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J. L. Lebowitz

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

At least 37 records · Page 2Linked to original sources
arXiv2010

Anomalous Fourier's law and long range correlations in a 1D non-momentum conserving mechanical model

We study by means of numerical simulations the velocity reversal model, a one-dimensional mechanical model of heat transport introduced in 1985 by Ianiro and Lebowitz. Our numerical results indicate that this model, although it does not conserve momentum, exhibits an anomalous Fourier's law similar to the ones previously observed in momentum-conserving models. This is contrary to what is obtained from the solution of the Boltzmann equation (BE) for this system. The pair correlation velocity field also looks very different from the correlations usually seen in diffusive systems, and shares some similarity with those of momentum-conserving heat transport models.

cond-mat.stat-mech↗
arXiv2010

A diffusive system driven by a battery or by a smoothly varying field

We consider the steady state of a one dimensional diffusive system, such as the symmetric simple exclusion process (SSEP) on a ring, driven by a battery at the origin or by a smoothly varying field along the ring. The battery appears as the limiting case of a smoothly varying field, when the field becomes a delta function at the origin. We find that in the scaling limit, the long range pair correlation functions of the system driven by a battery turn out to be very different from the ones known in the steady state of the SSEP maintained out of equilibrium by contact with two reservoirs, even when the steady state density profiles are identical in both models.

cond-mat.stat-mech↗
arXiv2010

Ionization of Coulomb systems in $\RR^3$ by time periodic forcings of arbitrary size

We analyze the long time behavior of solutions of the Schrödinger equation $iψ_t=(-Δ-b/r+V(t,x))ψ$, $x\in\RR^3$, $r=|x|$, describing a Coulomb system subjected to a spatially compactly supported time periodic potential $V(t,x)=V(t+2π/ω,x)$ with zero time average. We show that, for any $V(t,x)$ of the form $2Ω(r)\sin (ωt-θ)$, with $Ω(r)$ nonzero on its support, Floquet bound states do not exist. This implies that the system ionizes, {\em i.e.} $P(t,K)=\int_K|ψ(t,x)|^2dx\to 0$ as $t\to\infty$ for any compact set $K\subset\RR^3$. Furthermore, if the initial state is compactly supported and has only finitely many spherical harmonic modes, then $P(t,K)$ decays like $t^{-5/3}$ as $t \to \infty $. To prove these statements, we develop a rigorous WKB theory for infinite systems of ordinary differential equations.

math.AP↗
arXiv2009

Phase diagram of the ABC model on an interval

The three species asymmetric ABC model was initially defined on a ring by Evans, Kafri, Koduvely, and Mukamel, and the weakly asymmetric version was later studied by Clincy, Derrida, and Evans. Here the latter model is studied on a one-dimensional lattice of N sites with closed (zero flux) boundaries. In this geometry the local particle conserving dynamics satisfies detailed balance with respect to a canonical Gibbs measure with long range asymmetric pair interactions. This generalizes results for the ring case, where detailed balance holds, and in fact the steady state measure is known only for the case of equal densities of the different species: in the latter case the stationary states of the system on a ring and on an interval are the same. We prove that in the N to infinity limit the scaled density profiles are given by (pieces of) the periodic trajectory of a particle moving in a quartic confining potential. We further prove uniqueness of the profiles, i.e., the existence of a single phase, in all regions of the parameter space (of average densities and temperature) except at low temperature with all densities equal; in this case a continuum of phases, differing by translation, coexist. The results for the equal density case apply also to the system on the ring, and there extend results of Clincy et al.

cond-mat.stat-mech↗
arXiv2009

Droplet minimizers for the Gates-Lebowitz-Penrose free energy functional

We study the structure of the constrained minimizers of the Gates-Lebowitz-Penrose free-energy functional ${\mathcal F}_{\rm GLP}(m)$, non-local functional of a density field $m(x)$, $x\in {\mathcal T}_L$, a $d$-dimensional torus of side length $L$. At low temperatures, ${\mathcal F}_{\rm GLP}$ is not convex, and has two distinct global minimizers, corresponding to two equilibrium states. Here we constrain the average density $L^{-d}\int_{{\cal T}_L}m(x)\dd x$ to be a fixed value $n$ between the densities in the two equilibrium states, but close to the low density equilibrium value. In this case, a "droplet" of the high density phase may or may not form in a background of the low density phase, depending on the values $n$ and $L$. We determine the critical density for droplet formation, and the nature of the droplet, as a function of $n$ and $L$. The relation between the free energy and the large deviations functional for a particle model with long-range Kac potentials, proven in some cases, and expected to be true in general, then provides information on the structure of typical microscopic configurations of the Gibbs measure when the range of the Kac potential is large enough.

