not require assumptions of local or close to equilibrium. The argument remains exactly intact for diffusing fields where the fields correspond to macroscopic profiles of interacting particles under hydrodynamic fluctuations. We also show that the change of Shannon entropy is related to the antisymmetric part under a modified time-reversal of the time-integrated entropy flux. [ABSTRACT FROM AUTHOR]

Understanding the physics of nonequilibrium systems remains as one of the major challenges of modern theoretical physics. We believe nowadays that this problem can be cracked in part by investigating the macroscopic fluctuations of the currents characterizing nonequilibrium behavior, their statistics, associated structures and microscopic origin. This fundamental line of research has been severely hampered by the overwhelming complexity of this problem. However, during the last years two new powerful and general methods have appeared to investigate fluctuating behavior that are changing radically our understanding of nonequilibrium physics: a powerful macroscopic fluctuation theory (MFT) and a set of advanced computational techniques to measure rare events. In this work we study the statistics of current fluctuations in nonequilibrium diffusive systems, using macroscopic fluctuation theory as theoretical framework, and advanced Monte Carlo simulations of several stochastic lattice gases as a laboratory to test the emerging picture. Our quest will bring us from (1) the confirmation of an additivity conjecture in one and two dimensions, which considerably simplifies the MFT complex variational problem to compute the thermodynamics of currents, to (2) the discovery of novel isometric fluctuation relations, which opens an unexplored route toward a deeper understanding of nonequilibrium physics by bringing symmetry principles to the realm of fluctuations, and to (3) the observation of coherent structures in fluctuations, which appear via dynamic phase transitions involving a spontaneous symmetry breaking event at the fluctuating level. The clear-cut observation, measurement and characterization of these unexpected phenomena, well described by MFT, strongly support this theoretical scheme as the natural theory to understand the thermodynamics of currents in nonequilibrium diffusive media, opening new avenues of research in nonequilibrium physics. [ABSTRACT FROM AUTHOR]

A general method for deriving closed reduced models of Hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation. Given a vector of resolved variables, selected to describe the macroscopic state of the system, a family of quasi-equilibrium probability densities on phase space corresponding to the resolved variables is employed as a statistical model, and the evolution of the mean resolved vector is estimated by optimizing over paths of these densities. Specifically, a cost function is constructed to quantify the lack-of-fit to the microscopic dynamics of any feasible path of densities from the statistical model; it is an ensemble-averaged, weighted, squared-norm of the residual that results from submitting the path of densities to the Liouville equation. The path that minimizes the time integral of the cost function determines the best-fit evolution of the mean resolved vector. The closed reduced equations satisfied by the optimal path are derived by Hamilton-Jacobi theory. When expressed in terms of the macroscopic variables, these equations have the generic structure of governing equations for nonequilibrium thermodynamics. In particular, the value function for the optimization principle coincides with the dissipation potential that defines the relation between thermodynamic forces and fluxes. The adjustable closure parameters in the best-fit reduced equations depend explicitly on the arbitrary weights that enter into the lack-of-fit cost function. Two particular model reductions are outlined to illustrate the general method. In each example the set of weights in the optimization principle contracts into a single effective closure parameter. [ABSTRACT FROM AUTHOR]

We derive the equations governing the protocols minimizing the heat released by a continuous-time Markov jump process on a one-dimensional countable state space during a transition between assigned initial and final probability distributions in a finite time horizon. In particular, we identify the hypotheses on the transition rates under which the optimal control strategy and the probability distribution of the Markov jump problem obey a system of differential equations of Hamilton-Jacobi-Bellman-type. As the state-space mesh tends to zero, these equations converge to those satisfied by the diffusion process minimizing the heat released in the Langevin formulation of the same problem. We also show that in full analogy with the continuum case, heat minimization is equivalent to entropy production minimization. Thus, our results may be interpreted as a refined version of the second law of thermodynamics. [ABSTRACT FROM AUTHOR]

