arXiv subjects

J. L. Lebowitz

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

At least 73 records · Page 4Linked to original sources

arXiv2000

Evolution of a model quantum system under time periodic forcing: conditions for complete ionization

We analyze the time evolution of a one-dimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation $η(t)$. We show that for generic $η(t)$, which includes the sum of any finite number of harmonics, the system, started in a bound state will get fully ionized as $t\to\infty$. This is irrespective of the magnitude or frequency (resonant or not) of $η(t)$. There are however exceptional, very non-generic $η(t)$, that do not lead to full ionization, which include rather simple explicit periodic functions. For these $η(t)$ the system evolves to a nontrivial localized stationary state which is related to eigenfunctions of the Floquet operator.

math-ph↗

arXiv2000

Dynamics of mesoscopic precipitate lattices in phase separating alloys under external load

We investigate, via three-dimensional atomistic computer simulations, phase separation in an alloy under external load. A regular two-dimensional array of cylindrical precipitates, forming a mesoscopic precipitate lattice, evolves in the case of applied tensile stress by the movement of mesoscopic lattice defects. A striking similarity to ordinary crystals is found in the movement of "meso-dislocations", but new mechanisms are also observed. Point defects such as "meso-vacancies" or "meso-interstitials" are created or annihilated locally by merging and splitting of precipitates. When the system is subjected to compressive stress, we observe stacking faults in the mesoscopic one-dimensional array of plate-like precipitates.

cond-mat.mtrl-sci↗

arXiv2000

Spatial Structure in Low Dimensions for Diffusion Limited Two-Particle Reactions

Consider the system of particles on ${\Bbb Z}^d$ where particles are of two types, $A$ and $B$, and execute simple random walks in continuous time. Particles do not interact with their own type, but when a type $A$ particle meets a type $B$ particle, both disappear. Initially, particles are assumed to be distributed according to homogeneous Poisson random fields, with equal intensities for the two types. This system serves as a model for the chemical reaction $A+B\to inert$. In [BrLe91a], the densities of the two types of particles were shown to decay asymptotically like $1/t^{d/4}$ for $d<4$ and $1/t$ for $d\geq 4$, as $t\to\infty$. This change in behavior from low to high dimensions corresponds to a change in spatial structure. In $d<4$, particle types segregate, with only one type present locally. After suitable rescaling, the process converges to a limit, with density given by a Gaussian process. In $d>4$, both particle types are, at large times, present locally in concentrations not depending on the type, location or realization. In $d=4$, both particle types are present locally, but with varying concentrations. Here, we analyze this behavior in $d<4$; the behavior for $d\geq 4$ will be handled in a future work [BrLe99].

math-ph↗

arXiv2000

Macroscopic Evolution of Particle Systems with Short and Long Range Interactions

We consider a lattice gas with general short range interactions and a Kac potential $J_γ({r})$ of range $γ^{-1}$, $γ>0$, evolving via particles hopping to nearest neighbor empty sites with rates which satisfy detailed balance with respect to the equilibrium measure. Scaling space like $γ^{-1}$ and time like $γ^{-2}$, we prove that in the limit $γ\to 0$ the macroscopic density profile $ρ({r},t)$ satisfies a integro-differential equation which is in the form of the gradient flux of the energy functional $\cal F$, with a mobility given by the Einstein relation Beside a regularity condition on J, the only requirement for this result is that the reference system satisfy the hypotheses of the Varadhan--Yau Theorem leading to the equation for $J\equiv 0$. Therefore the equation holds also if $\cal F$ achieves its minimum on non constant density profiles and this includes the cases in which {\sl phase segregation} occurs. Using the same techniques we also derive hydrodynamic equations for the densities of a two component A-B mixture with long range repulsive interactions between A and B particles. The equations for the densities $ρ_A$ and $ρ_B$ are again in the form of the gradient flux. They describe, at low temperatures, the demixing transition in which segregation takes place via vacancies, i.e. jumps to empty sites. In the limit of very few vacancies the problem becomes similar to phase segregation in a continuum system in the so called incompressible limit.

cond-mat.stat-mech↗

arXiv2000

Properties of Stationary Nonequilibrium States in the Thermostatted Lorentz Gas I: the One Particle System

We study numerically and analytically the properties of the stationary state of a particle moving under the influence of an electric field $\bE$ in a two dimensional periodic Lorentz gas with the energy kept constant by a Gaussian thermostat. Numerically the current appears to be a continuous function of $\bE$ whose derivative varies very irregularly, possibly in a discontinuous manner. We argue for the non differentibility of the current as a function of $\bE$ utilizing a symbolic description of the dynamics based on the discontinuities of the collision map. The decay of correlations and the behavior of the diffusion constant are also investigated.

