Approximate thermodynamic structure for driven lattice gases in contact

Control of Metastable States by Heat Flux in the Hamiltonian Potts Model

The local equilibrium thermodynamics is a basic assumption of macroscopic descriptions of the out of equilibrium dynamics for Hamiltonian systems. We numerically analyze the Hamiltonian Potts model in two dimensions to study the violation of the assumption for phase coexistence in heat conduction. We observe that the temperature of the interface between ordered and disordered states deviates from the equilibrium transition temperature, indicating that metastable states at equilibrium are stabilized by the influence of a heat flux. We also find that the deviation is described by the formula proposed in an extended framework of the thermodynamics.

Steady-state thermodynamics: Description equivalence and violation of reservoir independence

For stochastic lattice models in spatially uniform nonequilibrium steady states, an effective thermodynamic temperature T and chemical potential μ can be defined via coexistence with heat and particle reservoirs. We verify that the probability distribution P_{N} for the number of particles in the driven lattice gas with nearest-neighbor exclusion in contact with a particle reservoir with dimensionless chemical potential µ^{*} possesses a large-deviation form in the thermodynamic limit. This implies that the thermodynamic properties determined in isolation (fixed particle number representation) and in contact with a particle reservoir (fixed dimensionless chemical potential representation) are equal. We refer to this as description equivalence. This finding motivates investigation of whether the effective intensive parameters so obtained depend on the nature of the exchange between system and reservoir. For example, a stochastic particle reservoir is usually taken to insert or remove a single particle in each exchange, but one may also consider a reservoir that inserts or removes a pair of particles in each event. In equilibrium, equivalence of pair and single-particle reservoirs is guaranteed by the canonical form of the probability distribution on configuration space. Remarkably, this equivalence is violated in nonequilibrium steady states, limiting the generality of steady-state thermodynamics based on intensive variables.

Analysis of entropy production in finitely slow processes between nonequilibrium steady states

We investigate entropy production in finitely slow transitions between nonequilibrium steady states in Markov jump processes using the improved adiabatic approximation method, which has a close relationship with the slow driving perturbation. This method provides systematic improvement of the adiabatic approximation on infinitely slow transitions from which we obtain nonadiabatic corrections. We analyze two types of excess entropy production and confirm that the leading adiabatic contribution reproduces known results, and then obtain nonadiabatic corrections written in terms of thermodynamic metrics defined in protocol parameter spaces. We also numerically study the resulting excess entropy production in a two-state system.

Stochastic order parameter dynamics for phase coexistence in heat conduction

We propose a stochastic order parameter model for describing phase coexistence in steady heat conduction near equilibrium. By analyzing the stochastic dynamics with a nonequilibrium adiabatic boundary condition, where total energy is conserved over time, we derive a variational principle that determines thermodynamic properties in nonequilibrium steady states. The resulting variational principle indicates that the temperature of the interface between the ordered region and the disordered region becomes greater (less) than the equilibrium transition temperature in the linear response regime when the thermal conductivity in the ordered region is less (greater) than that in the disordered region. This means that a superheated ordered (supercooled disordered) state appears near the interface, which was predicted by an extended framework of thermodynamics proposed in Nakagawa and Sasa [Liquid-Gas Transitions in Steady Heat Conduction, Phys. Rev. Lett. 119, 260602 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.260602.].

Nonequilibrium grand-canonical ensemble built from a physical particle reservoir

We introduce a nonequilibrium grand-canonical ensemble defined by considering the stationary state of a driven system of particles put in contact with a particle reservoir. When an additivity assumption holds for the large deviation function of density, a chemical potential of the reservoir can be defined. The grand-canonical distribution then takes a form similar to the equilibrium one. At variance with equilibrium, though, the probability weight is "renormalized" by a contribution coming from the contact, with respect to the canonical probability weight of the isolated system. A formal grand-canonical potential can be introduced in terms of a scaled cumulant generating function, defined as the Legendre-Fenchel transform of the large deviation function of density. The role of the formal Legendre parameter can be played, physically, by the chemical potential of the reservoir when the latter can be defined, or by a potential energy difference applied between the system and the reservoir. Static fluctuation-response relations naturally follow from the large deviation structure. Some of the results are illustrated on two different explicit examples, a gas of noninteracting active particles and a lattice model of interacting particles.

Nonequilibrium chemical potentials of steady-state lattice gas models in contact: A large-deviation approach

We introduce a general framework to describe the stationary state of two driven systems exchanging particles or mass through a contact, in a slow exchange limit. The definition of chemical potentials for the systems in contact requires that the large-deviation function describing the repartition of mass between the two systems is additive, in the sense of being a sum of contributions from each system. We show that this additivity property does not hold for an arbitrary contact dynamics, but is satisfied on condition that a macroscopic detailed balance condition holds at contact, and that the coarse-grained contact dynamics satisfies a factorization property. However, the nonequilibrium chemical potentials of the systems in contact keep track of the contact dynamics, and thus do not obey an equation of state. These nonequilibrium chemical potentials can be related either to the equilibrium chemical potential, or to the nonequilibrium chemical potential of the isolated systems. Results are applied both to an exactly solvable driven lattice gas model and to the Katz-Lebowitz-Spohn model using a numerical procedure to evaluate the chemical potential. The breaking of the additivity property is also illustrated on the exactly solvable model.

