arXiv subjects

J. L. Lebowitz

Publications and source records attributed to J. L. Lebowitz.

86 records · Page 5Linked to original sources

arXiv1998

Shift Equivalence of Measures and the Intrinsic Structure of Shocks in the Asymmetric Simple Exclusion Process

We investigate properties of non-translation-invariant measures, describing particle systems on $\bbz$, which are asymptotic to different translation invariant measures on the left and on the right. Often the structure of the transition region can only be observed from a point of view which is random---in particular, configuration dependent. Two such measures will be called shift equivalent if they differ only by the choice of such a viewpoint. We introduce certain quantities, called translation sums, which, under some auxiliary conditions, characterize the equivalence classes. Our prime example is the asymmetric simple exclusion process, for which the measures in question describe the microscopic structure of shocks. In this case we compute explicitly the translation sums and find that shocks generated in different ways---in particular, via initial conditions in an infinite system or by boundary conditions in a finite system---are described by shift equivalent measures. We show also that when the shock in the infinite system is observed from the location of a second class particle, treating this particle either as a first class particle or as an empty site leads to shift equivalent shock measures.

cond-mat↗

arXiv1998

(Global and Local) Fluctuations of Phase Space Contraction in Deterministic Stationary Non-equilibrium

We studied numerically the validity of the fluctuation theorem, introduced by Evans,Cohen and Morris and proved by Gallavotti and Cohen, for a 2-dimensional system of particles maintained in a steady shear flow by Maxwell daemon boundary conditions (see Chernov and Lebowitz). The theorem was found to hold if one considers the total phase space contraction $σ$ occuring at collisions with both walls: $σ=σ^\su+σ^\giu$. An attempt to extend it to more local quantities $σ^\su$ and $σ^\giu$, corresponding to the collisions with the top or bottom wall only, gave negative results. The time decay of the correlations in $σ^{\su,\giu}$ was very slow compared to that of $σ$.

chao-dyn↗

arXiv1997

Shock Profiles for the Asymmetric Simple Exclusion Process in One Dimension

The asymmetric simple exclusion process (ASEP) on a one-dimensional lattice is a system of particles which jump at rates $p$ and $1-p$ (here $p>1/2$) to adjacent empty sites on their right and left respectively. The system is described on suitable macroscopic spatial and temporal scales by the inviscid Burgers' equation; the latter has shock solutions with a discontinuous jump from left density $ρ_-$ to right density $ρ_+$, $ρ_-<ρ_+$, which travel with velocity $(2p-1)(1-ρ_+-ρ_-)$. In the microscopic system we may track the shock position by introducing a second class particle, which is attracted to and travels with the shock. In this paper we obtain the time invariant measure for this shock solution in the ASEP, as seen from such a particle. The mean density at lattice site $n$, measured from this particle, approaches $ρ_{\pm}$ at an exponential rate as $n\to\pm\infty$, with a characteristic length which becomes independent of $p$ when $p/(1-p)>\sqrt{ρ_+(1-ρ_-)/ρ_-(1-ρ_+)}$. For a special value of the asymmetry, given by $p/(1-p)=ρ_+(1-ρ_-)/ρ_-(1-ρ_+)$, the measure is Bernoulli, with density $ρ_-$ on the left and $ρ_+$ on the right. In the weakly asymmetric limit, $2p-1\to0$, the microscopic width of the shock diverges as $(2p-1)^{-1}$. The stationary measure is then essentially a superposition of Bernoulli measures, corresponding to a convolution of a density profile described by the viscous Burgers equation with a well-defined distribution for the location of the second class particle.

cond-mat↗

arXiv1997

Phase Transitions in the Multicomponent Widom-Rowlinson Model and in Hard Cubes on the BCC--Lattice

We use Monte Carlo techniques and analytical methods to study the phase diagram of the M--component Widom-Rowlinson model on the bcc-lattice: there are M species all with the same fugacity z and a nearest neighbor hard core exclusion between unlike particles. Simulations show that for M greater or equal 3 there is a ``crystal phase'' for z lying between z_c(M) and z_d(M) while for z > z_d(M) there are M demixed phases each consisting mostly of one species. For M=2 there is a direct second order transition from the gas phase to the demixed phase while for M greater or equal 3 the transition at z_d(M) appears to be first order putting it in the Potts model universality class. For M large, Pirogov-Sinai theory gives z_d(M) ~ M-2+2/(3M^2) + ... . In the crystal phase the particles preferentially occupy one of the sublattices, independent of species, i.e. spatial symmetry but not particle symmetry is broken. For M to infinity this transition approaches that of the one component hard cube gas with fugacity y = zM. We find by direct simulations of such a system a transition at y_c ~ 0.71 which is consistent with the simulation z_c(M) for large M. This transition appears to be always of the Ising type.

