Harmonically Confined Particles with Long-Range Repulsive Interactions.

We study an interacting system of N classical particles on a line at thermal equilibrium. The particles are confined by a harmonic trap and repel each other via pairwise interaction potential that behaves as a power law ∝∑[under i≠j][over 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→0^{+}), and the Calogero-Moser model (k=2). Because of 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.

Width Scaling of an Interface Constrained by a Membrane.

We investigate the shape of a growing interface in the presence of an impenetrable moving membrane. The two distinct geometrical arrangements of the interface and membrane, obtained by placing the membrane behind or ahead of the interface, are not symmetrically related. On the basis of numerical results and an exact calculation, we argue that these two arrangements represent two distinct universality classes for interfacial growth: while the well-established Kardar-Parisi-Zhang (KPZ) growth is obtained in the "ahead" arrangement, we find an arrested KPZ growth with a smaller roughness exponent in the "behind" arrangement. This suggests that the surface properties of growing cell membranes and expanding bacterial colonies, for example, are fundamentally distinct.

Driven tracers in narrow channels.

Steady-state properties of a driven tracer moving in a narrow two-dimensional (2D) channel of quiescent medium are studied. The tracer drives the system out of equilibrium, perturbs the density and pressure fields, and gives the bath particles a nonzero average velocity, creating a current in the channel. Three models in which the confining effect of the channel is probed are analyzed and compared in this study: the first is the simple symmetric exclusion process (SSEP), for which the stationary density profile and the pressure on the walls in the frame of the tracer are computed. We show that the tracer acts like a dipolar source in an average velocity field. The spatial structure of this 2D strip is then simplified to a one-dimensional (1D) SSEP, in which exchanges of position between the tracer and the bath particles are allowed. Using a combination of mean-field theory and exact solution in the limit where no exchange is allowed gives good predictions of the velocity of the tracer and the density field. Finally, we show that results obtained for the 1D SSEP with exchanges also apply to a gas of overdamped hard disks in a narrow channel. The correspondence between the parameters of the SSEP and of the gas of hard disks is systematic and follows from simple intuitive arguments. Our analytical results are checked numerically.

Nonequilibrium ensemble inequivalence and large deviations of the density in the ABC model.

We consider the one-dimensional driven ABC model under particle-conserving and particle-nonconserving processes. Two limiting cases are studied: (a) The rates of the nonconserving 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 nonconserving model, derived from G(r), is found to be different from the conserving one. This difference, which stems from the nonconvexity 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 nonconserving 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 nonconserving models do not coincide. These are new features which are present only when the rates of the conserving and nonconserving 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.

Phase diagram and density large deviations of a nonconserving ABC model.

The effect of particle-nonconserving processes on the steady state of driven diffusive systems is studied within the context of a generalized ABC model. It is shown that in the limit of slow nonconserving processes, the large deviation function of the overall particle density can be computed by making use of the steady-state density profile of the conserving model. In this limit one can define a chemical potential and identify first order transitions via Maxwell's construction, similarly to what is done in equilibrium systems. This method may be applied to other driven models subjected to slow nonconserving dynamics.

Long-range correlations and ensemble inequivalence in a generalized ABC model.

A generalization of the ABC model, a one-dimensional model of a driven system of three particle species with local dynamics, is introduced, in which the model evolves under either (i) density-conserving or (ii) nonconserving dynamics. For equal average densities of the three species, both dynamical models are demonstrated to exhibit detailed balance with respect to a Hamiltonian with long-range interactions. The model is found to exhibit two distinct phase diagrams, corresponding to the canonical (density-conserving) and grand canonical (density nonconserving) ensembles, as expected in long-range interacting systems. The implications of this result to nonequilibrium steady states, such as those of the ABC model with unequal average densities, are briefly discussed.

Supercoil formation in DNA denaturation.

We generalize the Poland-Scheraga model to the case of a circular DNA, taking into account the twisting of the two strains around each other. Guided by recent single-molecule experiments on DNA strands, we assume that the torsional stress induced by denaturation enforces the formation of supercoils whose writhe absorbs the linking number expelled by the loops. Our model predicts that when the entropy parameter of a loop satisfies c2, a first-order denaturation transition is consistent with our model and may take place in the actual system, as in the case with no supercoils. These results are in contrast with other treatments of circular DNA melting where denaturation is assumed to be accompanied by an increase in twist rather than writhe on the bound segments.

Dynamics of DNA melting.

The dynamics of loops at the DNA denaturation transition is studied. A scaling argument is used to evaluate the asymptotic behavior of the autocorrelation function of the state of complementary bases (either open or closed). The long-time asymptotic behavior of the autocorrelation function is expressed in terms of the entropy exponent, c, of a loop. The validity of the scaling argument is tested using a microscopic model of an isolated loop and a toy model of interacting loops. This suggests a method for measuring the entropy exponent using single-molecule experiments such as fluorescence correlation spectroscopy.

Loop dynamics in DNA denaturation.

The dynamics of a loop in DNA molecules at the denaturation transition is studied by scaling arguments and numerical simulations. The autocorrelation function of the state of complementary bases (either closed or open) is calculated. The long-time decay of the autocorrelation function is expressed in terms of the loop exponent c both for homopolymers and heteropolymers. This suggests an experimental method for measuring the exponent c using florescence correlation spectroscopy.