We rigorously show that the probability to have a specific trajectory of an externally perturbed classical open system satisfies a universal symmetry for Hamiltonian dynamics. It connects the ratio between the probabilities of time forward and reversed trajectories to a degree of the time reversal asymmetry of the final phase space distribution in a model-independent framework. Especially, it amounts to a nonequilibrium generalization of the detailed balance between the probabilities of the forward and reversed trajectories under the condition that the initial phase space distribution is described by an equilibrium ensemble. An expression of the microscopic reversibility for the subsystem is also derived based on this relation. [ABSTRACT FROM AUTHOR]

We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external electric field E as well as a Gaussian thermostat which keeps the speed | v| constant. We show that despite the singular nature of the SRB measure its projections on the space coordinates are absolutely continuous. We further show that these projections satisfy linear response laws for small E. Some of these projections are computed numerically. We compare these results with those obtained from simple models in which the collisions with the obstacles are replaced by random collisions. Similarities and differences are noted. [ABSTRACT FROM AUTHOR]

We consider a harmonic chain in contact with thermal reservoirs at different temperatures and subject to bulk noises of different types: velocity flips or self-consistent reservoirs. While both systems have the same covariances in the non-equilibrium stationary state (NESS) the measures are very different. We study hydrodynamical scaling, large deviations, fluctuations, and long range correlations in both systems. Some of our results extend to higher dimensions. [ABSTRACT FROM AUTHOR]

Recently, in their attempt to construct steady state thermodynamics (SST), Komatsu, Nakagawa, Sasa, and Tasaki found an extension of the Clausius relation to nonequilibrium steady states in classical stochastic processes. Here we derive a quantum mechanical version of the extended Clausius relation. We consider a small system of interest attached to large systems which play the role of heat baths. By only using the genuine quantum dynamics, we realize a heat conducting nonequilibrium steady state in the small system. We study the response of the steady state when the parameters of the system are changed abruptly, and show that the extended Clausius relation, in which 'heat' is replaced by the 'excess heat', is valid when the temperature difference is small. Moreover we show that the entropy that appears in the relation is similar to von Neumann entropy but has an extra symmetrization with respect to time-reversal. We believe that the present work opens a new possibility in the study of nonequilibrium phenomena in quantum systems, and also confirms the robustness of the approach by Komatsu et al. [ABSTRACT FROM AUTHOR]

Using the problem of ion channel thermodynamics as an example, we illustrate the idea of building up complex thermodynamic models by successively adding physical information. We present a new formulation of information algebra that generalizes methods of both information theory and statistical mechanics. From this foundation we derive a theory for ion channel kinetics, identifying a nonequilibrium 'process' free energy functional in addition to the well-known integrated work functionals. The Gibbs-Maxwell relation for the free energy functional is a Green-Kubo relation, applicable arbitrarily far from equilibrium, that captures the effect of non-local and time-dependent behavior from transient thermal and mechanical driving forces. Comparing the physical significance of the Lagrange multipliers to the canonical ensemble suggests definitions of nonequilibrium ensembles at constant capacitance or inductance in addition to constant resistance. Our result is that statistical mechanical descriptions derived from a few primitive algebraic operations on information can be used to create experimentally-relevant and computable models. By construction, these models may use information from more detailed atomistic simulations. Two surprising consequences to be explored in further work are that (in)distinguishability factors are automatically predicted from the problem formulation and that a direct analogue of the second law for thermodynamic entropy production is found by considering information loss in stochastic processes. The information loss identifies a novel contribution from the instantaneous information entropy that ensures non-negative loss. [ABSTRACT FROM AUTHOR]

We consider the symmetric simple exclusion process in the interval [− N, N] with additional birth and death processes respectively on ( N− K, N], K>0, and [− N,− N+ K). The exclusion is speeded up by a factor N, births and deaths by a factor N. Assuming propagation of chaos (a property proved in a companion paper, De Masi et al., ) we prove convergence in the limit N→∞ to the linear heat equation with Dirichlet condition on the boundaries; the boundary conditions however are not known a priori, they are obtained by solving a non-linear equation. The model simulates mass transport with current reservoirs at the boundaries and the Fourier law is proved to hold. [ABSTRACT FROM AUTHOR]