nlin.CD↗

arXiv2000

Fourier's Law: a Challenge for Theorists

We present a selective overview of the current state of our knowledge (more precisely of ourignorance) regarding the derivation of Fourier's Law, ${\bf J}(\br) =-κ{\bf \nabla}T(\br)$; ${\bf J}$ the heat flux, $T$ the temperature and $κ$, the heat conductivity. This law is empirically well tested for both fluids and crystals, when the temperature varies slowly on the microscopic scale, with $κ$ an intrinsic property which depends only on the system's equilibrium parameters, such as the local temperature and density. There is however at present no rigorous mathematical derivation of Fourier's law and ipso facto of Kubo's formula for $κ$, involving integrals over equilibrium time correlations, for any system (or model) with a deterministic, e.g. Hamiltonian, microscopic evolution.

math-ph↗

arXiv2000

Blocking measures for asymmetric exclusion processes via coupling

We give sufficient conditions on the rates of two asymmetric exclusion processes such that the existence of a blocking invariant measure for the first implies the existence of such a measure for the second. The main tool is a coupling between the two processes under which the first dominates the second in an appropriate sense. In an appendix we construct a class of processes for which the existence of a blocking measure can be proven directly; these are candidates for comparison processes in applications of the main result.

math.PR↗

arXiv2000

On the complete ionization of a periodically perturbed quantum system

We analyze the time evolution of a one-dimensional quantum system with zero range potential under time periodic parametric perturbation of arbitrary strength and frequency. We show that the projection of the wave function on the bound state vanishes, i.e. the system gets fully ionized, as time grows indefinitely.

math-ph↗

arXiv1999

Charge Fluctuations in the Two-Dimensional One-Component Plasma

We study, via computer simulations, the fluctuations in the net electric charge, in a two dimensional one component plasma (OCP) with uniform background charge density $-e ρ$, in a region $Λ$ inside a much larger overall neutral system. Setting $e=1$ this is the same as the fluctuations in $N_Λ$, the number of mobile particles of charge $e$. As expected the distribution of $ N_Λ$ has, for large $Λ$, a Gaussian form with a variance which grows only as $\hat κ|\partial Λ|$, where $|\partial Λ|$ is the length of the perimeter of $Λ$. The properties of this system depend only on the coupling parameter $Γ= kT$ which is the same as the reciprocal temperature in our units. Our simulations show that when the coupling parameter $Γ$ increases, $\hat κ(Γ)$ decreases to an asymptotic value $\hat κ(\infty) \sim \hat κ(2)/2$ which is equal (or very close) to that obtained for the corresponding variance of particles on a rigid triangular lattice. Thus, for large $Γ$, the characteristic length $ξ_L = 2\hat κ/ρ$ associated with charge fluctuations behaves very differently from that of the Debye length, $ξ_D \sim 1/\sqrt Γ$, which it approaches as $Γ\to 0$. The pair correlation function of the OCP is also studied.

cond-mat↗

arXiv1998

Modelling of Phase Separation in Alloys with Coherent Elastic Misfit

Elastic interactions arising from a difference of lattice spacing between two coherent phases can have a strong influence on the phase separation (coarsening) of alloys. If the elastic moduli are different in the two phases, the elastic interactions may accelerate, slow down or even stop the phase separation process. If the material is elastically anisotropic, the precipitates can be shaped like plates or needles instead of spheres and can form regular precipitate superlattices. Tensions or compressions applied externally to the specimen may have a strong effect on the shapes and arrangement of the precipitates. In this paper, we review the main theoretical approaches that have been used to model these effects and we relate them to experimental observations. The theoretical approaches considered are (i) `macroscopic' models treating the two phases as elastic media separated by a sharp interface (ii) `mesoscopic' models in which the concentration varies continuously across the interface (iii) `microscopic' models which use the positions of individual atoms.

math-ph↗

arXiv1998

Self-Diffusion in Simple Models: Systems with Long-Range Jumps

We review some exact results for the motion of a tagged particle in simple models. Then, we study the density dependence of the self diffusion coefficient, $D_N(ρ)$, in lattice systems with simple symmetric exclusion in which the particles can jump, with equal rates, to a set of $N$ neighboring sites. We obtain positive upper and lower bounds on $F_N(ρ)=N((1-\r)-[D_N(ρ)/D_N(0)])/(ρ(1-ρ))$ for $ρ\in [0,1]$. Computer simulations for the square, triangular and one dimensional lattice suggest that $F_N$ becomes effectively independent of $N$ for $N\ge 20$.

