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We investigate the full counting statistics (FCS) of a harmonically confined 1d short-range Riesz gas consisting of $N$ particles in equilibrium at finite temperature. The particles interact with each other through a repulsive power-law interaction with an exponent $k>1$ which includes the Calogero-Moser model for $k=2$. We examine the probability distribution of the number of particles in a finite domain $[-W, W]$ called number distribution, denoted by $\mathcal{N}(W, N)$. We analyze the probability distribution of $\mathcal{N}(W, N)$ and show that it exhibits a large deviation form for large $N$ characterised by a speed $N^{\frac{3k+2}{k+2}}$ and by a large deviation function of the fraction $c = \mathcal{N}(W, N)/N$ of the particles inside the domain and $W$. We show that the density profiles that create the large deviations display interesting shape transitions as one varies $c$ and $W$. This is manifested by a third-order phase transition exhibited by the large deviation function that has discontinuous third derivatives. Monte-Carlo (MC) simulations show good agreement with our analytical expressions for the corresponding density profiles. We find that the typical fluctuations of $\mathcal{N}(W, N)$, obtained from our field theoretic calculations are Gaussian distributed with a variance that scales as $N^{\nu_k}$, with $\nu_k = (2-k)/(2+k)$. We also present some numerical findings on the mean and the variance. Furthermore, we adapt our formalism to study the index distribution (where the domain is semi-infinite $(-\infty, W])$, linear statistics (the variance), thermodynamic pressure and bulk modulus., Comment: 36 pages, 7 figures

Ensemble inequivalence, i.e. the possibility of observing different thermodynamic properties depending on the statistical ensemble which describes the system, is one of the hallmarks of long-range physics, which has been demonstrated in numerous classical systems. Here, an example of ensemble inequivalence of a long-range quantum ferromagnet is presented. While the $T=0$ microcanonical quantum phase-diagram coincides with that of the canonical ensemble, the phase-diagrams of the two ensembles are different at finite temperature. This is in contrast with the common lore of statistical mechanics of systems with short-range interactions where thermodynamic properties are bound to coincide for macroscopic systems described by different ensembles. The consequences of these findings in the context of atomic, molecular and optical (AMO) setups are delineated., Comment: 5 pages, 3 figures

We study the stability of the ordered phase of flocking models with a scalar order parameter. Using both the active Ising model and a hydrodynamic description, we show that droplets of particles moving in the direction opposite to that of the ordered phase nucleate and grow. We characterize analytically this self-similar growth and demonstrate that droplets spread ballistically in all directions. Our results imply that, in the thermodynamic limit, discrete-symmetry flocks -- and, by extension, continuous-symmetry flocks with rotational anisotropy -- are metastable in all dimensions.

Mixed order phase transitions are transitions which have common features with both first order and second order transitions. I review some results obtained in the context of one of the prototypical models of mixed order transitions, the one-dimensional Ising model with long-range coupling that decays as truncated inverse square distance between spins. The correspondence between this model and the Poland Scheraga model of DNA denaturation, a subject to which Michael Fisher made substantial contribution, is then outlined., Comment: 10 pages, 5 figures, Michael Fisher Memorial volume References reordered and some comments added

We study the distribution of the position of the rightmost particle $x_{\max}$ in a $N$-particle Riesz gas in one dimension confined in a harmonic trap. The particles interact via long-range repulsive potential, of the form $r^{-k}$ with $-2-2$. We also find that these large deviation functions describe a pulled to pushed type phase transition as observed in Dyson's log-gas ($k\to 0$) and $1d$ one component plasma ($k=-1$). Remarkably, we find that the phase transition remains $3^{\rm rd}$ order for the entire regime. Our results demonstrate the striking universality of the $3^{\rm rd}$ order transition even in models that fall outside the paradigm of Coulomb systems and the random matrix theory. We numerically verify our analytical expressions of the large deviation functions via Monte Carlo simulation using an importance sampling algorithm., Comment: 42 pages, 13 figures

