arXiv subjects

Joel L. Lebowitz

Publications and source records attributed to Joel L. Lebowitz.

At least 91 records · Page 5Linked to original sources

arXiv2008

On the Two Species Asymmetric Exclusion Process with Semi-Permeable Boundaries

We investigate the structure of the nonequilibrium stationary state (NESS) of a system of first and second class particles, as well as vacancies (holes), on L sites of a one-dimensional lattice in contact with first class particle reservoirs at the boundary sites; these particles can enter at site 1, when it is vacant, with rate alpha, and exit from site L with rate beta. Second class particles can neither enter nor leave the system, so the boundaries are semi-permeable. The internal dynamics are described by the usual totally asymmetric exclusion process (TASEP) with second class particles. An exact solution of the NESS was found by Arita. Here we describe two consequences of the fact that the flux of second class particles is zero. First, there exist (pinned and unpinned) fat shocks which determine the general structure of the phase diagram and of the local measures; the latter describe the microscopic structure of the system at different macroscopic points (in the limit L going to infinity in terms of superpositions of extremal measures of the infinite system. Second, the distribution of second class particles is given by an equilibrium ensemble in fixed volume, or equivalently but more simply by a pressure ensemble, in which the pair potential between neighboring particles grows logarithmically with distance. We also point out an unexpected feature in the microscopic structure of the NESS for finite L: if there are n second class particles in the system then the distribution of first class particles (respectively holes) on the first (respectively last) n sites is exchangeable.

cond-mat.stat-mech↗

arXiv2008

Displacement convexity and minimal fronts at phase boundaries

We show that certain free energy functionals that are not convex with respect to the usual convex structure on their domain of definition, are strictly convex in the sense of displacement convexity under a natural change of variables. We use this to show that in certain cases, the only critical points of these functionals are minimizers. This approach based on displacement convexity permits us to treat multicomponent systems as well as single component systems. The developments produce new examples of displacement convex functionals, and, in the multi-component setting, jointly displacement convex functionals.

math.FA↗

arXiv2007

Periodic minimizers in 1D local mean field theory

Using reflection positivity techniques we prove the existence of minimizers for a class of mesoscopic free-energies representing 1D systems with competing interactions. All minimizers are either periodic, with zero average, or of constant sign. If the local term in the free energy satisfies a convexity condition, then all minimizers are either periodic or constant. Examples of both phenomena are given. This extends our previous work where such results were proved for the ground states of lattice systems with ferromagnetic nearest neighbor interactions and dipolar type antiferromagnetic long range interactions.

math-ph↗

arXiv2007

Erratum: Propagation Effects on the Breakdown of a Linear Amplifier Model: Complex-Mass Schrodinger Equation Driven by the Square of a Gaussian Field

The proof of the inequality $λ_{q}(x,t)\le (qμ_{x,t} -0^+)^{-1}$ [p 750, below Eq. (29)] is based on the statement that ${\cal E}(x,t;s)$ is an entire function of $s\in {\mathbb C}^M$ [see below Eq. (30)]. But according to Equation (9) and Lemma 1, all we know is that ${\cal E}(x,t;s)$ is an entire function of $k(s)\in {\mathbb R}^N$. Nevertheless, the above inequality holds, hence the proposition 1.

math-ph↗

arXiv2007

From Time-symmetric Microscopic Dynamics to Time-asymmetric Macroscopic Behavior: An Overview

Time-asymmetric behavior as embodied in the second law of thermodynamics is observed in {\it individual macroscopic} systems. It can be understood as arising naturally from time-symmetric microscopic laws when account is taken of a) the great disparity between microscopic and macroscopic scales, b) a low entropy state of the early universe, and c) the fact that what we observe is the behavior of systems coming from such an initial state--not all possible systems. The explanation of the origin of the second law based on these ingredients goes back to Maxwell, Thomson and particularly Boltzmann. Common alternate explanations, such as those based on the ergodic or mixing properties of probability distributions (ensembles) already present for chaotic dynamical systems having only a few degrees of freedom or on the impossibility of having a truly isolated system, are either unnecessary, misguided or misleading. Specific features of macroscopic evolution, such as the diffusion equation, do however depend on the dynamical instability (deterministic chaos) of trajectories of isolated macroscopic systems. The extensions of these classical notions to the quantum world is in many ways fairly direct. It does however also bring in some new problems. These will be discussed but not resolved.

