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.     

Exact Determination of the Phase Structure of a Multi-Species Asymmetric Exclusion Process

We consider a multi-species generalization of the Asymmetric Simple Exclusion Process on an open chain, in which particles hop with their characteristic hopping rates and fast particles can overtake slow ones. The number of species is arbitrary and the hopping rates can be selected from a discrete or continuous distribution. We determine exactly the phase structure of this model and show how the phase diagram of the 1-species ASEP is modified. Depending on the distribution of hopping rates, the system can exist in a three-phase regime or a two-phase regime. In the three-phase regime the phase structure is almost the same as in the one species case, that is, there are the low density, the high density and the maximal current phases, while in the two-phase regime there is no high-density phase.     

Hydrodynamic Limit of Brownian Particles Interacting with Short- and Long-Range Forces

We investigate the time evolution of a model system of interacting particles moving in a d-dimensional torus. The microscopic dynamics is first order in time with velocities set equal to the negative gradient of a potential energy term plus independent Brownian motions: is the sum of pair potentials, V(r)+ ^d J(r); the second term has the form of a Kac potential with inverse range . Using diffusive hydrodynamic scaling (spatial scale ^–1, temporal scale ^–2) we obtain, in the limit 0, a diffusive-type integrodifferential equation describing the time evolution of the macroscopic density profile.     

Hydrodynamic limit of a nongradient interacting particle process

A simple example of a nongradient stochastic interacting particle system is analyzed. In this model, symmetric simple exclusion in one dimension in a periodic environment, the dynamical term in the Green-Kubo formula contributes to the bulk diffusion constant. The law of large numbers for the density field and the central limit theorem for the density fluctuation field are proven, and the Green-Kubo expression for the diffusion constant is computed exactly. The hydrodynamic equation for the model turns out to be linear.     

Current Large Deviations for Asymmetric Exclusion Processes with Open Boundaries

We study the large deviation functional of the current for the Weakly Asymmetric Simple Exclusion Process in contact with two reservoirs. We compare this functional in the large drift limit to the one of the Totally Asymmetric Simple Exclusion Process, in particular to the Jensen-Varadhan functional. Conjectures for generalizing the Jensen-Varadhan functional to open systems are also stated. PACS: 02.50.-r, 05.40.-a, 05.70 Ln, 82.20-w     

Dynamics of a Tagged Particle in the Asymmetric Exclusion Process with the Step Initial Condition

The one-dimensional totally asymmetric simple exclusion process (TASEP) is considered. We study the time evolution property of a tagged particle in the TASEP with the step initial condition. Calculated is the multi-time joint distribution function of its position. Using the relation of the dynamics of the TASEP to the Schur process, we show that the function is represented as the Fredholm determinant. We also study the scaling limit. The universality of the largest eigenvalue in the random matrix theory is realized in the limit. When the hopping rates of all particles are the same, it is found that the joint distribution function converges to that of the Airy process after the time at which the particle begins to move. On the other hand, when there are several particles with small hopping rate in front of a tagged particle, the limiting process changes at a certain time from the Airy process to the process of the largest eigenvalue in the Hermitian multi-matrix model with external sources.     

Field Theory of Critical Behavior in Driven Diffusive Systems with Quenched Disorder

We present a field-theoretic renormalization-group study for the critical behavior of a uniformly driven diffusive system with quenched disorder, which is modeled by different kinds of potential barriers between sites. Due to their symmetry properties, these different realizations of the random potential barriers lead to three different models for the phase transition to transverse order and to one model for the phase transition to longitudinal order all belonging to distinct universality classes. In these four models, which have different upper critical dimensions d _c, we find the critical scaling behavior of the vertex functions in spatial dimensions d<d _c. The deviation from purely diffusive behavior is characterized by the anomaly exponent , which we calculate at first and second order, respectively, in =d _c–d. In each model turns out to be positive, which means superdiffusive spread of density fluctuations in the driving force direction.     

Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits

We present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter (r, t) of a binary alloy undergoing phase segregation. Our model is ad-dimensional lattice gas evolving via Kawasaki exchange with respect to the Gibbs measure for a Hamiltonian which includes both short-range (local) and long-range (nonlocal) interactions. The nonlocal part is given by a pair potential ^dJ(|x–y|), >0 x and y in ^d, in the limit 0. The macroscopic evolution is observed on the spatial scale ^–1 and time scale ^–2, i.e., the density (r, t) is the empirical average of the occupation numbers over a small macroscopic volume element centered atr=x. A rigorous derivation is presented in the case in which there is no local interaction. In a subsequent paper (Part II) we discuss the phase segregation phenomena in the model. In particular we argue that the phase boundary evolutions, arising as sharp interface limits of the family of equations derived in this paper, are the same as the ones obtained from the corresponding limits for the Cahn-Hilliard equation.     

The Hydrodynamic Limit of a Deterministic Particle System with Conservation of Mass and Momentum

We study the hydrodynamic limit of a deterministic one-dimensional particle system with nearest neighbour interaction and an additional regularizing force. Under its evolution mass and momentum are conserved. In the limit with Euler scaling their macroscopic distributions are shown to be governed by the compressible Navier–Stokes equations with a density dependent viscosity.     