A pump coupled to a conserved density generates long-range modulations, resulting from the non-equilibrium nature of the dynamics. We study how these modulations are modified at the critical point where the system exhibits intrinsic long-range correlations. To do so, we consider a pump in a diffusive fluid, which is known to generate a density profile in the form of an electric dipole potential and a current in the form of a dipolar field above the critical point. We demonstrate that while the current retains its form at the critical point, the density profile changes drastically. At criticality, in $d<4$ dimensions, the deviation of the density from the average is given by ${\rm sgn}(\cos(\theta))|\cos({\theta})/r^{(d-1)}|^{1/\delta}$ at large distance $r$ from the pump and angle $\theta$ with respect to the pump's orientation. At short distances, there is a crossover to a $\cos({\theta})/r^{d-3+\eta}$ profile. Here $\delta$ and $\eta$ are Ising critical exponents. The effect of the local pump on the domain wall structure below the critical point is also considered., Comment: 5 pages, 1 figure

In this paper, we compute exactly the average density of a harmonically confined Riesz gas of $N$ particles for large $N$ in the presence of a hard wall. In this Riesz gas, the particles repel each other via a pairwise interaction that behaves as $|x_i - x_j|^{-k}$ for $k>-2$, with $x_i$ denoting the position of the $i^{\rm th}$ particle. This density can be classified into three different regimes of $k$. For $k \geq 1$, where the interactions are effectively short-ranged, the appropriately scaled density has a finite support over $[-l_k(w),w]$ where $w$ is the scaled position of the wall. While the density vanishes at the left edge of the support, it approaches a nonzero constant at the right edge $w$. For $-1

Emergent bath-mediated attraction and condensation arise when multiple particles are simultaneously driven through an equilibrated bath under geometric constraints. While such scenarios are observed in a variety of non-equilibrium phenomena, with an abundance of experimental and numerical evidence, little quantitative understanding of how these interactions arise is currently available. Here we approach the problem by studying the behavior of two driven "tracer" particles, propagating through a bath in a 1D lattice with excluded-volume interactions. We analytically explore the mechanism responsible for the tracers' emergent interactions and compute the resulting effective attractive potential. This mechanism is then numerically shown to extend to a realistic model of hard driven Brownian discs confined to a narrow 2D channel., Comment: 12 pages

The dynamics of a driven tracer in a quiescent bath subject to geometric confinement effectively models a broad range of phenomena. We explore this dynamics in a 1D lattice model where geometric confinement is tuned by varying particle overtaking rates. Previous studies of the model's stationary properties on a ring of $L$ sites have revealed a phase in which the bath density profile extends over an $\sim \mathcal{O}(L)$ distance from the tracer and the tracer's velocity vanishes as $\sim 1/L$. Here, we study the model's dynamics in this phase as $L\rightarrow \infty$ and for long times. We show that the bath density profile evolves on a $\sim \sqrt{t}$ time-scale and, correspondingly, that the tracer's velocity decays as $\sim 1/\sqrt{t}$. Unlike the well-studied non-driven tracer, whose dynamics becomes diffusive whenever overtaking is allowed, we here find that driving the tracer preserves its hallmark sub-diffusive single-file dynamics, even in the presence of overtaking., Comment: 11 pages, 9 figures

The effect of particle overtaking on transport in a narrow channel is studied using a 1d model of a driven tracer in a quiescent bath. In contrast with the well-studied non-driven case, where the tracer's long-time dynamics changes from sub-diffusive to diffusive whenever overtaking is allowed, the driven tracer is shown to exhibit a phase transition at a finite overtaking rate. The transition separates a phase in which the stationary bath density profile, as seen in the tracer's frame, is extended, as in the non-overtaking case, to a phase with a localized bath density profile. In the extended phase the tracer velocity vanishes in the thermodynamic limit while it remains finite in the localized phase. The phase diagram of the model, as well as the tracer velocity and the bath density profile in both phases, are studied, demonstrating their distinct features., Comment: 9 pages and 6 figures