Five previously unknown inequalities relating equilibrium free energy differences and non-equilibrium work fluctuations are derived, and lucid path to derivation of many similar inequalities is presented. These results are based upon combined exploitation of the Jarzynski equality and the generalization of the scheme for producing uncertainty-type inequalities due to H. Weyl. The inequalities may possibly lead to better understanding of behavior of the equilibrium free-energy estimators from non-equilibrium experimental data in many important applications concerning biological, chemical, and physical molecular processes. [ABSTRACT FROM AUTHOR]

We elaborate and compare two approaches to nonequilibrium thermodynamics, the two-generator bracket formulation of time-evolution equations for averages and the macroscopic fluctuation theory, for a purely dissipative isothermal driven diffusive system under steady state conditions. The fluctuation dissipation relations of both approaches play an important role for a detailed comparison. The nonequilibrium Helmholtz free energies introduced in these two approaches differ as a result of boundary conditions. A Fokker-Planck equation derived by projection operator techniques properly reproduces long range fluctuations in nonequilibrium steady states and offers the most promising possibility to describe the physically relevant fluctuations around macroscopic averages for time-dependent nonequilibrium systems. [ABSTRACT FROM AUTHOR]

When a nonequilibrium growing interface in the presence of a wall is considered a nonequilibrium wetting transition may take place. This transition can be studied through Langevin equations or discrete growth models. In the first case, the Kardar-Parisi-Zhang equation, which defines a very robust universality class for nonequilibrium moving interfaces, with a soft-wall potential is considered. While in the second, microscopic models, in the corresponding universality class, with evaporation and deposition of particles in the presence of hard-wall are studied. Equilibrium wetting is related to a particular case of the problem, it corresponds to the Edwards-Wilkinson equation with a potential in the continuum approach or to the fulfillment of detailed balance in the microscopic models. In this review we present the analytical and numerical methods used to investigate the problem and the very rich behavior that is observed with them. [ABSTRACT FROM AUTHOR]

We consider a class of stochastic dynamical systems, called piecewise deterministic Markov processes, with states ( x, σ)∈Ω×Γ, Ω being a region in ℝ d or the d-dimensional torus, Γ being a finite set. The continuous variable x follows a piecewise deterministic dynamics, the discrete variable σ evolves by a stochastic jump dynamics and the two resulting evolutions are fully-coupled. We study stationarity, reversibility and time-reversal symmetries of the process. Increasing the frequency of the σ-jumps, the system behaves asymptotically as deterministic and we investigate the structure of its fluctuations (i.e. deviations from the asymptotic behavior), recovering in a non Markovian frame results obtained by Bertini et al. (Phys. Rev. Lett. 87(4):040601, ; J. Stat. Phys. 107(3–4):635–675, ; J. Stat. Mech. P07014, ; Preprint available online at , ), in the context of Markovian stochastic interacting particle systems. Finally, we discuss a Gallavotti–Cohen-type symmetry relation with involution map different from time-reversal. [ABSTRACT FROM AUTHOR]

In many experimental situations, a physical system undergoes stochastic evolution which may be described via random maps between two compact spaces. In the current work, we study the applicability of large deviations theory to time-averaged quantities which describe such stochastic maps, in particular time-averaged currents and density functionals. We derive the large deviations principle for these quantities, as well as for global topological currents, and formulate variational, thermodynamic relations to establish large deviation properties of the topological currents. We illustrate the theory with a nontrivial example of a Heisenberg spin-chain with a topological driving of the Wess-Zumino type. The Cramér functional of the topological current is found explicitly in the instanton gas regime for the spin-chain model in the weak-noise limit. In the context of the Morse theory, we discuss a general reduction of continuous stochastic models with weak noise to effective Markov chains describing transitions between stable fixed points. [ABSTRACT FROM AUTHOR]

