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 along one direction, while the other part is free of the drive. Thus the steady state represents coexistence between two subsystems, one far from equilibrium and the other in equilibrium, which exchange particles along the interfaces. The dimensionless chemical potential $μ^*(ρ,D) = μ/k_B T$ for the lattice gas with density $ρ$ is readily determined in studies of a uniform system. Under the nonuniform drive, however, equating the chemical potentials of the coexisting subsystems does not yield the coexisting densities. The steady state chemical potential is, moreover, different in the coexisting bulk regions, contrary to the basic principles of thermodynamics. These results cast serious doubt on the predictive value of SST.
The problem of particle fluctuations in arbitrary nonuniform systems with Bose-Einstein condensate is considered. This includes the case of trapped Bose atoms. It is shown that the correct description of particle fluctuations for any nonuniform system of interacting atoms always results in thermodynamically normal fluctuations.
We develop a controlled high-temperature expansion for nonequilibrium steady states of the driven lattice gas. We represent the steady state as $P(η)\propto e^{-H(η)-Ψ(η)}$, and evaluate the lowest order contribution to the nonequilibrium effective interaction $Ψ(η)$. We see that, in dimensions $d\ge2$, all models with nonsingular transition rates have the same $Ψ(η)$, which consists of a three-body effective interaction decaying like $1/r^{2d+1}$. The models with the Metropolis rule show exceptional behavior.
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…
A steady self-diffusion process in a gas of hard spheres at equilibrium is analyzed. The system exhibits a constant gradient of labeled particles. Neither the concentration of these particles nor its gradient are assumed to be small. It is shown that the Boltzmann-Enskog kinetic equation has an exact solution describing the state. The hydrodynamic transport equation for the density of labeled particles is derived, with an explicit expression for the involved self-diffusion transport coefficient. Also an approximated expression for the one-particle distribution function is obtained. The system …
The significance of the recent finding of the velocity distribution function of the steady-state Boltzmann equation under a steady heat current obtained by Kim and Haykawa (J. Phys. Soc. Jpn. {\bf 72}, 1904 (2003)) is discussed. Through the stability analysis, it seems that the steady solution is stable. One of possible applications to the nonequiliburium Knudsen effect in which one cell at equilibrium is connected to another cell under the steady heat conduction is discussed. This solution apparently shows that steady-state thermodynamics proposed by Sasa and Tasaki cannot be used in a naive …
We use the macroscopic fluctuation theory (MFT) to study large current fluctuations in non-stationary diffusive lattice gases. We identify two universality classes of these fluctuations which we call elliptic and hyperbolic. They emerge in the limit when the deterministic mass flux is small compared to the mass flux due to the shot noise. The two classes are determined by the sign of compressibility of \emph{effective fluid}, obtained by mapping the MFT into an inviscid hydrodynamics. An example of the elliptic class is the Symmetric Simple Exclusion Process where, for some initial conditions,…
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…
We numerically study a two-dimensional granular gas of rigid disks where an external driving force is applied to each particle in such a way that the system is driven into a steady state by balancing the energy input and the dissipation due to inelastic collisions. Two intruder particles embedded in this correlated medium experience a fluctuation-induced force -- that is itself a fluctuating quantity -- due to the confinement of the hydrodynamic fluctuations between them. We find that the probability distribution of this force is a Gaussian centered on a value that is proportional to the stead…
We employ Monte Carlo simulations to study the non-equilibrium relaxation of driven Ising lattice gases in two dimensions. Whereas the temporal scaling of the density auto-correlation function in the non-equilibrium steady state does not allow a precise measurement of the critical exponents, these can be accurately determined from the aging scaling of the two-time auto-correlations and the order parameter evolution following a quench to the critical point. We obtain excellent agreement with renormalization group predictions based on the standard Langevin representation of driven Ising lattice …
We report a general class of steady and transient states of granular gases. We find that the kinetic theory of inelastic gases admits stationary solutions with a power-law velocity distribution, f(v) ~ v^(-sigma). The exponent sigma is found analytically and depends on the spatial dimension, the degree of inelasticity, and the homogeneity degree of the collision rate. Driven steady-states, with the same power-law tail and a cut-off can be maintained by injecting energy at a large velocity scale, which then cascades to smaller velocities where it is dissipated. Associated with these steady-stat…
We study the effect of quenched spatial disorder on the steady states of driven systems of interacting particles. Two sorts of models are studied: disordered drop-push processes and their generalizations, and the disordered asymmetric simple exclusion process. We write down the exact steady-state measure, and consequently a number of physical quantities explicitly, for the drop-push dynamics in any dimensions for arbitrary disorder. We find that three qualitatively different regimes of behaviour are possible in 1-$d$ disordered driven systems. In the Vanishing-Current regime, the steady-state …
In particle systems subject to a nonuniform drive, particle migration is observed from the driven to the non--driven region and vice--versa, depending on details of the hopping dynamics, leading to apparent violations of Fick's law and of steady--state thermodynamics. We propose and discuss a very basic model in the framework of independent random walkers on a pair of rings, one of which features biased hopping rates, in which this phenomenon is observed and fully explained.