The canonical phase diagram of the Blume-Emery-Griffiths (BEG) model with infinite-range interactions is known to exhibit a fourth order critical point at some negative value of the bi-quadratic interaction $K<0$. Here we study the microcanonical phase diagram of this model for $K<0$, extending previous studies which were restricted to positive $K$. A fourth order critical point is found to exist at coupling parameters which are different from those of the canonical ensemble. The microcanonical phase diagram of the model close to the fourth order critical point is studied in detail revealing some distinct features from the canonical counterpart., Comment: 9 pages, 10 figures

We study an interacting system of $N$ classical particles on a line at thermal equilibrium. The particles are confined by a harmonic trap and repelling each other via pairwise interaction potential that behaves as a power law $\propto \sum_{\substack{i\neq j}}^N|x_i-x_j|^{-k}$ (with $k>-2$) of their mutual distance. This is a generalization of the well known cases of the one component plasma ($k=-1$), Dyson's log-gas ($k\to 0^+$), and the Calogero-Moser model ($k=2$). Due to the competition between harmonic confinement and pairwise repulsion, the particles spread over a finite region of space for all $k>-2$. We compute exactly the average density profile for large $N$ for all $k>-2$ and show that while it is independent of temperature for sufficiently low temperature, it has a rich and nontrivial dependence on $k$ with distinct behavior for $-21$ and $k=1$., Comment: Main text: 6 pages + 1 Fig., Supp. Mat.: 11 pages + 3 Figs. Accepted for publication in Physical Review Letters

Mixed order transitions are those which show a discontinuity of the order parameter as well as a divergent correlation length. We show that the behaviour of the order parameter correlation function along the transition line of mixed order transitions can change from normal critical behaviour with power law decay, to fluctuation-dominated phase ordering as a parameter is varied. The defining features of fluctuation-dominated order are anomalous fluctuations which remain large in the thermodynamic limit, and correlation functions which approach a finite value through a cusp singularity as the separation scaled by the system size approaches zero. We demonstrate that fluctuation-dominated order sets in along a portion of the transition line of an Ising model with truncated long-range interactions which was earlier shown to exhibit mixed order transitions, and also argue that this connection should hold more generally.

A recently developed non-linear fluctuating hydrodynamics theory has been quite successful in describing various features of anomalous energy transport. However the diffusion and the noise terms present in this theory are not derived from microscopic descriptions but rather added phenomenologically. We here derive these hydrodynamic equations with explicit calculation of the diffusion and noise terms in a one-dimensional model. We show that in this model the energy current scales anomalously with system size $L$ as $\sim L^{-2/3}$ in the leading order with a diffusive correction of order $\sim L^{-1}$. The crossover length $\ell_{c}$ from diffusive to anomalous transport is expressed in terms of microscopic parameters. Our theoretical predictions are verified numerically., Comment: 12 pages, 3 figures

We study extreme value statistics (EVS) for spatially extended models exhibiting mixed order phase transitions (MOT). These are phase transitions which exhibit features common to both first order (discontinuity of the order parameter) and second order (diverging correlation length) transitions. We consider here the truncated inverse distance squared Ising (TIDSI) model which is a prototypical model exhibiting MOT, and study analytically the extreme value statistics of the domain lengths. The lengths of the domains are identically distributed random variables except for the global constraint that their sum equals the total system size $L$. In addition, the number of such domains is also a fluctuating variable, and not fixed. In the paramagnetic phase, we show that the distribution of the largest domain length $l_{\max}$ converges, in the large $L$ limit, to a Gumbel distribution. However, at the critical point (for a certain range of parameters) and in the ferromagnetic phase, we show that the fluctuations of $l_{\max}$ are governed by novel distributions which we compute exactly. Our main analytical results are verified by numerical simulations., Comment: 25 pages, 6 figures, 1 table