cond-mat.stat-mech↗

arXiv1997

Ordering and Demixing Transitions in Multicomponent Widom-Rowlinson Models

We use Monte Carlo techniques and analytical methods to study the phase diagram of multicomponent Widom-Rowlinson models on a square lattice: there are M species all with the same fugacity z and a nearest neighbor hard core exclusion between unlike particles. Simulations show that for M between two and six there is a direct transition from the gas phase at z < z_d (M) to a demixed phase consisting mostly of one species at z > z_d (M) while for M \geq 7 there is an intermediate ``crystal phase'' for z lying between z_c(M) and z_d(M). In this phase, which is driven by entropy, particles, independent of species, preferentially occupy one of the sublattices, i.e. spatial symmetry but not particle symmetry is broken. The transition at z_d(M) appears to be first order for M \geq 5 putting it in the Potts model universality class. For large M the transition between the crystalline and demixed phase at z_d(M) can be proven to be first order with z_d(M) \sim M-2 + 1/M + ..., while z_c(M) is argued to behave as μ_{cr}/M, with μ_{cr} the value of the fugacity at which the one component hard square lattice gas has a transition, and to be always of the Ising type. Explicit calculations for the Bethe lattice with the coordination number q=4 give results similar to those for the square lattice except that the transition at z_d(M) becomes first order at M>2. This happens for all q, consistent with the model being in the Potts universality class.

cond-mat.stat-mech↗

arXiv1997

Kinetics of a Model Weakly Ionized Plasma in the Presence of Multiple Equilibria

We study, globaly in time, the velocity distribution $f(v,t)$ of a spatially homogeneous system that models a system of electrons in a weakly ionized plasma, subjected to a constant external electric field $E$. The density $f$ satisfies a Boltzmann type kinetic equation containing a full nonlinear electron-electron collision term as well as linear terms representing collisions with reservoir particles having a specified Maxwellian distribution. We show that when the constant in front of the nonlinear collision kernel, thought of as a scaling parameter, is sufficiently strong, then the $L^1$ distance between $f$ and a certain time dependent Maxwellian stays small uniformly in $t$. Moreover, the mean and variance of this time dependent Maxwellian satisfy a coupled set of nonlinear ODE's that constitute the ``hydrodynamical'' equations for this kinetic system. This remain true even when these ODE's have non-unique equilibria, thus proving the existence of multiple stabe stationary solutions for the full kinetic model. Our approach relies on scale independent estimates for the kinetic equation, and entropy production estimates. The novel aspects of this approach may be useful in other problems concerning the relation between the kinetic and hydrodynamic scales globably in time.

physics.plasm-ph↗

arXiv1997

Metastability in the two-dimensional Ising model with free boundary conditions

We investigate metastability in the two dimensional Ising model in a square with free boundary conditions at low temperatures. Starting with all spins down in a small positive magnetic field, we show that the exit from this metastable phase occurs via the nucleation of a critical droplet in one of the four corners of the system. We compute the lifetime of the metastable phase analytically in the limit $T\to 0$, $h\to 0$ and via Monte Carlo simulations at fixed values of $T$ and $h$ and find good agreement. This system models the effects of boundary domains in magnetic storage systems exiting from a metastable phase when a small external field is applied.

cond-mat.stat-mech↗

arXiv1997

Phase Segregation Dynamics In Particle Systems with Long Range Interactions II: Interface motion

We study properties of the solutions of a family of second order integro-differential equations, which describe the large scale dynamics of a class of microscopic phase segregation models with particle conserving dynamics. We first establish existence and uniqueness as well as some properties of the instantonic solutions. Then we concentrate on formal asymptotic (sharp interface) limits. We argue that the obtained interface evolution laws (a Stefan-like problem and the Mullins-Sekerka solidification model) coincide with the ones which can be obtained in the analogous limits from the Cahn-Hilliard equation, the fourth order PDE which is the standard macroscopic model for phase segregation with one conservation law.

patt-sol↗

arXiv1997

Phase Segregation Dynamics in Particle Systems with Long Range Interactions I: Macroscopic Limits

We present and discuss the derivation of a nonlinear non-local integro-differential equation for the macroscopic time evolution of the conserved order parameter of a binary alloy undergoing phase segregation. Our model is a d-dimensional lattice gas evolving via Kawasaki exchange dynamics, i.e. a (Poisson) nearest-neighbor exchange process, reversible with respect to the Gibbs measure for a Hamiltonian which includes both short range (local) and long range (nonlocal) interactions. 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.