cond-mat.stat-mech↗

arXiv1998

Improved Peierls Argument for High Dimensional Ising Models

We consider the low temperature expansion for the Ising model on $\Z^d$, $d \ge 2$, with ferromagnetic nearest neighbor interactions in terms of Peierls contours. We prove that the expansion converges for all temperatures smaller than $C d (\log d)^{-1}$, which is the correct order in $d$.

cond-mat↗

arXiv1998

Liquid-Vapor Phase Transitions for Systems with Finite Range Interactions

We consider particles in $\R^d, d \geq 2$ interacting via attractive pair and repulsive four-body potentials of the Kac type. Perturbing about mean field theory, valid when the interaction range becomes infinite, we prove rigorously the existence of a liquid-gas phase transition, when the interaction range is finite but long compared to the interparticle spacing for a range of temperature.

cond-mat↗

arXiv1998

Rigorous Proof of a Liquid-Vapor Phase Transition in a Continuum Particle System

We consider particles in ${\Bbb R}^d, d \geq 2$, interacting via attractive pair and repulsive four-body potentials of the Kac type. Perturbing about mean field theory, valid when the interaction range becomes infinite, we prove rigorously the existence of a liquid-gas phase transition when the interaction range is finite but long compared to the interparticle spacing.

cond-mat↗

arXiv1998

Bounds on the Mobility of Electrons in Weakly Ionized Plasmas

We obtain exact upper and lower bounds on the steady state drift velocity, and kinetic energy of electrons, driven by an external field in a weakly ionized plasma (swarm approximation). The scattering is assumed to be elastic with simplified velocity dependence of the collision cross sections. When the field is large the bounds are close to each other and to the results obtained from the conventional approximation of the Boltzmann equation in which one keeps only the first two terms of a Legendre expansion. The bounds prove rigorously that it is possible to increase the electron mobility by the addition of suitably chosen scatterers to the system as predicted by the Druyvesteyn approximation and found in experiments.

physics.plasm-ph↗

arXiv1998

Exact Large Deviation Function in the Asymmetric Exclusion Process

By an extension of the Bethe ansatz method used by Gwa and Spohn, we obtain an exact expression for the large deviation function of the time averaged current for the fully asymmetric exclusion process in a ring containing $N$ sites and $p$ particles. Using this expression we easily recover the exact diffusion constant obtained earlier and calculate as well some higher cumulants. The distribution of the deviation $y$ of the average current is, in the limit $N \to \infty$, skew and decays like $\exp - (A y^{5/2})$ for $y \to + \infty$ and $\exp - (A' |y|^{3/2})$ for $y \to -\infty$. Surprisingly, the large deviation function has an expression very similar to the pressure (as a function of the density) of an ideal Bose or Fermi gas in $3d$.

cond-mat↗

arXiv1998

Hydrodynamical Equation for Electron Swarms

We study the long time behavior of light particles, e.g. an electron swarm in which Coulomb interactions are unimportant, subjected to an external field and elastic collisions with an inert neutral gas. The time evolution of the velocity and position distribution function is described by a linear Boltzmann equation (LBE). The small ratio of electron to neutral masses, $ε$, makes the energy transfer between them very inefficient. We show that under suitable scalings the LBE reduces, in the limit $ε\to 0$, to a formally exact equation for the speed (energy) and position distribution of the electrons which contains mixed spatial and speed derivatives. When the system is spatially homogeneous this equation reduces to and thus justifies, for $ε$ small enough, the commonly used ``two-term'' approximation.

physics.plasm-ph↗

arXiv1998

Remark on the (Non)convergence of Ensemble Densities in Dynamical Systems

We consider a dynamical system with state space $M$, a smooth, compact subset of some ${\Bbb R}^n$, and evolution given by $T_t$, $x_t = T_t x$, $x \in M$; $T_t$ is invertible and the time $t$ may be discrete, $t \in {\Bbb Z}$, $T_t = T^t$, or continuous, $t \in {\Bbb R}$. Here we show that starting with a continuous positive initial probability density $ρ(x,0) > 0$, with respect to $dx$, the smooth volume measure induced on $M$ by Lebesgue measure on ${\Bbb R}^n$, the expectation value of $\log ρ(x,t)$, with respect to any stationary (i.e. time invariant) measure $ν(dx)$, is linear in $t$, $ν(\log ρ(x,t)) = ν(\log ρ(x,0)) + Kt$. $K$ depends only on $ν$ and vanishes when $ν$ is absolutely continuous wrt $dx$.

math-ph↗