math-ph↗
arXiv2008

Effect of phonon-phonon interactions on localization

We study the heat current J in a classical one-dimensional disordered chain with on-site pinning and with ends connected to stochastic thermal reservoirs at different temperatures. In the absence of anharmonicity all modes are localized and there is a gap in the spectrum. Consequently J decays exponentially with system size N. Using simulations we find that even a small amount of anharmonicity leads to a J~1/N dependence, implying diffusive transport of energy.

cond-mat.dis-nn↗
arXiv2007

On a Random Matrix Models of Quantum Relaxation

Earlier two of us (J.L. and L.P.) considered a matrix model for a two-level system interacting with a $n\times n$ reservoir and assuming that the interaction is modelled by a random matrix. We presented there a formula for the reduced density matrix in the limit $n\to \infty $ as well as several its properties and asymptotic forms in various regimes. In this paper we give the proofs of the assertions, and present also a new fact about the model.

math-ph↗
arXiv2007

Exact Results for Ionization of Model Atomic Systems

We present rigorous results for quantum systems with both bound and continuum states subjected to an arbitrary strength time-periodic field. We prove that the wave function takes the form of a sum of time-periodic resonant states with complex quasi-energies and dispersive part of the the solution given by a power series in t^{-1/2}. Generally, the imaginary part of each resonance is negative, leading to ionization of the atom, but we also give examples where the ionization rate is zero implying the existence of a time-periodic Floquet bound state. The complex quasi-energy has a convergent perturbation expansion for small field strengths.

physics.atom-ph↗
arXiv2007

Ionization in a 1-Dimensional Dipole Model

We study the evolution of a one dimensional model atom with $δ$-function binding potential, subjected to a dipole radiation field $E(t) x$ with $E(t)$ a $2π/ω$-periodic real-valued function. Starting with $ψ(x,t=0)$ an initially localized state and $E(t)$ a trigonometric polynomial, complete ionization occurs; the probability of finding the electron in any fixed region goes to zero. For $ψ(x,0)$ compactly supported and general periodic fields, we construct a resonance expansion. Each resonance is given explicitly as a Gamow vector, and is $2π/ω$ periodic in time and behaves like the exponentially growing Green's function near $x=\pm \infty$. The remainder is given by an asymptotic power series in $t^{-1/2}$ with coefficients varying with $x$.

math-ph↗
arXiv2007

Percolation Phenomena in Low and High Density Systems

We consider the 2D quenched--disordered $q$--state Potts ferromagnets and show that at self--dual points any amalgamation of $q-1$ species will fail to percolate despite an overall (high) density of $1-q^{-1}$. Further, in the dilute bond version of these systems, if the system is just above threshold, then throughout the low temperature phase there is percolation of a single species despite a correspondingly small density. Finally, we demonstrate both phenomena in a single model by considering a ``perturbation'' of the dilute model that has a self--dual point. We also demonstrate that these phenomena occur, by a similar mechanism, in a simple coloring model invented by O. Häggström.

cond-mat.stat-mech↗
arXiv2007

Entropy of Open Lattice Systems

We investigate the behavior of the Gibbs-Shannon entropy of the stationary nonequilibrium measure describing a one-dimensional lattice gas, of L sites, with symmetric exclusion dynamics and in contact with particle reservoirs at different densities. In the hydrodynamic scaling limit, L to infinity, the leading order (O(L)) behavior of this entropy has been shown by Bahadoran to be that of a product measure corresponding to strict local equilibrium; we compute the first correction, which is O(1). The computation uses a formal expansion of the entropy in terms of truncated correlation functions; for this system the k-th such correlation is shown to be O(L^{-k+1}). This entropy correction depends only on the scaled truncated pair correlation, which describes the covariance of the density field. It coincides, in the large L limit, with the corresponding correction obtained from a Gaussian measure with the same covariance.