This is the second part of the study by the author on the symmetry of the linearized Boltzmann equation. The issue of the present part is the entropy production and the Onsager–Casimir reciprocity relation in the steady non-equilibrium systems. After the discussions on the definition of the entropy, entropy flow, and entropy production in the non-equilibrium gas systems, the expression of the entropy production in the steady state is presented. Then, for the systems weakly perturbed from a uniform equilibrium state, the entropy production is shown to be expressed in terms of the solution of the linearized Boltzmann equation. The thermodynamic forces and fluxes and the kinetic coefficients are defined solely from the expression of the entropy production. The conventional-type Onsager–Casimir relation is shown to hold for the entire range of the Knudsen number in bounded- and unbounded-domain systems, provided that the state of the gas in a far field is a local Maxwellian satisfying the Boltzmann equation for the latter. As to the other unbounded-domain systems, a nonconventional reciprocity is shown to hold. [ABSTRACT FROM AUTHOR]

In this paper we present a self-contained macroscopic description of diffusive systems interacting with boundary reservoirs and under the action of external fields. The approach is based on simple postulates which are suggested by a wide class of microscopic stochastic models where they are satisfied. The description however does not refer in any way to an underlying microscopic dynamics: the only input required are transport coefficients as functions of thermodynamic variables, which are experimentally accessible. The basic postulates are local equilibrium which allows a hydrodynamic description of the evolution, the Einstein relation among the transport coefficients, and a variational principle defining the out of equilibrium free energy. Associated to the variational principle there is a Hamilton-Jacobi equation satisfied by the free energy, very useful for concrete calculations. Correlations over a macroscopic scale are, in our scheme, a generic property of nonequilibrium states. Correlation functions of any order can be calculated from the free energy functional which is generically a non local functional of thermodynamic variables. Special attention is given to the notion of equilibrium state from the standpoint of nonequilibrium. [ABSTRACT FROM AUTHOR]

We analyze a systematic algorithm for the exact computation of the current cumulants in stochastic nonequilibrium systems, recently discussed in the framework of full counting statistics for mesoscopic systems. This method is based on identifying the current cumulants from a Rayleigh-Schrödinger perturbation expansion for the generating function. Here it is derived from a simple path-distribution identity and extended to the joint statistics of multiple currents. For a possible thermodynamical interpretation, we compare this approach to a generalized Onsager-Machlup formalism. We present calculations for a boundary driven Kawasaki dynamics on a one-dimensional chain, both for attractive and repulsive particle interactions. [ABSTRACT FROM AUTHOR]

We obtain explicit expressions for the long range correlations in the ABC model and in diffusive models conditioned to produce an atypical current of particles. In both cases, the two-point correlation functions allow one to detect the occurrence of a phase transition as they become singular when the system approaches the transition. [ABSTRACT FROM AUTHOR]

An abstract dynamical system is formulated from three features extracted from classical hydrodynamics. One its particular realization is then the classical hydrodynamics, other possible realizations are extensions of the classical hydrodynamics. The three features entering the formulation of the abstract dynamical system are the conservation laws, the compatibility with equilibrium thermodynamics, and the compatibility with classical mechanics in the limit of no dissipation. The particular extensions on which we illustrate the process of constructing different realizations are those arising when dealing with fluids in the vicinity of gas-liquid phase transitions (i.e. fluids involving large spatial inhomogeneities and large fluctuations). [ABSTRACT FROM AUTHOR]

Applying the well-known Feynman-Kac formula of inhomogeneous case, an interesting and rigorous mathematical proof of generalized Jarzynski’s equality of inhomogeneous multidimensional diffusion processes is presented, followed by an extension of the second law of thermodynamics. Then, we explain its physical meaning and applications, extending Hummer and Szabo’s work (Proc. Natl. Acad. Sci. USA 98(7):3658–3661, []) and Hatano-Sasa equality of steady state thermodynamics (Phys. Rev. Lett. 86:3463–3466, []) to the general multidimensional case. [ABSTRACT FROM AUTHOR]

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 an extended 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 and heat satisfy multiple versions of the first and second laws of thermodynamics as well as of their fluctuation theorems. Unique laws and relations appear only to be obtainable for concretely defined systems, using physical arguments to choose the relevant physical quantities. This is illustrated on a number of systems, including a Brownian particle in an electric field, a driven torsion pendulum, electric circuits and an energy transfer driven by a temperature difference. [ABSTRACT FROM AUTHOR]