cond-mat.stat-mech↗

arXiv2007

Striped phases in two dimensional dipole systems

We prove that a system of discrete 2D in-plane dipoles with four possible orientations, interacting via a 3D dipole-dipole interaction plus a nearest neighbor ferromagnetic term, has periodic striped ground states. As the strength of the ferromagnetic term is increased, the size of the stripes in the ground state increases, becoming infinite, i.e., giving a ferromagentic ground state, when the ferromagentic interaction exceeds a certain critical value. We also give a rigorous proof of the reorientation transition in the ground state of a 2D system of discrete dipoles with six possible orientations, interacting via a 3D dipole-dipole interaction plus a nearest neighbor antiferromagnetic term. As the strength of the antiferromagnetic term is increased the ground state flips from being striped and in-plane to being staggered and out-of-plane. An example of a rotator model with a sinusoidal ground state is also discussed.

cond-mat.stat-mech↗

arXiv2006

Ising models with long-range dipolar and short-range ferromagnetic interactions

We study the ground state of a $d$--dimensional Ising model with both long range (dipole--like) and nearest neighbor ferromagnetic (FM) interactions. The long range interaction is equal to $r^{-p}$, $p>d$, while the FM interaction has strength $J$. If $p>d+1$ and $J$ is large enough the ground state is FM, while if $d 1$ the ground state has a series of transitions from an antiferromagnetic state of period 2 to $2h$--periodic states of blocks of sizes $h$ with alternating sign, the size $h$ growing when the FM interaction strength $J$ is increased (a generalization of this result to the case $0<p\le 1$ is also discussed). In $d\ge 2$ we prove, for $d<p\le d+1$, that the dominant asymptotic behavior of the ground state energy agrees for large $J$ with that obtained from a periodic striped state conjectured to be the true ground state. The geometry of contours in the ground state is discussed.

cond-mat.stat-mech↗

arXiv2006

Percolation in the Harmonic Crystal and Voter Model in three dimensions

We investigate the site percolation transition in two strongly correlated systems in three dimensions: the massless harmonic crystal and the voter model. In the first case we start with a Gibbs measure for the potential, $U=\frac{J}{2} \sum_{ } (ϕ(x) - ϕ(y))^2$, $x,y \in \mathbb{Z}^3$, $J > 0$ and $ϕ(x) \in \mathbb{R}$, a scalar height variable, and define occupation variables $ρ_h(x) =1,(0)$ for $ϕ(x) > h (<h)$. The probability $p$ of a site being occupied, is then a function of $h$. In the voter model we consider the stationary measure, in which each site is either occupied or empty, with probability $p$. In both cases the truncated pair correlation of the occupation variables, $G(x-y)$, decays asymptotically like $|x-y|^{-1}$. Using some novel Monte Carlo simulation methods and finite size scaling we find accurate values of $p_c$ as well as the critical exponents for these systems. The latter are different from that of independent percolation in $d=3$, as expected from the work of Weinrib and Halperin [WH] for the percolation transition of systems with $G(r) \sim r^{-a}$ [A. Weinrib and B. Halperin, Phys. Rev. B 27, 413 (1983)]. In particular the correlation length exponent $ν$ is very close to the predicted value of 2 supporting the conjecture by WH that $ν= \frac{2}{a}$ is exact.

cond-mat.stat-mech↗

arXiv2006

Propagation Effects on the Breakdown of a Linear Amplifier Model: Complex-Mass Schrodinger Equation Driven by the Square of a Gaussian Field

