Nonequilibrium steady states in contact: approximate thermodynamic structure and zeroth law for driven lattice gases

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.

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.

Einstein relation and hydrodynamics of nonequilibrium mass transport processes

We derive hydrodynamics of paradigmatic conserved-mass transport processes on a ring. The systems, governed by chipping, diffusion, and coalescence of masses, eventually reach a nonequilibrium steady state, having nontrivial correlations, with steady-state measures in most cases not known. In these processes, we analytically calculate two transport coefficients, bulk-diffusion coefficient and conductivity. Remarkably, the two transport coefficients obey an equilibrium-like Einstein relation even when the microscopic dynamics violates detailed balance and systems are far from equilibrium. Moreover, we show, using a macroscopic fluctuation theory, that the probability of large deviation in density, obtained from the above hydrodynamics, is in complete agreement with the same derived earlier by Das et al. [Phys. Rev. E 93, 062135 (2016)2470-004510.1103/PhysRevE.93.062135] using an additivity property.

Spatial correlations, additivity, and fluctuations in conserved-mass transport processes

We exactly calculate two-point spatial correlation functions in steady state in a broad class of conserved-mass transport processes, which are governed by chipping, diffusion, and coalescence of masses. We find that the spatial correlations are in general short-ranged and, consequently, on a large scale, these transport processes possess a remarkable thermodynamic structure in the steady state. That is, the processes have an equilibrium-like additivity property and, consequently, a fluctuation-response relation, which help us to obtain subsystem mass distributions in the limit of subsystem size large.

Additivity, density fluctuations, and nonequilibrium thermodynamics for active Brownian particles

Using an additivity property, we study particle-number fluctuations in a system of interacting self-propelled particles, called active Brownian particles (ABPs), which consists of repulsive disks with random self-propulsion velocities. From a fluctuation-response relation, a direct consequence of additivity, we formulate a thermodynamic theory which captures the previously observed features of nonequilibrium phase transition in the ABPs from a homogeneous fluid phase to an inhomogeneous phase of coexisting gas and liquid. We substantiate the predictions of additivity by analytically calculating the subsystem particle-number distributions in the homogeneous fluid phase away from criticality where analytically obtained distributions are compatible with simulations in the ABPs.

Additivity property and emergence of power laws in nonequilibrium steady states

We show that an equilibriumlike additivity property can remarkably lead to power-law distributions observed frequently in a wide class of out-of-equilibrium systems. The additivity property can determine the full scaling form of the distribution functions and the associated exponents. The asymptotic behavior of these distributions is solely governed by branch-cut singularity in the variance of subsystem mass. To substantiate these claims, we explicitly calculate, using the additivity property, subsystem mass distributions in a wide class of previously studied mass aggregation models as well as in their variants. These results could help in the thermodynamic characterization of nonequilibrium critical phenomena.

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.

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.

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.

Approximate thermodynamic structure for driven lattice gases in contact

For a class of nonequilibrium systems, called driven lattice gases, we study what happens when two systems are kept in contact and allowed to exchange particles with the total number of particles conserved. For both attractive and repulsive nearest-neighbor interactions among particles and for a wide range of parameter values, we find that, to a good approximation, one could define an intensive thermodynamic variable, such as the equilibrium chemical potential, that determines the final steady state for two initially separated driven lattice gases brought into contact. However, due to nontrivial contact dynamics, there are also observable deviations from this simple thermodynamic law. To illustrate the role of the contact dynamics, we study a variant of the zero-range process and discuss how the deviations could be explained by a modified large-deviation principle. We identify an additional contribution to the large-deviation function, which we call the excess chemical potential, for the variant of the zero-range process as well as the driven lattice gases. The excess chemical potential depends on the specifics of the contact dynamics and is in general a priori unknown. A contact dependence implies that, even though an intensive variable may equalize, the zeroth law could still be violated.