Critical phase in nonconserving zero-range processes and rewiring networks.

Zero-range processes, in which particles hop between sites on a lattice, are closely related to rewiring networks, in which rewiring of links between nodes takes place. Both systems exhibit a condensation transition for appropriate choices of the dynamical rules. The transition results in a macroscopically occupied site for zero-range processes and a macroscopically connected node for networks. Criticality, characterized by a scale-free distribution, is obtained only at the transition point. This is in contrast with the widespread scale-free complex networks. Here we propose a generalization of these models whereby criticality is obtained throughout an entire phase, and the scale-free distribution does not depend on any fine-tuned parameter.

Breaking of ergodicity and long relaxation times in systems with long-range interactions.

The thermodynamic and dynamical properties of an Ising model with both short-range and long-range, mean-field-like, interactions are studied within the microcanonical ensemble. It is found that the relaxation time of thermodynamically unstable states diverges logarithmically with system size. This is in contrast with the case of short-range interactions where this time is finite. Moreover, at sufficiently low energies, gaps in the magnetization interval may develop to which no microscopic configuration corresponds. As a result, in local microcanonical dynamics the system cannot move across the gap, leading to breaking of ergodicity even in finite systems. These are general features of systems with long-range interactions and are expected to be valid even when the interaction is slowly decaying with distance.

Directed percolation with long-range interactions: Modeling nonequilibrium wetting.

It is argued that some phase transitions observed in models of nonequilibrium wetting phenomena are related to contact processes with long-range interactions. This is investigated by introducing a model where the activation rate of a site at the edge of an inactive island of length l is 1+a l(-sigma) . Mean-field analysis and numerical simulations indicate that for sigma>1 the transition is continuous and belongs to the universality class of directed percolation, while for 0

Hard disks in narrow channels.

The thermodynamic and dynamical behavior of a gas of hard disks in a narrow channel is studied theoretically and numerically. Using a virial expansion, we find that the pressure and collision frequency curves exhibit a singularity at a channel width corresponding to twice the disk diameter. As expected, the maximum Lyapunov exponent is also found to display a similar behavior. At high density, these curves are dominated by solidlike configurations which are different from the bulk ones, due to the channel boundary conditions.

Wetting under nonequilibrium conditions.

We report a detailed account of the phase diagram of a recently introduced model for nonequilibrium wetting in (1+1) dimensions [H. Hinrichsen, R. Livi, D. Mukamel, and A. Politi, Phys. Rev. Lett. 79, 2710 (1997)]. A mean-field approximation is shown to reproduce the main features of the phase diagram, while providing indications for the behavior of the wetting transition in higher dimensions. The mean-field phase diagram is found to exhibit an extra transition line which does not exist in (1+1) dimensions. The line separates a phase in which the interface height distribution decays exponentially at large heights from a superexponentially decaying phase. Implications to wetting in dimensions higher than (1+1) are discussed.

Phase-separation transition in one-dimensional driven models.

A class of models of two-species driven diffusive systems which is shown to exhibit phase separation in d=1 dimensions is introduced. Unlike previously studied models exhibiting similar phenomena, here the relative density of the two species is fluctuating within the macroscopic domain of the phase separtated state. The nature of the phase transition from the homogeneous to the phase-separated state is discussed in view of a recently introduced criterion for phase separation in one-dimensional driven systems.

Griffiths singularities in unbinding of strongly disordered polymers.

Griffiths singularities occurring in the unbinding of strongly disordered heteropolymers are studied. A model with two randomly distributed binding energies, -1 and -v, is introduced and studied analytically by analyzing the Lee-Yang zeros of the partition sum. It is demonstrated that in the limit v--> infinity the model exhibits a Griffiths type singularity at a temperature T(G)=O(1) corresponding to melting of long homogeneous domains of the low binding energy. For finite v>>1 the model is expected to exhibit an additional, unbinding, transition at a high temperature T(M)=O(v).

Interstrand distance distribution of DNA near melting.

The distance distribution between complementary base pairs of the two strands of a DNA molecule is studied near the melting transition. Scaling arguments are presented for a generalized Poland-Scheraga-type model that includes self-avoiding interactions. At the transition temperature and for a large distance r, the distribution decays as 1/r(kappa) with kappa=1+(c-2)/nu. Here nu is the self-avoiding walk correlation length exponent and c is the exponent associated with the entropy of an open loop in the chain. Results for the distribution function just below the melting point are also presented. Numerical simulations that fully take into account the self-avoiding interactions are in good agreement with the scaling approach.

From multiplicative noise to directed percolation in wetting transitions.

A simple one-dimensional microscopic model of the depinning transition of an interface from an attractive hard wall is introduced and investigated. Upon varying a control parameter, the critical behavior observed along the transition line changes from a directed-percolation type to a multiplicative-noise type. Numerical simulations allow for a quantitative study of the multicritical point separating the two regions. Mean-field arguments and the mapping on yet a simpler model provide some further insight on the overall scenario.

Criterion for phase separation in one-dimensional driven systems.

A general criterion for the existence of phase separation in driven density-conserving one-dimensional systems is proposed. It is suggested that phase separation is related to the size dependence of the steady-state currents of domains in the system. A quantitative criterion for the existence of phase separation is conjectured using a correspondence made between driven diffusive models and zero-range processes. The criterion is verified in all cases where analytical results are available, and predictions for other models are provided.