Factorised steady states and condensation transitions in nonequilibrium systems

Systems driven out of equilibrium can often exhibit behaviour not seen in systems in thermal equilibrium —for example phase transitions in one-dimensional systems. In this talk I will review a simple model of a nonequilibrium system known as the ‘zero-range process’ and its recent developments. The nonequilibrium stationary state of this model factorises and this property allows a detailed analysis of several ‘condensation’ transitions wherein a finite fraction of the constituent particles condenses onto a single lattice site. I will then consider a more general class of mass transport models, encompassing continuous mass variables and discrete time updating, and present a necessary and sufficient condition for the steady state to factorise. The property of factorisation again allows an analysis of the condensation transitions which may occur.     

Quantum Macrostatistical Picture of Nonequilibrium Steady States

We apply our quantum macrostatistical treatment of irreversible processes to prove that, in nonequilibrium steady states, (a) the hydrodynamical observables execute a generalised Onsager–Machlup process and (b) the spatial correlations of these observables are generically of long range. The key assumptions behind these results are a nonequilibrium version of Onsager regression hypothesis, together with certain hypotheses of chaoticity and local equilibrium for hydrodynamical fluctuations.     

Einstein Relation for Nonequilibrium Steady States

The Einstein relation, relating the steady state fluctuation properties to the linear response to a perturbation, is considered for steady states of stochastic models with a finite state space. We show how an Einstein relation always holds if the steady state satisfies detailed balance. More generally, we consider nonequilibrium steady states where detailed balance does not hold and show how a generalisation of the Einstein relation may be derived in certain cases. In particular, for the asymmetric simple exclusion process and a driven diffusive dimer model, the external perturbation creates and annihilates particles thus breaking the particle conservation of the unperturbed model.     

Hydrodynamics of the Zero-Range Process in the Condensation Regime

We argue that the coarse-grained dynamics of the zero-range process in the condensation regime can be described by an extension of the standard hydrodynamic equation obtained from Eulerian scaling even though the system is not locally stationary. Our result is supported by Monte Carlo simulations.     

Condensation in the quantum spherical spin-glass

The quantum spherical spin-glass is analyzed by using a replica symmetric theory (RS) and the physical properties of the system can be explained by the existence of a zero frequency condensate below the critical temperature. The free energy coincides with previous results obtained with an annealed average of the partition function. A calculation with one step replica symmetry breaking (1S-RSB) shows that the only solution is the replica symmetric one.     

Representation of Nonequilibrium Steady States in Large Mechanical Systems

Recently a novel concise representation of the probability distribution of heat conducting nonequilibrium steady states was derived. The representation is valid to the second order in the “degree of nonequilibrium”, and has a very suggestive form where the effective Hamiltonian is determined by the excess entropy production. Here we extend the representation to a wide class of nonequilibrium steady states realized in classical mechanical systems where baths (reservoirs) are also defined in terms of deterministic mechanics. The present extension covers such nonequilibrium steady states with a heat conduction, with particle flow (maintained either by external field or by particle reservoirs), and under an oscillating external field. We also simplify the derivation and discuss the corresponding representation to the full order.     

Comments on the Entropy of Nonequilibrium Steady States

We discuss the entropy of nonequilibrium steady states. We analyze the so-called spontaneous production of entropy in certain reversible deterministic nonequilibrium system, and its link with the collapse of such systems towards an attractor that is of lower dimension than the dimension of phase space. This means that in the steady state limit, the Gibbs entropy diverges to negative infinity. We argue that if the Gibbs entropy is expanded in a series involving 1, 2,... body terms, the divergence of the Gibbs entropy is manifest only in terms involving integrals whose dimension is higher than, approximately, the Kaplan–Yorke dimension of the steady state attractor. All the low order terms are finite and sum in the weak field limit to the local equilibrium entropy of linear irreversible thermodynamics.     

First-Order Phase Transitions in One-Dimensional Steady States

The steady states of the two-species (positive and negative particles) asymmetric exclusion model of Evans, Foster, Godrèche, and Mukamel are studied using Monte Carlo simulations. We show that mean-field theory does not give the correct phase diagram. On the first-order phase transition line which separates the CP-symmetric phase from the broken phase, the density profiles can be understood through an unexpected pattern of shocks. In the broken phase the free energy functional is not a convex function, but looks like a standard Ginzburg–Landau picture. If a symmetry-breaking term is introduced in the boundaries, the Ginzburg–Landau picture remains and one obtains spinodal points. The spectrum of the Hamiltonian associated with the master equation was studied using numerical diagonalization. There are massless excitations on the first-order phase transition fine with a dynamical critical exponent z = 2, as expected from the existence of shocks, and at the spinodal points, where we find z = 1. It is the first time that this value, which characterizes conformal invariant equilibrium problems, appears in stochastic processes.     

