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Joel L. Lebowitz

Publications and source records attributed to Joel L. Lebowitz.

At least 55 records · Page 3Linked to original sources
arXiv2016

Fowler-Nordheim emission modified by laser pulses in the adiabatic regime

We investigate enhanced field emission due to a continuous or pulsed oscillating field added to a constant electric field $E$ at the emitter surface. When the frequency of oscillation, field strength, and property of the emitter material satisfy the Keldysh condition $γ<1/2$ one can use the adiabatic approximation for treating the oscillating field, i.e. consider the tunneling through the instantaneous Fowler-Nordheim barrier created by both fields. Due to the great sensitivity of the emission to the field strength the average tunneling current can be much larger than the current produced by only the constant field. We carry out the computations for arbitrary strong constant electric fields, beyond the commonly used Fowler-Nordheim approximation which exhibit in particular an important property of the wave function inside the potential barrier where it is found to be monotonically decreasing without oscillations.

physics.plasm-ph↗
arXiv2015

Thermal equilibrium of a macroscopic quantum system in a pure state

We consider the notion of thermal equilibrium for an individual closed macroscopic quantum system in a pure state, i.e., described by a wave function. The macroscopic properties in thermal equilibrium of such a system, determined by its wave function, must be the same as those obtained from thermodynamics, e.g., spatial uniformity of temperature and chemical potential. When this is true we say that the system is in macroscopic thermal equilibrium (MATE). Such a system may however not be in microscopic thermal equilibrium (MITE). The latter requires that the reduced density matrices of small subsystems be close to those obtained from the microcanonical, equivalently the canonical, ensemble for the whole system. The distinction between MITE and MATE is particularly relevant for systems with many-body localization (MBL) for which the energy eigenfunctions fail to be in MITE while necessarily most of them, but not all, are in MATE. We note however that for generic macroscopic systems, including those with MBL, most wave functions in an energy shell are in both MATE and MITE. For a classical macroscopic system, MATE holds for most phase points on the energy surface, but MITE fails to hold for any phase point.

cond-mat.stat-mech↗
arXiv2015

On the Equilibrium State of a Small System with Random Matrix Coupling to Its Environment

We consider a random matrix model of interaction between a small $n$-level system, $S$, and its environment, a $N$-level heat reservoir, $R$. The interaction between $S$ and $R$ is modeled by a tensor product of a fixed $% n\times n$ matrix and a $N\times N$ hermitian Gaussian random matrix. We show that under certain "macroscopicity" conditions on $R$, the reduced density matrix of the system $ρ_{S}=\mathrm{Tr}_{R}ρ_{S\cup R}^{(eq)} $, is given by $ρ_{S}^{(c)}\sim \exp {\{-βH_{S}\}}$, where $H_{S}$ is the Hamiltonian of the isolated system. This holds for all strengths of the interaction and thus gives some justification for using $% ρ_{S}^{(c)}$ to describe some nano-systems, like biopolymers, in equilibrium with their environment \cite{Se:12}. Our results extend those obtained previously in \cite{Le-Pa:03,Le-Co:07} for a special two-level system.

math-ph↗
arXiv2015

Universal Probability Distribution for the Wave Function of a Quantum System Entangled with Its Environment

A quantum system (with Hilbert space $\mathscr{H}_1$) entangled with its environment (with Hilbert space $\mathscr{H}_2$) is usually not attributed a wave function but only a reduced density matrix $ρ_1$. Nevertheless, there is a precise way of attributing to it a random wave function $ψ_1$, called its conditional wave function, whose probability distribution $μ_1$ depends on the entangled wave function $ψ\in\mathscr{H}_1\otimes\mathscr{H}_2$ in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of $\mathscr{H}_2$ but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about $μ_1$, e.g., that if the environment is sufficiently large then for every orthonormal basis of $\mathscr{H}_2$, most entangled states $ψ$ with given reduced density matrix $ρ_1$ are such that $μ_1$ is close to one of the so-called GAP (Gaussian adjusted projected) measures, $GAP(ρ_1)$. We also show that, for most entangled states $ψ$ from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval $[E,E+δE]$) and most orthonormal bases of $\mathscr{H}_2$, $μ_1$ is close to $GAP(\mathrm{tr}_2 ρ_{mc})$ with $ρ_{mc}$ the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then $μ_1$ is close to $GAP(ρ_β)$ with $ρ_β$ the canonical density matrix on $\mathscr{H}_1$ at inverse temperature $β=β(E)$. This provides the mathematical justification of our claim in [J. Statist. Phys. 125:1193 (2006), http://arxiv.org/abs/quant-ph/0309021] that $GAP$ measures describe the thermal equilibrium distribution of the wave function.