We study the effect of a one-dimensional driving field on the interface between two coexisting phases in a two dimensional model. This is done by considering an Ising model on a cylinder with Glauber dynamics in all sites and additional biased Kawasaki dynamics in the central ring. Based on the exact solution of the two-dimensional Ising model, we are able to compute the phase diagram of the driven model within a special limit of fast drive and slow spin flips in the central ring. The model is found to exhibit two phases where the interface is pinned to the central ring: one in which it fluctuates symmetrically around the central ring and another where it fluctuates asymmetrically. In addition, we find a phase where the interface is centered in the bulk of the system, either below or above the central ring of the cylinder. In the latter case, the symmetry breaking is `stronger' than that found in equilibrium when considering a repulsive potential on the central ring. This equilibrium model is analyzed here by using a restricted solid-on-solid model.

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 figures

We consider a simple one dimensional stochastic model of heat transport which locally conserves both energy and momentum and which is coupled to heat reservoirs with different temperatures at its two ends. The steady state is analyzed and the model is found to obey the Fourier law with finite heat conductivity. In the infinite length limit, the steady state is described locally by an equilibrium Gibbs state. However finite size corrections to this local equilibrium state are present. We analyze these finite size corrections by calculating the on-site fluctuations of the momentum and the two point correlation of the momentum and energy. These correlations are long ranged and have scaling forms which are computed explicitly. We also introduce a multi-lane variant of the model in which correlations vanish in the steady state. The deviation from local equilibrium in this model as expressed in terms of the on-site momentum fluctuations is calculated in the large length limit., Comment: 20 pages, 8 figures

We study statistics of the gaps in Random Average Process (RAP) on a ring with particles hopping symmetrically, except one tracer particle which could be driven. These particles hop either to the left or to the right by a random fraction $\eta$ of the space available till next particle in the respective directions. The random fraction $\eta \in [0,~1)$ is chosen from a distribution $R(\eta)$. For non-driven tracer, when $R(\eta)$ satisfies a necessary and sufficient condition, the stationary joint distribution of the gaps between successive particles takes an universal form that is factorized except for a global constraint. Some interesting explicit forms of $R(\eta)$ are found which satisfy this condition. In case of driven tracer, the system reaches a current-carrying steady state where such factorization does not hold. Analytical progress has been made in the thermodynamic limit, where we computed the single site mass distribution inside the bulk. We have also computed the two point gap-gap correlation exactly in that limit. Numerical simulations support our analytical results., Comment: 19 pages, 6 figures

We study the asymmetric zero-range process (ZRP) with L sites and open boundaries, conditioned to carry an atypical current. Using a generalized Doob h-transform we compute explicitly the transition rates of an effective process for which the conditioned dynamics are typical. This effective process is a zero-range process with renormalized hopping rates, which are space dependent even when the original rates are constant. This leads to non-trivial density profiles in the steady state of the conditioned dynamics, and, under generic conditions on the jump rates of the unconditioned ZRP, to an intriguing supercritical bulk region where condensates can grow. These results provide a microscopic perspective on macroscopic fluctuation theory (MFT) for the weakly asymmetric case: It turns out that the predictions of MFT remain valid in the non-rigorous limit of finite asymmetry. In addition, the microscopic results yield the correct scaling factor for the asymmetry that MFT cannot predict., Comment: 26 pages, 4 figures

In equilibrium, the effect of a spatially localised perturbation is typically confined around the perturbed region. Quite contrary to this, in a non-equilibrium stationary state often the entire system is affected. This appears to be a generic feature of non-equilibrium. We study such non-local response in the stationary state of a lattice gas with a shear drive at the boundary which keeps the system out of equilibrium. We show that a perturbation in the form of a localised blockage at the boundary, induces algebraically decaying density and current profile. In two examples, non-interacting particles and particles with simple exclusion, we analytically derive the power-law tail of the profiles., Comment: 32 pages, 7 figures