We develop n-cluster mean-field theories (0 < n < 5) for calculating the flow properties of the non-equilibrium steady-states of the Katz-Lebowitz-Spohn model of the driven diffusive lattice gas, with attractive and repulsive inter-particle interactions, in both one and two dimensions for arbitrary particle densities, temperature as well as the driving field. We compare our theoretical results with the corresponding numerical data we have obtained from the computer simulations to demonstrate the level of accuracy of our theoretical predictions. We also compare our results with those for some o…
The steady states of two vibrated granular gases separated by an adiabatic piston are investigated. The system exhibits a non-equilibrium phase transition with an spontaneous symmetry breaking. Even if the gases at both sides of the piston have the same number of particles and are mechanically identical, their steady volumes and temperatures can be rather different. The transition can be explained by a simple kinetic theory model expressing mechanical equilibrium and the energy balance occurring in the system. The model predictions are in good agreement with molecular dynamics simulation resul…
We demonstrate here a series of exact mappings between particular cases of four statistical physics models: equilibrium 1-dimensional lattice gas with nearest-neighbor repulsion, $(1+1)$-dimensional combinatorial heap of pieces, random walks on half-plane and totally asymmetric simple exclusion process (TASEP) in one dimension (1D). In particular, we show that generating function of a steady state of one-dimensional TASEP with open boundaries can be interpreted as a quotient of partition functions of 1D hard-core lattice gases with one adsorbing lattice site and negative fugacity. This result …
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. Both for 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, like equilibrium chemical potential, which determines the final steady state for two initially separated driven lattice gases brought into contact. However, due to nontrivial con…
We use an extension of fundamental measure theory to lattice hard-core fluids to study the phase diagram of two different systems. First, two-dimensional parallel hard squares with edge-length $σ=2$ in a simple square lattice. This system is equivalent to the lattice gas with first and second neighbor exclusion in the same lattice, and has the peculiarity that its close packing is degenerated (the system orders in sliding columns). A comparison with other theories is discussed. Second, a three-dimensional binary mixture of parallel hard cubes with $σ_{\rm{L}}=6$ and $σ_{\rm{S}}=2$. Previous si…
We consider the inelastic Maxwell model, which consists of a collection of particles that are characterized by only their velocities, and evolving through binary collisions and external driving. At any instant, a particle is equally likely to collide with any of the remaining particles. The system evolves in continuous time with mutual collisions and driving taken to be point processes with rates $τ_c^-{1}$ and $τ_w^{-1}$ respectively. The mutual collisions conserve momentum and are inelastic, with a coefficient of restitution $r$. The velocity change of a particle with velocity $v$, due to dr…
A model of homogeneously driven dissipative system, consisting of a collection of $N$ particles that are characterized by only their velocities, is considered. Adopting a discrete time dynamics, at each time step, a pair of velocities is randomly selected. They undergo inelastic collision with probability $p$. With probability $(1-p)$, energy of the system is changed by changing the velocities of both the particles independently according to $v\rightarrow -r_w v +η$, where $η$ is a Gaussian noise drawn independently for each particle as well as at each time steps. For the case $r_w=- 1$, altho…
The steady states of two gases of hard spheres or disks separated by an adiabatic piston in presence of a temperature gradient are discussed. The temperature field is generated by two thermal walls at different temperatures, each of them in contact with one of the gases. The presence of the piston strongly affects the hydrodynamic fields, inducing a jump in its vicinity. A simple kinetic theory model is formulated. Its predictions are shown to be in good agreement with molecular dynamics simulation results. The applicability of the minimum entropy production principle is analyzed, and it is fo…