math-ph↗
arXiv2015

Any Orthonormal Basis in High Dimension is Uniformly Distributed over the Sphere

Let X be a real or complex Hilbert space of finite but large dimension d, let S(X) denote the unit sphere of X, and let u denote the normalized uniform measure on S(X). For a finite subset B of S(X), we may test whether it is approximately uniformly distributed over the sphere by choosing a partition A_1,...,A_m of S(X) and checking whether the fraction of points in B that lie in A_k is close to u(A_k) for each k=1,...,m. We show that if B is any orthonormal basis of X and m is not too large, then, if we randomize the test by applying a random rotation to the sets A_1,...,A_m, B will pass the random test with probability close to 1. This statement is related to, but not entailed by, the law of large numbers. An application of this fact in quantum statistical mechanics is briefly described.

math.PR↗
arXiv2014

Green-Kubo formula for weakly coupled system with dynamical noise

We study the Green-Kubo (GK) formula $κ(\varepsilon, ξ)$ for the heat conductivity of an infinite chain of $d$-dimensional finite systems (cells) coupled by a smooth nearest neighbour potential $\varepsilon V$. The uncoupled systems evolve according to Hamiltonian dynamics perturbed stochastically by an energy conserving noise of strength $ξ$. Noting that $κ(\varepsilon, ξ)$ exists and is finite whenever $ξ> 0$, we are interested in what happens when the strength of the noise $ξ\to 0$. For this, we start in this work by formally expanding $κ(\varepsilon, ξ)$ in a power series in $\varepsilon$, $κ(\varepsilon, ξ) = \varepsilon^2 \sum_{n\ge 2} \varepsilon^{n-2} κ_n (ξ)$ and investigating the (formal) equations satisfied by $κ_n (ξ$. We show in particular that $κ_2 (ξ)$ is well defined when no pinning potential is present, and coincides formally with the heat conductivity obtained in the weak coupling (van Hove) limit, where time is rescaled as $\varepsilon^{-2}t$, for the cases where the latter has been established \cite{LO, DL}. For one-dimensional systems, we investigate $κ_2 (ξ)$ as $ξ\to 0$ in three cases: the disordered harmonic chain, the rotor chain and a chain of strongly anharmonic oscillators. Moreover, we formally identify $κ_2 (ξ)$ with the conductivity obtained by having the chain between two reservoirs at temperature $T$ and $T+δT$, in the limit $δT\to 0$, $N \to \infty$, $\varepsilon \to 0$.

cond-mat.stat-mech↗
arXiv2014

Exponential approach to, and properties of, a non-equilibrium steady state in a dilute gas

We investigate a kinetic model of a system in contact with several thermal reservoirs at different temperatures $T_α$. Our system is a spatially uniform dilute gas whose internal dynamics is described by the nonlinear Boltzmann equation with Maxwellian collisions. Similarly, the interaction with reservoir $α$ is represented by a Markovian process that has the Maxwellian $M_{T_α}$ as its stationary state. We prove existence and uniqueness of a non-equilibrium steady state (NESS) and show exponential convergence to this NESS in a metric on probability measures introduced into the study of Maxwellian collisions by Gabetta, Toscani and Wenberg (GTW). This shows that the GTW distance between the current velocity distribution to the steady-state velocity distribution is a Lyapunov functional for the system. We also derive expressions for the entropy production in the system plus the reservoirs which is always positive.

math-ph↗
arXiv2014

Local Central Limit Theorem for Determinantal Point Processes

We prove a local central limit theorem (LCLT) for the number of points $N(J)$ in a region $J$ in $\mathbb R^d$ specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of $N(J)$ tends to infinity as $|J| \to \infty$. This extends a previous result giving a weaker central limit theorem (CLT) for these systems. Our result relies on the fact that the Lee-Yang zeros of the generating function for $\{E(k;J)\}$ --- the probabilities of there being exactly $k$ points in $J$ --- all lie on the negative real $z$-axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble (GSE). A LCLT is also established for the probability density function of the $k$-th largest eigenvalue at the soft edge, and of the spacing between $k$-th neigbors in the bulk.