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 authors

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 figures

We show that the presence of a driven bond in an otherwise diffusive lattice gas with simple exclusion interaction results in long-range density-density correlation in its stationary state. In dimensions $d>1$ we show that in the thermodynamic limit this correlation decays as $C(r,s)\sim (r^2+s^2)^{-d}$ at large distances $r$ and $s$ away from the drive with $|r-s|>>1$. This is derived using an electrostatic analogy whereby $C(r,s)$ is expressed as the potential due to a configuration of electrostatic charges distributed in $2d$-dimension. At bulk density $\rho=1/2$ we show that the potential is that of a localized quadrupolar charge. At other densities the same is correct in leading order in the strength of the drive and is argued numerically to be valid at higher orders., Comment: 5 pages, 4 figures

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 material

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 version

We examine the effect of spatial correlations on the phenomenon of real-space condensation in driven mass-transport systems. We suggest that in a broad class of models with a spatially correlated steady state, the condensate drifts with a non-vanishing velocity. We present a robust mechanism leading to this condensate drift. This is done within the framework of a generalized zero-range process (ZRP) in which, unlike the usual ZRP, the steady state is not a product measure. The validity of the mechanism in other mass-transport models is discussed., Comment: 5 pages, 2 figures. For supplementary material see http://www.weizmann.ac.il/weizsites/mukamel/sm/cond-drift/

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 figures

The denaturation transition of circular DNA is studied within a Poland-Scheraga type approach, generalized to account for the fact that the total linking number (LK), which measures the number of windings of one strand around the other, is conserved. In the model the LK conservation is maintained by invoking both overtwisting and writhing (supercoiling) mechanisms. This generalizes previous studies which considered each mechanism separately. The phase diagram of the model is analyzed as a function of the temperature and the elastic constant $\kappa$ associated with the overtwisting energy for any given loop entropy exponent, $c$. As is the case where the two mechanisms apply separately, the model exhibits no denaturation transition for $c \le 2$. For $c>2$ and $\kappa=0$ we find that the model exhibits a first order transition. The transition becomes of higher order for any $\kappa>0$. We also calculate the contribution of the two mechanisms separately in maintaining the conservation of the linking number and find that it is weakly dependent on the loop exponent $c$., Comment: 10 pages, 6 figures

The effect of a localized drive on the steady state of an interface separating two phases in coexistence is studied. This is done using a spin conserving kinetic Ising model on a two dimensional lattice with cylindrical boundary conditions, where a drive is applied along a single ring on which the interface separating the two phases is centered. The drive is found to induce an interface spontaneous symmetry breaking whereby the magnetization of the driven ring becomes non-zero. The width of the interface becomes finite and its fluctuations around the driven ring are non-symmetric. The dynamical origin of these properties is analyzed in an adiabatic limit which allows the evaluation of the large deviation function of the driven-ring magnetization., Comment: 5 pages, 7 figures

Condensation transition in a non-Markovian zero-range process is studied in one and higher dimensions. In the mean-field approximation, corresponding to infinite range hopping, the model exhibits condensation with a stationary condensate, as in the Markovian case, but with a modified phase diagram. In the case of nearest-neighbor hopping, the condensate is found to drift by a "slinky" motion from one site to the next. The mechanism of the drift is explored numerically in detail. A modified model with nearest-neighbor hopping which allows exact calculation of the steady state is introduced. The steady state of this model is found to be a product measure, and the condensate is stationary., Comment: 31 pages, 9 figures

The statistical mechanics of DNA denaturation under fixed linking number is qualitatively different from that of the unconstrained DNA. Quantitatively different melting scenarios are reached from two alternative assumptions, namely, that the denatured loops are formed in expense of 1) overtwist, 2) supercoils. Recent work has shown that the supercoiling mechanism results in a BEC-like picture where a macroscopic loop appears at Tc and grows steadily with temperature, while the nature of the denatured phase for the overtwisting case has not been studied. By extending an earlier result, we show here that a macroscopic loop appears in the overtwisting scenario as well. We calculate its size as a function of temperature and show that the fraction of the total sum of microscopic loops decreases above Tc, with a cusp at the critical point., Comment: 5 pages, 3 figures, submitted for publication