Fluctuation-Induced First-Order Transition in a Nonequilibrium Steady State

We present the first example of a phase transition in a nonequilibrium steady state that can be argued analytically to be first order. The system of interest is a two-species reaction-diffusion problem whose control parameter is the total density . Mean-field theory predicts a second-order transition between two stationary states at a critical density = _c. We develop a phenomenological picture that instead predicts a first-order transition below the upper critical dimension d _c=4. This picture is confirmed by hysteresis found in numerical simulations, and by the study of a renormalization-group improved equation of state. The latter approach is inspired by the Coleman–Weinberg mechanism in QED.     

Phase separation transition in a driven diffusive system with anti-ferromagnetic interaction

One-dimensional non-equilibrium systems with short-range interaction can undergo phase transitions from homogeneous states to phase separated states as interaction (?) among particles is increased. One of the model systems where such a transition has been observed is the extended Katz-Lebowitz-Spohn (KLS) model with ferromagnetically interacting particles at ?=4/5. Here, the system remains homogeneous for small interaction strength (?<4/5), and for anti-ferromagnetic interactions (?<0). We show that the phase separation transitions can also occur in anti-ferromagnetic systems if interaction among particles depends explicitly on the size of the block (n) they belong to. We study this transition in detail for a specific case ?=δ/n, where phase separation occurs for δ<−1.     

Finite-size effects for phase segregation in a two-dimensional asymmetric exclusion model with two species

We investigate the stationary states of a two-dimensional lattice gas model with exclusion, in the presence of an external field. The lattice is populated by equal numbers of positively and negatively charged particles. An analytical mean-field approach and Monte Carlo simulations give strong evidence of the fact that at any finite density the only relevant stationary state of the system in the thermodynamic limit is inhomogeneous, consisting of a strip of particles transverse to the field. In the inhomogeneous phase, the density profiles and the current measured by Monte Carlo simulations are closely related to those found in mean field. The same is true for the finite-size behavior of the system.     

On the solutions and the steady states of a master equation

A complete characterization of the time behavior of the means and variance of a stochastic process which is generated by a finite number of independent systems is presented based on the master equation for the conditional probability. It is found that the means and variance relax to a steady state and that the steady state will be independent of the initial state if and only if a matrix related to the transition matrix is nonsingular. Finally, the result that the variance approaches its steady-state form at twice the rate of the means is shown to depend on the nonsingularity of the same matrix.     

Steady State Properties of One-Dimensional Non-uniform Granular Gases Subjected to Gaussian White Noise Driving

We present a model of non-uniform granular gases in one-dimensional case, whose granularity distribution has the fractal characteristic. We have studied the nonequilibrium properties of the system by means of Monte Carlo method. When the typical relaxation time T of the Brownian process is greater than the mean collision time To, the energy evolution of the system exponentially decays, with a tendency to achieve a stable asymptotic value, and the system finally reaches a nonequilibrium steady state in which the velocity distribution strongly deviates from the Gaussian one. Three other aspects have also been studied for the steady state： the visualized change of the particle density, the entropy of the system and the correlations in the velocity of particles. And the results of simulations indicate that the system has strong spatial clustering; Furthermore, the influence of the inelasticity and inhomogeneity on dynamic behaviors have also been extensively investigated, especially the dependence of the entropy and the correlations in the velocity of particles on the restitute coefficient e and the fractal dimension D.     

Generation of W States with Many Superconducting-Quantum-Interference-Devices in Cavity QED

We propose a scheme to generate the W states with manySQUIDs (superconducting-quantum-interference-devices) in cavity QED viaRaman transition. In this scheme, the transfer of quantum informationbetween the SQUIDs and cavity is not required. And the cavity field is onlyvirtually excited, thus the cavity decay is suppressed during the W statesgeneration. The SQUIDs are always populated in the two ground states.Therefore, the scheme is insensitive to the spontaneous emission of theexcited level of the SQUID and cavity decay.     