Flux and storage of energy in nonequilibrium stationary states

Systems kept out of equilibrium in stationary states by an external source of energy store an energy ΔU=U-U_{0}. U_{0} is the internal energy at equilibrium state, obtained after the shutdown of energy input. We determine ΔU for two model systems: ideal gas and a Lennard-Jones fluid. ΔU depends not only on the total energy flux, J_{U}, but also on the mode of energy transfer into the system. We use three different modes of energy transfer where the energy flux per unit volume is (i) constant, (ii) proportional to the local temperature, and (iii) proportional to the local density. We show that ΔU/J_{U}=τ is minimized in the stationary states formed in these systems, irrespective of the mode of energy transfer. τ is the characteristic timescale of energy outflow from the system immediately after the shutdown of energy flux. We prove that τ is minimized in stable states of the Rayleigh-Benard cell.

Steady-state entropy: A proposal based on thermodynamic integration

Defining an entropy function out of equilibrium is an outstanding challenge. For stochastic lattice models in spatially uniform nonequilibrium steady states, definitions of temperature T and chemical potential μ have been verified using coexistence with heat and particle reservoirs. For an appropriate choice of exchange rates, T and μ satisfy the zeroth law, marking an important step in the development of steady-state thermodynamics. These results suggest that an associated steady-state entropy S_{th} be constructed via thermodynamic integration, using relations such as (∂S/∂E)_{V,N}=1/T, ensuring that derivatives of S_{th} with respect to energy and particle number yield the expected intensive parameters. We determine via direct calculation the stationary nonequilibrium probability distribution of the driven lattice gas with nearest-neighbor exclusion, the Katz-Lebowitz-Spohn driven lattice gas, and a two-temperature Ising model so that we may evaluate the Shannon entropy S_{S} as well as S_{th} defined above. Although the two entropies are identical in equilibrium (as expected), they differ out of equilibrium; for small values of the drive, D, we find |S_{S}-S_{th}|∝D^{2}, as expected on the basis of symmetry. We verify that S_{th} is not a state function: changes ΔS_{th} depend not only on the initial and final points, but also on the path in parameter space. The inequivalence of S_{S} and S_{th} implies that derivatives of S_{S} are not predictive of coexistence. In other words, a nonequilibrium steady state is not determined by maximizing the Shannon entropy. Our results cast doubt on the possibility of defining a state function that plays the role of a thermodynamic entropy for nonequilibrium steady states.

Driven Widom-Rowlinson lattice gas

In the Widom-Rowlinson lattice gas, two particle species (A, B) diffuse freely via particle-hole exchange, subject to both on-site exclusion and prohibition of A-B nearest-neighbor pairs. As an athermal system, the overall densities are the only control parameters. As the densities increase, an entropically driven phase transition occurs, leading to ordered states with A- and B-rich domains separated by hole-rich interfaces. Using Monte Carlo simulations, we analyze the effect of imposing a drive on this system, biasing particle moves along one direction. Our study parallels that for a driven Ising lattice gas, the Katz-Lebowitz-Spohn (KLS) model, which displays atypical collective behavior, e.g., structure factors with discontinuity singularities and ordered states with domains only parallel to the drive. Here, other interesting features emerge, including structure factors with kink singularities (best fitted to |q|), maxima at nonvanishing wave-vector values, oscillating correlation functions, and ordering into multiple striped domains perpendicular to the drive, with a preferred wavelength depending on density and drive intensity. Moreover, the (hole-rich) interfaces between the domains are statistically rough (whether driven or not), in sharp contrast with those in the KLS model, in which the drive suppresses interfacial roughness. Defining an order parameter that accounts for the emergence of multistripe states, we map out the phase diagram in the density-drive plane and present preliminary evidence for a critical phase in this driven lattice gas.

Liquid-Gas Transitions in Steady Heat Conduction

We study liquid-gas transitions of heat conduction systems in contact with two heat baths under constant pressure in the linear response regime. On the basis of local equilibrium thermodynamics, we propose an equality with a global temperature, which determines the volume near the equilibrium liquid-gas transition. We find that the formation of the liquid-gas interface is accompanied by a discontinuous change in the volume when increasing the mean temperature of the baths. A supercooled gas near the interface is observed as a stable steady state.