The equation which describes a particle diffusing in a logarithmic potential arises in diverse physical problems such as momentum diffusion of atoms in optical traps, condensation processes, and denaturation of DNA molecules. A detailed study of the approach of such systems to equilibrium via a scaling analysis is carried out, revealing three surprising features: (i) the solution is given by two distinct scaling forms, corresponding to a diffusive (x ~ \sqrt{t}) and a subdiffusive (x >> \sqrt{t}) length scales, respectively; (ii) the scaling exponents and scaling functions corresponding to both regimes are selected by the initial condition; and (iii) this dependence on the initial condition manifests a "phase transition" from a regime in which the scaling solution depends on the initial condition to a regime in which it is independent of it. The selection mechanism which is found has many similarities to the marginal stability mechanism which has been widely studied in the context of fronts propagating into unstable states. The general scaling forms are presented and their practical and theoretical applications are discussed., Comment: 42 pages

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., Comment: 5 pages, 3 figures, submitted to Physics Review Letters

We investigate a new symmetry of the large deviation function of certain time-integrated currents in non-equilibrium systems. The symmetry is similar to the well-known Gallavotti-Cohen-Evans-Morriss-symmetry for the entropy production, but it concerns a different functional of the stochatic trajectory. The symmetry can be found in a restricted class of Markov jump processes, where the network of microscopic transitions has a particular structure and the transition rates satisfy certain constraints. We provide three physical examples, where time-integrated observables display such a symmetry. Moreover, we argue that the origin of the symmetry can be traced back to time-reversal if stochastic trajectories are grouped appropriately., Comment: 18 pages, 8 figures

The denaturation transition which takes place in circular DNA is analyzed by extending the Poland-Scheraga model to include the winding degrees of freedom. We consider the case of a homopolymer whereby the winding number of the double stranded helix, released by a loop denaturation, is absorbed by \emph{supercoils}. We find that as in the case of linear DNA, the order of the transition is determined by the loop exponent $c$. However the first order transition displayed by the PS model for $c>2$ in linear DNA is replaced by a continuous transition with arbitrarily high order as $c$ approaches 2, while the second-order transition found in the linear case in the regime $1

The ABC model is a driven diffusive exclusion model, composed of three species of particles that hop on a ring with local asymmetric rates. In the weak asymmetry limit, where the asymmetry vanishes with the length of the system, the model exhibits a phase transition between a homogenous state and a phase separated state. We derive the exact solution for the density profiles of the three species in the hydrodynamic limit for arbitrary average densities. The solution yields the complete phase diagram of the model and allows the study of the nature of the first order phase transition found for average densities that deviate significantly from the equal densities point., Comment: 19 pages, 6 figures, submitted to J. Phys. A

We show that the presence of a localized drive in an otherwise diffusive system results in steady-state density and current profiles that decay algebraically to their global average value, away from the drive in two or higher dimensions. An analogy to an electrostatic problem is established, whereby the density profile induced by a driving bond maps onto the electrostatic potential due to an electric dipole located along the bond. The dipole strength is proportional to the drive, and is determined self-consistently by solving the electrostatic problem. The profile resulting from a localized configuration of more than one driving bond can be straightforwardly determined by the superposition principle of electrostatics. This picture is shown to hold even in the presence of exclusion interaction between particles., Comment: Authors name has been corrected

The late-time distribution function P(x,t) of a particle diffusing in a one-dimensional logarithmic potential is calculated for arbitrary initial conditions. We find a scaling solution with three surprising features: (i) the solution is given by two distinct scaling forms, corresponding to a diffusive (x ~ t^(1/2)) and a subdiffusive (x ~ t^{\gamma} with a given {\gamma} < 1/2) length scale, respectively, (ii) the overall scaling function is selected by the initial condition, and (iii) depending on the tail of the initial condition, the scaling exponent which characterizes the scaling function is found to exhibit a transition from a continuously varying to a fixed value., Comment: 4 pages, 3 figures; Published version