We consider the two dimensional (2D) classical lattice Coulomb gas as a model for magnetic field induced vortices in 2D superconducting networks. Two different dynamical rules are introduced to investigate driven diffusive steady states far from equilibrium as a function of temperature and driving force. The resulting steady states differ dramatically depending on which dynamical rule is used. We show that the commonly used driven diffusive Metropolis Monte Carlo dynamics contains unphysical intrinsic randomness that destroys the spatial ordering present in equilibrium (the vortex lattice) ove…
A lattice gas with infinite repulsion between particles separated by $\leq 1$ lattice spacing, and nearest-neighbor hopping dynamics, is subject to a drive favoring movement along one axis of the square lattice. The equilibrium (zero drive) transition to a phase with sublattice ordering, known to be continuous, shifts to lower density, and becomes discontinuous for large bias. In the ordered nonequilibrium steady state, both the particle and order-parameter densities are nonuniform, with a large fraction of the particles occupying a jammed strip oriented along the drive. The relaxation exhibit…
We discuss stationary aspects of a set of driven lattice gases in which hard-core particles with spatial extent, covering more than one lattice site, diffuse and reconstruct in one dimension under nearest-neighbor interactions. As in the uncoupled case [M. Barma et al., J. Phys. Condens. Matter 19, 065112 (2007)], the dynamics of the phase space breaks up into an exponentially large number of mutually disconnected sectors labeled by a non-local construct, the irreducible string. Depending on whether the particle couplings are taken attractive or repulsive, simulations in most of the studied se…
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 mu, and an effective temperature T_e, are possible. mu and T_e 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 d…
In this paper, we investigate vortex dynamics in a two-dimensional Bose-Hubbard model coupled with a weak artificial magnetic field, a random white noise and a dissipation. Origin of the noise and dissipation is considered as thermal fluctuations of atoms that do not participate the Bose-Einstein condensation (BEC). Solving a stochastic Gross-Pitaevskii equation to this system, we show that the interplay of the magnetic field and the white noise generates vortices in the bulk of the BEC and stable steady states of vortices form after a transition period. We calculate the incompressible part of…
We study inelastic gases in two dimensions using event-driven molecular dynamics simulations. Our focus is the nature of the stationary state attained by rare injection of large amounts of energy to balance the dissipation due to collisions. We find that under such extreme driving, with the injection rate much smaller than the collision rate, the velocity distribution has a power-law high energy tail. The numerically measured exponent characterizing this tail is in excellent agreement with predictions of kinetic theory over a wide range of system parameters. We conclude that driving by rare bu…
One of the main features of statistical systems out of equilibrium is the currents they exhibit in their stationary state: microscopic currents of probability between configurations, which translate into macroscopic currents of mass, charge, etc. Understanding the general behaviour of these currents is an important step towards building a universal framework for non-equilibrium steady states akin to the Gibbs-Boltzmann distribution for equilibrium systems. In this review, we consider one-dimensional bulk-driven particle gases, and in particular the asymmetric simple exclusion process (ASEP) wi…
We study a dilute granular gas immersed in a thermal bath made of smaller particles with masses not much smaller than the granular ones in this work. Granular particles are assumed to have inelastic and hard interactions, losing energy in collisions as accounted by a constant coefficient of normal restitution. The interaction with the thermal bath is modeled by a nonlinear drag force plus a white-noise stochastic force. The kinetic theory for this system is described by an Enskog--Fokker--Planck equation for the one-particle velocity distribution function. To get explicit results of the temper…
Mean-Field Convective Phase Separation under Thermal Gradients
King’s College London（伦敦国王学院）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences（理化学研究所交叉理论与数学科学中心）；RIKEN Pioneering Research Institute（理化学研究所先驱研究 institute）