Exact results for the asymmetric simple exclusion process with a blockage

We present new results for the current as a function of transmission rate in the one-dimensional totally asymmetric simple exclusion process (TASEP) with a blockage that lowers the jump rate at one site from one tor<1. Exact finitevolume results serve to bound the allowed values for the current in the infinite system. This proves the existence of a nonequilibrium phase transition, corresponding to an immiscibility gap in the allowed values of the asymptotic densities which the infinite system can have in a stationary state. A series expansion inr, derived from the finite systems, is proven to be asymptotic for all sufficiently large systems. Padé approximants based on this series, which make specific assumptions about the nature of the singularity atr=1, match numerical data for the infinite system to 1 part in 10⁴.     

A Markov Process Associated with a Boltzmann Equation Without Cutoff and for Non-Maxwell Molecules

Tanaka,⁽¹⁸⁾ showed a way to relate the measure solution {P _t } _t of a spatially homogeneous Boltzmann equation of Maxwellian molecules without angular cutoff to a Poisson-driven stochastic differential equation: {P _t } is the flow of time marginals of the solution of this stochastic equation. In the present paper, we extend this probabilistic interpretation to much more general spatially homogeneous Boltzmann equations. Then we derive from this interpretation a numerical method for the concerned Boltzmann equations, by using easily simulable interacting particle systems.     

Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws

We introduce a general framework for kinetic BGK models. We assume to be given a system of hyperbolic conservation laws with a family of Lax entropies, and we characterize the BGK models that lead to this system in the hydrodynamic limit, and that are compatible with the whole family of entropies. This is obtained by a new characterization of Maxwellians as entropy minimizers that can take into account the simultaneous minimization problems corresponding to the family of entropies. We deduce a general procedure to construct such BGK models, and we show how classical examples enter the framework. We apply our theory to isentropic gas dynamics and full gas dynamics, and in both cases we obtain new BGK models satisfying all entropy inequalities.     

Correlation Functions by Cluster Variation Method for Ising Model with NN, NNN, and Plaquette Interactions

We consider a procedure for calculating the pair correlation function in the context of cluster variation methods (CVM). As specific cases, we study the pair correlation function in the paramagnetic phase of the Ising model with nearest neighbors, next to nearest neighbors, and plaquette interactions in two and three dimensions. In presence of competing interactions, the so-called disorder line separates in the paramagnetic phase a region where the correlation function has the usual exponential behavior from a region where the correlation has an oscillating, exponentially damped behavior. In two dimensions, using the plaquette as the maximal cluster of the CVM approximation, we calculate the phase diagram and the disorder line for a case where a comparison is possible with known results for the eight-vertex model. In three dimensions, in the CVM cube approximation, we calculate the phase diagram and the disorder line in some cases of particular interest. The relevance of our results for experimental systems like mixtures of oil, water, and surfactant is also discussed.     

A New Macro Model for Traffic Flow on a Highway with Ramps and Numerical Tests

In this paper, we present a new macro model for traffic flow on a highway with ramps based on the existing models. We use the new model to study the effects of on-off-ramp on the main road traffic during the morning rush period and the evening rush period. Numerical tests show that, during the two rush periods, these effects are often different and related to the status of the main road traffic. If the main road traffic flow is uniform, then ramps always produce stop-and-go traffic when the main road density is between two critical values, and ramps have little effect on the main road traffic when the main road density is less than the smaller critical value or greater than the larger critical value. If a small perturbation appears on the main road, ramp may lead to stop-and-go traffic, or relieve or even eliminate the stop-and-go traffic, under different circumstances. These results are consistent with real traffic, which shows that the new model is reasonable.     

Delay times in a cellular traffic flow model for road sections with periodic outflow

A simple cellular automaton to model traffic flow is used to analyze the impact of traffic lights on travel times. The model is investigated on a stretch of road with open boundaries, where the inflow is driven stochastically, but the outflow is controlled by a traffic light with periodically changing outflow rates. Especially in the transition regime from free flow to oversaturated flow, the detailed space-time structure can be used to derive analytical lower and upper boundaries for the travel time function. The analytical results have been verified by a simulation study. The function displays a strong non-trivial dependency on the length of the system, which is at odds with a well-known formula derived almost 50 years ago by Webster. Such a model is interesting both from a theoretical point of view and for applications.     

Entropies and IPR as Markers for a Phase Transition in a Two-Level Model for Atom–Diatomic Molecule Coexistence

A quantum phase transition (QPT) in a simple model that describes the coexistence of atoms and diatomic molecules is studied. The model, which is briefly discussed, presents a second-order ground state phase transition in the thermodynamic (or large particle number) limit, changing from a molecular condensate in one phase to an equilibrium of diatomic molecules–atoms in coexistence in the other one. The usual markers for this phase transition are the ground state energy and the expected value of the number of atoms (alternatively, the number of molecules) in the ground state. In this work, other markers for the QPT, such as the inverse participation ratio (IPR), and particularly, the Rényi entropy, are analyzed and proposed as QPT markers. Both magnitudes present abrupt changes at the critical point of the QPT.