Long-lived quasistationary states, associated with stationary stable solutions of the Vlasov equation, are found in systems with long-range interactions. Studies of the relaxation time in a model of $N$ globally coupled particles moving on a ring, the Hamiltonian Mean Field model (HMF), have shown that it diverges as $N^\gamma$ for large $N$, with $\gamma \simeq 1.7$ for some initial conditions with homogeneously distributed particles. We propose a method for identifying exact inhomogeneous steady states in the thermodynamic limit, based on analysing models of uncoupled particles moving in an external field. For the HMF model, we show numerically that the relaxation time of these states diverges with $N$ with the exponent $\gamma \simeq 1$. The method, applicable to other models with globally coupled particles, also allows an exact evaluation of the stability limit of homogeneous steady states. In some cases it provides a good approximation for the correspondence between the initial condition and the final steady state., Comment: 5 pages, 4 figures, revised, accepted for publication on Phys. Rev. E

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 revision

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 version

We introduce a model of uncoupled pendula, which mimics the dynamical behavior of the Hamiltonian Mean Field (HMF) model. This model has become a paradigm for long-range interactions, like Coulomb or dipolar forces. As in the HMF model, this simplified integrable model is found to obey the Vlasov equation and to exhibit Quasi Stationary States (QSS), which arise after a "collisionless" relaxation process. Both the magnetization and the single particle distribution function in these QSS can be predicted using Lynden-Bell's theory. The existence of an extra conserved quantity for this model, the energy distribution function, allows us to understand the origin of some discrepancies of the theory with numerical experiments. It also suggests us an improvement of Lynden-Bell's theory, which we fully implement for the zero field case., Comment: 15 pages, 8 figures, accepted in Philosophical Transactions of the Royal Society A

Quasistationary states are long-lived nonequilibrium states, observed in some systems with long-range interactions under deterministic Hamiltonian evolution. These intriguing non-Boltzmann states relax to equilibrium over times which diverge algebraically with the system size. To test the robustness of this phenomenon to non-deterministic dynamical processes, we have generalized the paradigmatic model exhibiting such a behavior, the Hamiltonian Mean-Field model, to include energy-conserving stochastic processes. Analysis, based on the Boltzmann equation, a scaling approach and numerical studies, demonstrates that in the long time limit, the system relaxes to the equilibrium state on timescales which do not diverge algebraically with the system size. Thus, quasistationarity takes place only as a crossover phenomenon on times determined by the strength of the stochastic process., Comment: 4+ pages, 1 figure; v2: minor changes, published version

A model for epidemic spreading on rewiring networks is introduced and analyzed for the case of scale free steady state networks. It is found that contrary to what one would have naively expected, the rewiring process typically tends to suppress epidemic spreading. In particular it is found that as in static networks, rewiring networks with degree distribution exponent $\gamma >3$ exhibit a threshold in the infection rate below which epidemics die out in the steady state. However the threshold is higher in the rewiring case. For $2<\gamma \leq 3$ no such threshold exists, but for small infection rate the steady state density of infected nodes (prevalence) is smaller for rewiring networks., Comment: 7 pages, 7 figures

Thermodynamic and dynamical properties of systems with long-range pairwise interactions (LRI), which decay as $1/r^{d+\sigma}$ at large distances $r$ in $d$ dimensions, are reviewed. Two broad classes of such systems are discussed. (i) Systems with a slow decay of the interactions, termed "strong" LRI, where the energy is super-extensive. These systems are characterized by unusual properties such as inequivalence of ensembles, negative specific heat, slow decay of correlations, anomalous diffusion and ergodicity breaking. (ii) Systems with faster decay of the interaction potential, where the energy is additive, thus resulting in less dramatic effects. These interactions affect the thermodynamic behavior of systems near phase transitions, where long-range correlations are naturally present. Long-range correlations are often present in systems driven out of equilibrium when the dynamics involves conserved quantities. Steady state properties of driven systems with local dynamics are considered within the framework outlined above., Comment: Partly based on lectures given by one of the authors (DM) at the school "Fundamental Problems in Statistical Physics: XII" held at Leuven, Belgium (2009). 37 pages, 7 figures. v2: minor changes, published version