We present the results of a detailed study of energy correlations at steady state for a 1-D model of coupled energy and matter transport. Our aim is to discover—via theoretical arguments, conjectures, and numerical simulations—how spatial covariances scale with system size, their relations to local thermodynamic quantities, and the randomizing effects of heat baths. Among our findings are that short-range covariances respond quadratically to local temperature gradients, and long-range covariances decay linearly with macroscopic distance. These findings are consistent with exact results for the simple exclusion and KMP models. [ABSTRACT FROM AUTHOR]

We investigate the large deviation function π∞( w) for the fluctuations of the power W( t) = wt, integrated over a time t, injected by a homogeneous random driving into a granular gas, in the infinite time limit. Our analytical study starts from a generalized Liouville equation and exploits a Molecular Chaos-like assumption. We obtain an equation for the generating function of the cumulants μ(λ) which appears as a generalization of the inelastic Boltzmann equation and has a clear physical interpretation. Reasonable assumptions are used to obtain μ(λ) in a closed analytical form. A Legendre transform is sufficient to get the large deviation function π∞( w). Our main result, apart from an estimate of all the cumulants of W( t) at large times t, is that π∞ has no negative branch. This immediately results in the inapplicability of the Gallavotti-Cohen Fluctuation Relation (GCFR), that in previous studies had been suggested to be valid for injected power in driven granular gases. We also present numerical results, in order to discuss the finite time behavior of the fluctuations of W ( t) . We discover that their probability density function converges extremely slowly to its asymptotic scaling form: the third cumulant saturates after a characteristic time τ larger than ∼50 mean free times and the higher order cumulants evolve even slower. The asymptotic value is in good agreement with our theory. Remarkably, a numerical check of the GCFR is feasible only at small times (at most τ/10), since negative events disappear at larger times. At such small times this check leads to the misleading conclusion that GCFR is satisfied for π∞( w). We offer an explanation for this remarkable apparent verification. In the inelastic Maxwell model, where a better statistics can be achieved, we are able to numerically observe the “failure” of GCFR. [ABSTRACT FROM AUTHOR]

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 through operational procedures which can be (in principle) realized experimentally. Based on SST thus constructed, we make some nontrivial predictions, including an extension of Einstein's formula on density fluctuation, an extension of the minimum work principle, the existence of a new osmotic pressure of a purely nonequilibrium origin, and a shift of coexistence temperature. All these predictions may be checked experimentally to test SST for its quantitative validity. [ABSTRACT FROM AUTHOR]

Thermodynamic fluctuations in systems that are in nonequilibrium steady states are always spatially long ranged, in contrast to fluctuations in thermodynamic equilibrium. In the present paper we consider a fluid subjected to a stationary temperature gradient. Two different physical mechanisms have been identified by which the temperature gradient causes long-ranged fluctuations. One cause is the presence of couplings between fluctuating fields. Secondly, spatial variation of the strength of random forces, resulting from the local version of the fluctuation-dissipation theorem, has also been shown to generate long-ranged fluctuations. We evaluate the contributions to the long-ranged temperature fluctuations due to both mechanisms. While the inhomogeneously correlated Langevin noise does lead to long-ranged fluctuations, in practice, they turn out to be negligible as compared to nonequilibrium temperature fluctuations resulting from the coupling between temperature and velocity fluctuations. [ABSTRACT FROM AUTHOR]

We compute the first-order correction to the correlation functions of the stationary state of a stochastically forced harmonic chain out of equilibrium when a small on-site anharmonic potential is added. This is achieved by deriving a suitable formula for the covariance matrix of the invariant state. We find that the first-order correction of the heat current does not depend on the size of the system. Second, the temperature profile is linear when the harmonic part of the on-site potential is zero. The sign of the gradient of the profile, however, is opposite to the sign of the temperature difference of the two heat baths. [ABSTRACT FROM AUTHOR]