Condensation for a Fixed Number of Independent Random Variables

A family of m independent identically distributed random variables indexed by a chemical potential φ∈[0,γ] represents piles of particles. As φ increases to γ, the mean number of particles per site converges to a maximal density ρ _c <∞. The distribution of particles conditioned on the total number of particles equal to n does not depend on φ (canonical ensemble). For fixed m, as n goes to infinity the canonical ensemble measure behave as follows: removing the site with the maximal number of particles, the distribution of particles in the remaining sites converges to the grand canonical measure with density ρ _c ; the remaining particles concentrate (condensate) on a single site.     

Onsager-Machlup Theory for Nonequilibrium Steady States and Fluctuation Theorems

A generalization of the Onsager-Machlup theory from equilibrium to nonequilibrium steady states and its connection with recent fluctuation theorems are discussed for a dragged particle restricted by a harmonic potential in a heat reservoir. Using a functional integral approach, the probability functional for a path is expressed in terms of a Lagrangian function from which an entropy production rate and dissipation functions are introduced, and nonequilibrium thermodynamic relations like the energy conservation law and the second law of thermodynamics are derived. Using this Lagrangian function we establish two nonequilibrium detailed balance relations, which not only lead to a fluctuation theorem for work but also to one related to energy loss by friction. In addition, we carried out the functional integral for heat explicitly, leading to the extended fluctuation theorem for heat. We also present a simple argument for this extended fluctuation theorem in the long time limit. PACS numbers: 05.70.Ln, 05.40.-a, 05.10.Gg.     

A non-equilibrium Ising model of turbulence

We introduce a model of interacting lattices at different resolutions driven by the two-dimensional Ising dynamics with a nearest-neighbor interaction. We study this model both with tools borrowed from equilibrium statistical mechanics as well as non-equilibrium thermodynamics. Our findings show that this model keeps the signature of the equilibrium phase transition. The critical temperature of the equilibrium models corresponds to the state maximizing the entropy and delimits two out-of-equilibrium regimes, one satisfying the Onsager relations for systems close to equilibrium and one resembling convective turbulent states. Since the model preserves the entropy and energy fluxes in the scale space, it seems a good candidate for parametric studies of out-of-equilibrium turbulent systems.     

Analysis of Vector Meson Transitions Among Heavy Quarkonium States

In this article,we study the vector meson transitions among the charmonium and bottomonium states with the heavy quark effective theory in a systematic way,and make predictions for the ratios among the vector meson decay widths of a special multiplet to another multiplet.The predictions can be confronted with the experimental data in the future.     

The Equilibrium States for a Model with Two Kinds of Bose Condensation

We study the equilibrium Gibbs states for a Boson gas model, defined by Bru and Zagrebnov, which has two phase transitions of the Bose condensation type. The two phase transitions correspond to two distinct mechanisms by which these condensations can occur. The first (non-conventional) Bose condensation is mediated by a zero-mode interaction term in the Hamiltonian. The second is a transition due to saturation quite similar to the conventional Bose–Einstein (BE) condensation in the ideal Bose gas. Due to repulsive interaction in non-zero modes the model manifests a generalized type III; i.e., non-extensive BE condensation. Our main result is that, as in the ideal Bose gas, the conventional condensation is accompanied by a loss of strong equivalence of the canonical and grand canonical ensembles whereas the non-conventional one, due to the interaction, does not break the equivalence of ensembles, at least not on the level of the gauge invariant states. It is also interesting to note that the type of (generalized) condensate, I, II, or III (in the terminology of van den Berg, Lewis, and Pulé), has no effect on the equivalence of ensembles. These results are proved by computing the generating functional of the cyclic representation of the Canonical Commutation Relation (CCR) for the corresponding equilibrium Gibbs states.     

Computer simulation of shock waves in the completely asymmetric simple exclusion process

We study the evolution of the completely asymmetric simple exclusion process in one dimension, with particles moving only to the right, for initial configurations corresponding to average density _– ( ₊) left (right) of the origin, _{– +}. The microscopic shock position is identified by introducing a second-class particle. Results indicate that the shock profile is stable, and that the distribution as seen from the shock positionN(t) tends, as time increases, to a limiting distribution, which is locally close to an equilibrium distribution far from the shock. Moreover , withV=1– _–– ₊, as predicted, and the dispersion ofN(t), ²(t), behaves linearly, for not too small values of ₊– _–, i.e., , whereS is equal, up to a scaling factor, to the valueS _WA predicted in the weakly asymmetric case. For ₊= _– we find agreement with the conjecture .Dedicated to the memory of Paola Calderoni.