Scaling and Selection in Cellular Structures and Living Polymers

The dynamical behavior of two types of non-equilibrium systems is discussed: $(a)$ two-dimensional cellular structures, and $(b)$ living polymers. Simple models governing their evolution are introduced and steady state distributions (cell side in the case of cellular structures and length in the case of living polymers) are calculated. In both cases the models possess a one parameter family of steady state distributions. Selection mechanism by which a particular distribution is dynamically selected is discussed.Comment: 12 pages, 2 figures, two seminars presented at the 1994 Les Houches Summer School ``Fluctuating Geometries in Statistical Mechanics and Field Theory.'' (also available at http://xxx.lanl.gov/lh94/

Non-equilibrium ensemble inequivalence and density large deviations in the ABC model

We consider the one-dimensional driven ABC model under particle-conserving and particle-non-conserving processes. Two limiting cases are studied: (a) the rates of the non-conserving processes are vanishingly slow compared with the conserving processes in the thermodynamic limit and (b) the two rates are comparable. For case (a) we provide a detailed analysis of the phase diagram and the large deviations function of the overall density, G(r). The phase diagram of the non-conserving model, derived from G(r), is found to be different from the conserving one. This difference, which stems from the non-convexity of G(r), is analogous to ensemble inequivalence found in equilibrium systems with long-range interactions. An outline of the analysis of case (a) was given in an earlier letter. For case (b) we show that unlike the conserving model, the non-conserving model exhibits a moving density profile in the steady-state with a velocity that remains finite in the thermodynamic limit. Moreover, in contrast with case (a), the critical lines of the conserving and non-conserving models do not coincide. These are new features which are present only when the rates of the conserving and non-conserving processes are comparable. In addition, we analyze G(r) in case (b) using macroscopic fluctuations theory. Much of the derivation presented in this paper is applicable to any driven-diffusive system coupled to an external particle-bath via a slow dynamics.Comment: 19 pages, 8 figure

Quasistationarity in a model of classical spins with long-range interactions

Systems with long-range interactions, while relaxing towards equilibrium, sometimes get trapped in long-lived non-Boltzmann quasistationary states (QSS) which have lifetimes that grow algebraically with the system size. Such states have been observed in models of globally coupled particles that move under Hamiltonian dynamics either on a unit circle or on a unit spherical surface. Here, we address the ubiquity of QSS in long-range systems by considering a different dynamical setting. Thus, we consider an anisotropic Heisenberg model consisting of classical Heisenberg spins with mean-field interactions and evolving under classical spin dynamics. Our analysis of the corresponding Vlasov equation for time evolution of the phase space distribution shows that in a certain energy interval, relaxation of a class of initial states occurs over a timescale which grows algebraically with the system size. We support these findings by extensive numerical simulations. This work further supports the generality of occurrence of QSS in long-range systems evolving under Hamiltonian dynamics.Comment: 12 pages, 3 figures; v2: minor revisio

Relaxation dynamics of stochastic long-range interacting systems

Long-range interacting systems, while relaxing towards equilibrium, may get trapped in nonequilibrium quasistationary states (QSS) for a time which diverges algebraically with the system size. These intriguing non-Boltzmann states have been observed under deterministic Hamiltonian evolution of a paradigmatic system, the Hamiltonian Mean-Field (HMF) model. We study here the robustness of QSS with respect to stochastic processes beyond deterministic dynamics within a microcanonical ensemble. To this end, we generalize the HMF model by allowing for stochastic three-particle collision dynamics in addition to the deterministic ones. By analyzing the resulting Boltzmann equation for the phase space density, we demonstrate that in the presence of stochasticity, QSS occur only as a crossover phenomenon over a finite time determined by the strength of the stochastic process. In particular, we argue that the relaxation time to equilibrium does not scale algebraically with the system size. We propose a scaling form for the relaxation time which is in very good agreement with results of extensive numerical simulations. The broader validity of these results is tested on a different stochastic HMF model involving microcanonical Monte Carlo dynamical moves.Comment: 23 pages, 6 figures. v2: minor changes with added references, published versio

Mixed order transition and condensation in exactly soluble one dimensional spin model