Zeroth law and nonequilibrium thermodynamics for steady states in contact

We ask what happens when two nonequilibrium systems in 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 a certain intensive thermodynamic variable, like equilibrium chemical potential, equalizes following the zeroth law of thermodynamics and, if so, under what conditions is it possible? We argue that an equilibriumlike thermodynamic structure can be extended to nonequilibrium steady states having short-ranged spatial correlations, provided that the systems interact weakly to exchange mass with rates satisfying a balance condition-reminiscent of a detailed balance condition in equilibrium. The short-ranged correlations would lead to subsystem factorization on a coarse-grained level and the balance condition ensures both equalization of an intensive thermodynamic variable as well as ensemble equivalence, which are crucial for construction of a well-defined nonequilibrium thermodynamics. This proposition is proved and demonstrated in various conserved-mass transport processes having nonzero spatial correlations.

Failure of steady-state thermodynamics in nonuniform driven lattice gases

To be useful, steady-state thermodynamics (SST) must be self-consistent and have predictive value. Consistency of SST was recently verified for driven lattice gases under global weak exchange. Here I verify consistency of SST under local (pointwise) exchange, but only in the limit of a vanishing exchange rate; for a finite exchange rate the coexisting systems have different chemical potentials. I consider the lattice gas with nearest-neighbor exclusion on the square lattice, with nearest-neighbor hopping, and with hopping to both nearest and next-nearest neighbors. I show that SST does not predict the coexisting densities under a nonuniform drive or in the presence of a nonuniform density provoked by a hard wall or nonuniform transition rates. The steady-state chemical potential profile is, moreover, nonuniform at coexistence, contrary to the basic principles of thermodynamics. Finally, I discuss examples of a pair of systems possessing identical steady states but which do not coexist when placed in contact. The results of these studies confirm the validity of SST for coexistence between spatially uniform systems but cast serious doubt on its consistency and predictive value in systems with a finite rate of particle exchange between coexisting regions exhibiting a nonuniform particle density.

Inconsistencies in steady-state thermodynamics

We address the issue of extending thermodynamics to nonequilibrium steady states. Using driven stochastic lattice gases, we ask whether consistent definitions of an effective chemical potential μ, and an effective temperature Te, are possible. μ and Te are determined via coexistence, i.e., zero flux of particles and energy between the driven system and a reservoir. In the lattice gas with nearest-neighbor exclusion, temperature is not relevant, and we find that the effective chemical potential, a function of density and drive strength, satisfies the zeroth law, and correctly predicts the densities of coexisting systems. In the Katz-Lebowitz-Spohn driven lattice gas both μ and Te need to be defined. We show analytically that in this case the zeroth law is violated for Metropolis exchange rates, and determine the size of the violations numerically. The zeroth law appears to be violated for generic exchange rates. Remarkably, the system-reservoir coupling proposed by Sasa and Tasaki [J. Stat. Phys. 125, 125 (2006)] is free of inconsistencies, and the zeroth law holds. This is because the rate depends only on the state of the donor system, and is independent of that of the acceptor.

Gammalike mass distributions and mass fluctuations in conserved-mass transport processes

We show that, in conserved-mass transport processes, the steady-state distribution of mass in a subsystem is uniquely determined from the functional dependence of variance of the subsystem mass on its mean, provided that the joint mass distribution of subsystems is factorized in the thermodynamic limit. The factorization condition is not too restrictive as it would hold in systems with short-ranged spatial correlations. To demonstrate the result, we revisit a broad class of mass transport models and its generic variants, and show that the variance of the subsystem mass in these models is proportional to the square of its mean. This particular functional form of the variance constrains the subsystem mass distribution to be a gamma distribution irrespective of the dynamical rules.

Heat conduction induced by non-Gaussian athermal fluctuations

We study the properties of heat conduction induced by non-Gaussian noises from athermal environments. We find that new terms should be added to the conventional Fourier law and the fluctuation theorem for the heat current, where its average and fluctuation are determined not only by the noise intensities but also by the non-Gaussian nature of the noises. Our results explicitly show the absence of the zeroth law of thermodynamics in athermal systems.

Stochastic thermodynamics, fluctuation theorems and molecular machines

Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation-dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production.

Phase diagram and density large deviations of a nonconserving ABC model

The effect of particle-nonconserving processes on the steady state of driven diffusive systems is studied within the context of a generalized ABC model. It is shown that in the limit of slow nonconserving processes, the large deviation function of the overall particle density can be computed by making use of the steady-state density profile of the conserving model. In this limit one can define a chemical potential and identify first order transitions via Maxwell's construction, similarly to what is done in equilibrium systems. This method may be applied to other driven models subjected to slow nonconserving dynamics.

Geometrical expression of excess entropy production

We derive a geometrical expression of the excess entropy production for quasistatic 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 the thermodynamics of nonequilibrium steady states.

Thermodynamic theory of phase transitions in driven lattice gases

We formulate an approximate thermodynamic theory of the phase transition in driven lattice gases with attractive nearest-neighbor interactions. We construct the van der Waals equation of state for a driven system where a nonequilibrium chemical potential can be expressed as a function of density and driving field. A Maxwell's construction leads to the phase transition from a homogeneous fluid phase to the coexisting phases of gas and liquid.