cond-mat.stat-mech↗
arXiv2007

Realizability of point processes

There are various situations in which it is natural to ask whether a given collection of $k$ functions, $ρ_j(\r_1,...,\r_j)$, $j=1,...,k$, defined on a set $X$, are the first $k$ correlation functions of a point process on $X$. Here we describe some necessary and sufficient conditions on the $ρ_j$'s for this to be true. Our primary examples are $X=\mathbb{R}^d$, $X=\matbb{Z}^d$, and $X$ an arbitrary finite set. In particular, we extend a result by Ambartzumian and Sukiasian showing realizability at sufficiently small densities $ρ_1(\mathbf{r})$. Typically if any realizing process exists there will be many (even an uncountable number); in this case we prove, when $X$ is a finite set, the existence of a realizing Gibbs measure with $k$ body potentials which maximizes the entropy among all realizing measures. We also investigate in detail a simple example in which a uniform density $ρ$ and translation invariant $ρ_2$ are specified on $\mathbb{Z}$; there is a gap between our best upper bound on possible values of $ρ$ and the largest $ρ$ for which realizability can be established.

math-ph↗
arXiv2006

Time asymptotics of the Schroedinger wave function in time-periodic potentials

We study the transition to the continuum of an initially bound quantum particle in $\RR^d$, $d=1,2,3$, subjected, for $t\ge 0$, to a time periodic forcing of arbitrary magnitude. The analysis is carried out for compactly supported potentials, satisfying certain auxiliary conditions. It provides complete analytic information on the time Laplace transform of the wave function. From this, comprehensive time asymptotic properties (Borel summable transseries) follow. We obtain in particular a criterion for whether the wave function gets fully delocalized (complete ionization). This criterion shows that complete ionization is generic and provides a convenient test for particular cases. When satisfied it implies absence of discrete spectrum and resonances of the associated Floquet operator. As an illustration we show that the parametric harmonic perturbation of a potential chosen to be any nonzero multiple of the characteristic function of a measurable compact set has this property.

math-ph↗
arXiv2006

Decay versus survival of a localized state subjected to harmonic forcing: exact results

We investigate the survival probability of a localized 1-d quantum particle subjected to a time dependent potential of the form $rU(x)\sin{ωt}$ with $U(x)=2δ(x-a)$ or $U(x)= 2δ(x-a)-2δ(x+a)$. The particle is initially in a bound state produced by the binding potential $-2δ(x)$. We prove that this probability goes to zero as $t\to\infty$ for almost all values of $r$, $ω$, and $a$. The decay is initially exponential followed by a $t^{-3}$ law if $ω$ is not close to resonances and $r$ is small; otherwise the exponential disappears and Fermi's golden rule fails. For exceptional sets of parameters $r,ω$ and $a$ the survival probability never decays to zero, corresponding to the Floquet operator having a bound state. We show similar behavior even in the absence of a binding potential: permitting a free particle to be trapped by harmonically oscillating delta function potential.

math-ph↗
arXiv2005

Product Measure Steady States of Generalized Zero Range Processes

We establish necessary and sufficient conditions for the existence of factorizable steady states of the Generalized Zero Range Process. This process allows transitions from a site $i$ to a site $i+q$ involving multiple particles with rates depending on the content of the site $i$, the direction $q$ of movement, and the number of particles moving. We also show the sufficiency of a similar condition for the continuous time Mass Transport Process, where the mass at each site and the amount transferred in each transition are continuous variables; we conjecture that this is also a necessary condition.

cond-mat.stat-mech↗
arXiv2004

On the Construction of Particle Distributions with Specified Single and Pair Densities

We discuss necessary conditions for the existence of probability distribution on particle configurations in $d$-dimensions i.e. a point process, compatible with a specified density $ρ$ and radial distribution function $g({\bf r})$. In $d=1$ we give necessary and sufficient criteria on $ρg({\bf r})$ for the existence of such a point process of renewal (Markov) type. We prove that these conditions are satisfied for the case $g(r) = 0, r < D$ and $g(r) = 1, r > D$, if and only if $ρD \leq e^{-1}$: the maximum density obtainable from diluting a Poisson process. We then describe briefly necessary and sufficient conditions, valid in every dimension, for $ρg(r)$ to specify a determinantal point process for which all $n$-particle densities, $ρ_n({\bf r}_1, ..., {\bf r}_n)$, are given explicitly as determinants. We give several examples.

cond-mat.stat-mech↗
arXiv2004

Space charge limited flow of a thin electron beam confined by a strong magnetic field

An approximate analytic theory is developed and implemented numerically for calculating the space charge limited current and electric field of a thin cylindrical beam or current sheet between two wide parallel electrodes. The flow is confined by a sufficiently strong magnetic field. Assuming that the potential and current density are almost homogeneous in the direction transversal to the flow we compute the beam current and profile by a variational method. We find that the average current density scales as the reciprocal of the beam width when the latter becomes very small. The total cylindrical beam current thus decreases proportionly to its diameter while the total current of a sheet becomes almost independent of the width in this regime.

physics.plasm-ph↗