Solutions to the equation $\partial_t{\cal E}(x,t)-\frac{i}{2m}Δ{\cal E}(x,t)=λ| S(x,t)|^2{\cal E}(x,t)$ are investigated, where $S(x,t)$ is a complex Gaussian field with zero mean and specified covariance, and $m\ne 0$ is a complex mass with ${\rm Im}(m)\ge 0$. For real $m$ this equation describes the backscattering of a smoothed laser beam by an optically active medium. Assuming that $S(x,t)$ is the sum of a finite number of independent complex Gaussian random variables, we obtain an expression for the value of $λ$ at which the $q$-th moment of $| {\cal E}(x,t)|$ w.r.t. the Gaussian field $S$ diverges. This value is found to be less or equal for all $m\ne 0$, ${\rm Im}(m)\ge 0$ and $| m| <+\infty$ than for $| m| =+\infty$, i.e. when the $Δ{\cal E}$ term is absent. Our solution is based on a distributional formulation of the Feynman path-integral and the Paley-Wiener theorem.

math-ph↗

arXiv2006

Canonical Typicality

It is well known that a system, S, weakly coupled to a heat bath, B, is described by the canonical ensemble when the composite, S+B, is described by the microcanonical ensemble corresponding to a suitable energy shell. This is true both for classical distributions on the phase space and for quantum density matrices. Here we show that a much stronger statement holds for quantum systems. Even if the state of the composite corresponds to a single wave function rather than a mixture, the reduced density matrix of the system is canonical, for the overwhelming majority of wave functions in the subspace corresponding to the energy interval encompassed by the microcanonical ensemble. This clarifies, expands and justifies remarks made by Schrödinger in 1952.

cond-mat.stat-mech↗

arXiv2005

On the Distribution of the Wave Function for Systems in Thermal Equilibrium

For a quantum system, a density matrix rho that is not pure can arise, via averaging, from a distribution mu of its wave function, a normalized vector belonging to its Hilbert space H. While rho itself does not determine a unique mu, additional facts, such as that the system has come to thermal equilibrium, might. It is thus not unreasonable to ask, which mu, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix rho, a natural measure on the unit sphere in H, denoted GAP(rho). We do this using a suitable projection of the Gaussian measure on H with covariance rho. We establish some nice properties of GAP(rho) and show that this measure arises naturally when considering macroscopic systems. In particular, we argue that it is the most appropriate choice for systems in thermal equilibrium, described by the canonical ensemble density matrix rho_beta = (1/Z) exp(- beta H). GAP(rho) may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems, where distributions on H are often used.

quant-ph↗

arXiv2005

Population Dynamics in Spatially Heterogeneous Systems with Drift: the generalized contact process

We investigate the time evolution and stationary states of a stochastic, spatially discrete, population model (contact process) with spatial heterogeneity and imposed drift (wind) in one- and two-dimensions. We consider in particular a situation in which space is divided into two regions: an oasis and a desert (low and high death rates). Carrying out computer simulations we find that the population in the (quasi) stationary state will be zero, localized, or delocalized, depending on the values of the drift and other parameters. The phase diagram is similar to that obtained by Nelson and coworkers from a deterministic, spatially continuous model of a bacterial population undergoing convection in a heterogeneous medium.

cond-mat.stat-mech↗

arXiv2005

Large deviations for a stochastic model of heat flow

We investigate a one dimensional chain of $2N$ harmonic oscillators in which neighboring sites have their energies redistributed randomly. The sites $-N$ and $N$ are in contact with thermal reservoirs at different temperature $τ_-$ and $τ_+$. Kipnis, Marchioro, and Presutti \cite{KMP} proved that this model satisfies {}Fourier's law and that in the hydrodynamical scaling limit, when $N \to \infty$, the stationary state has a linear energy density profile $\bar θ(u)$, $u \in [-1,1]$. We derive the large deviation function $S(θ(u))$ for the probability of finding, in the stationary state, a profile $θ(u)$ different from $\bar θ(u)$. The function $S(θ)$ has striking similarities to, but also large differences from, the corresponding one of the symmetric exclusion process. Like the latter it is nonlocal and satisfies a variational equation. Unlike the latter it is not convex and the Gaussian normal fluctuations are enhanced rather than suppressed compared to the local equilibrium state. We also briefly discuss more general model and find the features common in these two and other models whose $S(θ)$ is known.