comp-gas↗

arXiv1997

Spinodal Decomposition in Binary Gases

We carried out three-dimensional simulations, with about 1.4 million particles, of phase segregation in a low density binary fluid mixture, described mesoscopically by energy and momentum conserving Boltzmann-Vlasov equations. Using a combination of Direct Simulation Monte Carlo(DSMC) for the short range collisions and a version of Particle-In-Cell(PIC) evolution for the smooth long range interaction, we found dynamical scaling after the ratio of the interface thickness(whose shape is described approximately by a hyperbolic tangent profile) to the domain size is less than ~0.1. The scaling length R(t) grows at late times like t^alpha, with alpha=1 for critical quenches and alpha=1/3 for off-critical ones. We also measured the variation of temperature, total particle density and hydrodynamic velocity during the segregation process.

patt-sol↗

arXiv1997

Kinetics of Joint Ordering and Decomposition in Binary Alloys

We study phase segregation in a model alloy undergoing both ordering and decomposition, using computer simulations of Kawasaki exchange dynamics on a square lattice. Following a quench into the miscibility gap we observe an early stage in which ordering develops while the composition remains almost uniform. Then decomposition starts with segregation into ordered and disordered phases. The two spherically averaged structure functions, related to decomposition and to ordering, were both observed to obey scaling rules in the late coarsening stage where the time increase of the characteristic lengths was consistent with $a(t^{1/3} + b)$. While $a$ was similar for ordering and decomposition at low concentration of the minority component, it showed an increase (decrease) with concentration for ordering (decomposition). The domain morphology was found to depend on the concentration of the minority component, in a way that suggests a wetting of antiphase boundaries in the ordered domains by the disordered phase.

cond-mat.stat-mech↗

arXiv1997

On the uniqueness of Gibbs states in the Pirogov-Sinai theory

We prove that, for low-temperature systems considered in the Pirogov-Sinai theory, uniqueness in the class of translation-periodic Gibbs states implies global uniqueness, i.e. the absence of any non-periodic Gibbs state. The approach to this infinite volume state is exponentially fast.

cond-mat.stat-mech↗

arXiv1996

Numerical study of a non-equilibrium interface model

We have carried out extensive computer simulations of one-dimensional models related to the low noise (solid-on-solid) non-equilibrium interface of a two dimensional anchored Toom model with unbiased and biased noise. For the unbiased case the computed fluctuations of the interface in this limit provide new numerical evidence for the logarithmic correction to the subnormal L^(1/2) variance which was predicted by the dynamic renormalization group calculations on the modified Edwards-Wilkinson equation. In the biased case the simulations are in close quantitative agreement with the predictions of the Collective Variable Approximation (CVA), which gives the same L^(2/3) behavior of the variance as the KPZ equation.

cond-mat↗

arXiv1993

Steady-State Electrical Conduction in the Periodic Lorentz Gas

We study nonequilibrium steady states in the Lorentz gas of periodic scatterers when an external field is applied and the particle kinetic energy is held fixed by a ``thermostat'' constructed according to Gauss' principle of least constraint (a model problem previously studied numerically by Moran and Hoover). The resulting dynamics is reversible and deterministic, but does not preserve Liouville measure. For a sufficiently small field, we prove the following results: (1) existence of a unique stationary, ergodic measure obtained by forward evolution of initial absolutely continuous distributions, for which the Pesin entropy formula and Young's expression for the fractal dimension are valid; (2) exact identity of the steady-state thermodyamic entropy production, the asymptotic decay of the Gibbs entropy for the time-evolved distribution, and minus the sum of the Lyapunov exponents; (3) an explicit expression for the full nonlinear current response (Kawasaki formula); and (4) validity of linear response theory and Ohm's transport law, including the Einstein relation between conductivity and diffusion matrices. Results (2) and (4) yield also a direct relation between Lyapunov exponents and zero-field transport (=diffusion) coefficients. Although we restrict ourselves here to dimension $d=2,$ the results carry over to higher dimensions and to some other physical situations: e.g. with additional external magnetic fields. The proofs use a well-developed theory of small perturbations of hyperbolic dynamical systems and the method of Markov sieves, an approximation of Markov partitions. In our context we discuss also the van Kampen objection to linear response theory, which, we point out, overlooks the ``structural stability'' of strongly hyperbolic flows.

chao-dyn↗