A simple two-species asymmetric exclusion model in one dimension with bulk and boundary exchanges of particles is investigated for the existence of spontaneous symmetry breaking. The model is a generalization of the bridge model for which earlier studies have confirmed the existence of symmetry-broken phases, and the motivation here is to check the robustness of the observed symmetry breaking with respect to additional dynamical moves, in particular, the boundary exchange of the two species of particles. Our analysis, based on general considerations, mean-field approximation and numerical simulations, shows that the symmetry breaking in the bridge model is sustained for a range of values of the boundary exchange rate. Moreover, the mechanism through which symmetry is broken is similar to that in the bridge model. Our analysis allows us to plot the complete phase diagram of the model, demarcating regions of symmetric and symmetry-broken phases., Comment: 26 pages, 12 figures, v2: minor changes with an added appendix, published version

Condensation phenomena in non-equilibrium systems have been modeled by the zero-range process, which is a model of particles hopping between boxes with Markovian dynamics. In many cases, memory effects in the dynamics cannot be neglected. In an attempt to understand the possible impact of temporal correlations on the condensate, we introduce and study a process with non-Markovian zero-range dynamics. We find that memory effects have significant impact on the condensation scenario. Specifically, two main results are found: (1) In mean-field dynamics, the steady state corresponds to that of a Markovian ZRP, but with modified hopping rates which can affect condensation, and (2) for nearest-neighbor hopping in one dimension, the condensate occupies two adjacent sites on the lattice and drifts with a finite velocity. The validity of these results in a more general context is discussed., Comment: 4 pages, 3 figures

Thermodynamic and dynamical properties of systems with long range pairwise interactions (LRI) which decay as 1/r^{d+\sigma} at large distances r in $d$ dimensions are reviewed in these Notes. Two broad classes of such systems are identified: (a) systems with a slow decay of the interactions, termed "strong" LRI, where the energy is super-extensive. These systems are characterized by unusual properties such as inequivalence of ensembles, negative specific heat, slow decay of correlations and ergodicity breaking. And (b) systems with faster decay of the interaction potential where the energy is additive, thus resulting in less dramatic effects. These interaction affect the thermodynamic behavior of systems near phase transitions, where long range correlations are naturally present. Long range correlations are often present in systems driven out of equilibrium when the dynamics involves conserved quantities. Steady state properties of driven systems are considered within the framework outlined above., Comment: Notes of lectures given at a Summer School in Les Houches (France), 4-29 August 2008

Recent theoretical studies of statistical mechanical properties of systems with long range interactions are briefly reviewed. In these systems the interaction potential decays with a rate slower than 1/r^d at large distances r in d dimensions. As a result, these systems are non-additive and they display unusual thermodynamic and dynamical properties which are not present in systems with short range interactions. In particular, the various statistical mechanical ensembles are not equivalent and the microcanonical specific heat may be negative. Long range interactions may also result in breaking of ergodicity, making the maximal entropy state inaccessible from some regions of phase space. In addition, in many cases long range interactions result in slow relaxation processes, with time scales which diverge in the thermodynamic limit. Various models which have been found to exhibit these features are discussed., Comment: Published in AIP Conference Proceedings 970 "Dynamics and Thermodynamics of Systems with Long-Range Interactions: Theory and Experiments", Assisi, Italy 4-8 July 2007, editors A. Campa, A. Giansanti, G. Morigi and F. Sylos Labini, p. 22 (2008)