cond-mat.stat-mech↗

arXiv2004

Pair approximation of the stochastic susceptible-infected-recovered-susceptible epidemic model on the hypercubic lattice

We investigate the time-evolution and steady states of the stochastic susceptible-infected-recovered-susceptible(SIRS) epidemic model on one- and two- dimensional lattices. We compare the behavior of this system, obtained from computer simulations, with those obtained from the mean-field approximation(MFA) and pair-approximation(PA). The former(latter) approximates higher order moments in terms of first(second) order ones. We find that the PA gives consistently better results than the MFA. In one dimension the improvement is even qualitative.

q-bio.PE↗

arXiv2004

Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation

We investigate saturation effects in susceptible-infected-susceptible (SIS) models of the spread of epidemics in heterogeneous populations. The structure of interactions in the population is represented by networks with connectivity distribution $P(k)$,including scale-free(SF) networks with power law distributions $P(k)\sim k^{-γ}$. Considering cases where the transmission of infection between nodes depends on their connectivity, we introduce a saturation function $C(k)$ which reduces the infection transmission rate $λ$ across an edge going from a node with high connectivity $k$. A mean field approximation with the neglect of degree-degree correlation then leads to a finite threshold $λ_{c}>0$ for SF networks with $2<γ\leq 3$. We also find, in this approximation, the fraction of infected individuals among those with degree $k$ for $λ$ close to $λ_{c}$. We investigate via computer simulation the contact process on a heterogeneous regular lattice and compare the results with those obtained from mean field theory with and without neglect of degree-degree correlations.

cond-mat.stat-mech↗

arXiv2004

Note on a diffraction-amplification problem

We investigate the solution of the equation \partial_t E(x,t)-iD\partial_x^2 E(x,t)= λ|S(x,t)|^2 E(x,t)$, for x in a circle and S(x,t) a Gaussian stochastic field with a covariance of a particular form. It is shown that the coupling λ_c at which <|E|> diverges for t>=1 (in suitable units), is always less or equal for D>0 than D=0.

math-ph↗

arXiv2003

Fourier's Law for a Harmonic Crystal with Self-consistent Stochastic Reservoirs

We consider a d-dimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the "exterior" left and right heat baths are at specified values T_L and T_R, respectively, while the temperatures of the "interior" baths are chosen self-consistently so that there is no average flux of energy between them and the system in the steady state. We prove that this requirement uniquely fixes the temperatures and the self consistent system has a unique steady state. For the infinite system this state is one of local thermal equilibrium. The corresponding heat current satisfies Fourier's law with a finite positive thermal conductivity which can also be computed using the Green-Kubo formula. For the harmonic chain (d=1) the conductivity agrees with the expression obtained by Bolsterli, Rich and Visscher in 1970 who first studied this model. In the other limit, d>>1, the stationary infinite volume heat conductivity behaves as 1/(l_d*d) where l_d is the coupling to the intermediate reservoirs. We also analyze the effect of having a non-uniform distribution of the heat bath couplings. These results are proven rigorously by controlling the behavior of the correlations in the thermodynamic limit.

math-ph↗

arXiv2003

On the (Boltzmann) Entropy of Nonequilibrium Systems

Boltzmann defined the entropy of a macroscopic system in a macrostate $M$ as the $\log$ of the volume of phase space (number of microstates) corresponding to $M$. This agrees with the thermodynamic entropy of Clausius when $M$ specifies the locally conserved quantities of a system in local thermal equilibrium (LTE). Here we discuss Boltzmann's entropy, involving an appropriate choice of macro-variables, for systems not in LTE. We generalize the formulas of Boltzmann for dilute gases and of Resibois for hard sphere fluids and show that for macro-variables satisfying any deterministic autonomous evolution equation arising from the microscopic dynamics the corresponding Boltzmann entropy must satisfy an ${\cal H}$-theorem.

cond-mat.stat-mech↗