Mixed order phase transitions (MOT), which display discontinuous order parameter and diverging correlation length, appear in several seemingly unrelated settings ranging from equilibrium models with long-range interactions to models far from thermal equilibrium. In a recent paper [1] an exactly soluble spin model with long-range interactions that exhibits MOT was introduced and analyzed both by a grand canonical calculation and a renormalization group analysis. The model was shown to lay a bridge between two classes of one dimensional models exhibiting MOT, namely between spin models with inverse distance square interactions and surface depinning models. In this paper we elaborate on the calculations done in [1]. We also analyze the model in the canonical ensemble, which yields a better insight into the mechanism of MOT. In addition, we generalize the model to include Potts and general Ising spins, and also consider a broader class of interactions which decay with distance with a power law different from 2.Comment: 36 pages, 11 figures; Updated affiliation and email of author

Ensemble inequivalence : Landau theory and the ABC model

It is well known that systems with long-range interactions may exhibit different phase diagrams when studied within two different ensembles. In many of the previously studied examples of ensemble inequivalence, the phase diagrams differ only when the transition in one of the ensembles is first order. By contrast, in a recent study of a generalized ABC model, the canonical and grand-canonical ensembles of the model were shown to differ even when they both exhibit a continuous transition. Here we show that the order of the transition where ensemble inequivalence may occur is related to the symmetry properties of the order parameter associated with the transition. This is done by analyzing the Landau expansion of a generic model with long-range interactions. The conclusions drawn from the generic analysis are demonstrated for the ABC model by explicit calculation of its Landau expansion.Comment: 23 pages, 6 figure

Quasistationarity in a long-range interacting model of particles moving on a sphere

We consider a long-range interacting system of $N$ particles moving on a spherical surface under an attractive Heisenberg-like interaction of infinite range, and evolving under deterministic Hamilton dynamics. The system may also be viewed as one of globally coupled Heisenberg spins. In equilibrium, the system has a continuous phase transition from a low-energy magnetized phase, in which the particles are clustered on the spherical surface, to a high-energy homogeneous phase. The dynamical behavior of the model is studied analytically by analyzing the Vlasov equation for the evolution of the single-particle distribution, and numerically by direct simulations. The model is found to exhibit long lived non-magnetized quasistationary states (QSSs) which in the thermodynamic limit are dynamically stable within an energy range where the equilibrium state is magnetized. For finite $N$, these states relax to equilibrium over a time that increases algebraically with $N$. In the dynamically unstable regime, non-magnetized states relax fast to equilibrium over a time that scales as $\log N$. These features are retained in presence of a global anisotropy in the magnetization.Comment: 9 pages, 4 figures; v2: refs. added, published versio

Condensation transition and drifting condensates in the accelerated exclusion process

Recently, it was shown that spatial correlations may have a drastic effect on the dynamics of real-space condensates in driven mass-transport systems: in models with a spatially correlated steady state, the condensate is quite generically found to drift with a non-vanishing velocity. Here we examine the condensate dynamics in the accelerate exclusion process (AEP), where spatial correlations are present. This model is a "facilitated" generalization of the totally asymmetric simple exclusion process (TASEP) where each hopping particle may trigger another hopping event. Within a mean-field approach that captures some of the effects of correlations, we calculate the phase diagram of the AEP, analyze the nature of the condensation transition, and show that the condensate drifts, albeit with a velocity that vanishes in the thermodynamic limit. Numerical simulations are consistent with the mean-field phase diagram.Comment: 16 pages, 10 figure

Mixed order phase transition in a one dimensional model

We introduce and analyze an exactly soluble one-dimensional Ising model with long range interactions which exhibits a mixed order transition (MOT), namely a phase transition in which the order parameter is discontinuous as in first order transitions while the correlation length diverges as in second order transitions. Such transitions are known to appear in a diverse classes of models which are seemingly unrelated. The model we present serves as a link between two classes of models which exhibit MOT in one dimension, namely, spin models with a coupling constant which decays as the inverse distance squared and models of depinning transitions, thus making a step towards a unifying framework.Comment: 6 pages, 4 figures, includes supplementary materia

Asymmetric exclusion model for mixed ionic condustors

The ionic conductivity of mixed alkali glasses exhibits a deep minimum as a function of the relative concentrations of the two alkali ions. To study this behaviour we consider a simple one-dimensional model for asymmetric diffusion of two kinds of particles. Different particles are assumed to repulse each other. We consider two versions of the model: with or without overtaking of particles. For the case of perfect repulsion we find exact expressions for the stationary current. The model with weaker repulsion is studied by means of numerical simulations. The stationary current as a function of the ratio of particle concentrations is found to exhibit a minimum, related to correlations existing in this system.Comment: revised version (Some minor errors were corrected and some changes made.), 16 pages RevTeX and 3 postscript figure