math-ph↗
arXiv2013

Harmonic Systems With Bulk Noises

We consider a harmonic chain in contact with thermal reservoirs at different temperatures and subject to bulk noises of different types: velocity flips or self-consistent reservoirs. While both systems have the same covariances in the nonequilibrium stationary state (NESS) the measures are very different. We study hydrodynamical scaling, large deviations, fluctuations, and long range correlations in both systems. Some of our results extend to higher dimensions.

cond-mat.stat-mech↗
arXiv2013

On an Extension Problem for Density Matrices

We investigate the problem of the existence of a density matrix rho on the product of three Hilbert spaces with given marginals on the pair (1,2) and the pair (2,3). While we do not solve this problem completely we offer partial results in the form of some necessary and some sufficient conditions on the two marginals. The quantum case differs markedly from the classical (commutative) case, where the obvious necessary compatibility condition suffices, namely, trace_1 (rho_{12}) = \trace_3 (rho_{23}).

quant-ph↗
arXiv2012

Location of the Lee-Yang zeros and absence of phase transitions in some Ising spin systems

We consider a class of Ising spin systems on a set Λof sites. The sites are grouped into units with the property that each site belongs to either one or two units, and the total internal energy of the system is the sum of the energies of the individual units, which in turn depend only on the number of up spins in the unit. We show that under suitable conditions on these interactions none of the |Λ| Lee-Yang zeros in the complex z = exp{2βh} plane, where βis the inverse temperature and h the uniform magnetic field, touch the positive real axis, at least for large values of β. In some cases one obtains, in an appropriately taken βto infinity limit, a gas of hard objects on a set Λ'; the fugacity for the limiting system is a rescaling of z and the Lee-Yang zeros of the new partition function also avoid the positive real axis. For certain forms of the energies of the individual units the Lee-Yang zeros of both the finite- and zero-temperature systems lie on the negative real axis for all β. One zero-temperature limit of this type, for example, is a monomer-dimer system; our results thus generalize, to finite β, a well-known result of Heilmann and Lieb that the Lee-Yang zeros of monomer-dimer systems are real and negative.

cond-mat.stat-mech↗
arXiv2011

Proof of Rounding by Quenched Disorder of First Order Transitions in Low-Dimensional Quantum Systems

We prove that for quantum lattice systems in d<=2 dimensions the addition of quenched disorder rounds any first order phase transition in the corresponding conjugate order parameter, both at positive temperatures and at T=0. For systems with continuous symmetry the statement extends up to d<=4 dimensions. This establishes for quantum systems the existence of the Imry-Ma phenomenon which for classical systems was proven by Aizenman and Wehr. The extension of the proof to quantum systems is achieved by carrying out the analysis at the level of thermodynamic quantities rather than equilibrium states.

cond-mat.stat-mech↗
arXiv2011

Nonequilibrium Stationary States of Harmonic Chains with Bulk Noises

We consider a chain composed of $N$ coupled harmonic oscillators in contact with heat baths at temperature $T_\ell$ and $T_r$ at sites 1 and $N$ respectively. The oscillators are also subjected to non-momentum conserving bulk stochastic noises. These make the heat conductivity satisfy Fourier's law. Here we describe some new results about the hydrodynamical equations for typical macroscopic energy and displacement profiles, as well as their fluctuations and large deviations, in two simple models of this type.

cond-mat.stat-mech↗
arXiv2011

Necessary and sufficient conditions for realizability of point processes

We give necessary and sufficient conditions for a pair of (generalized) functions $ρ_1(\mathbf{r}_1)$ and $ρ_2(\mathbf{r}_1,\mathbf{r}_2)$, $\mathbf{r}_i\in X$, to be the density and pair correlations of some point process in a topological space $X$, for example, $\mathbb {R}^d$, $\mathbb {Z}^d$ or a subset of these. This is an infinite-dimensional version of the classical "truncated moment" problem. Standard techniques apply in the case in which there can be only a bounded number of points in any compact subset of $X$. Without this restriction we obtain, for compact $X$, strengthened conditions which are necessary and sufficient for the existence of a process satisfying a further requirement---the existence of a finite third order moment. We generalize the latter conditions in two distinct ways when $X$ is not compact.

math.PR↗
arXiv2011

Spatial Structure of Stationary Nonequilibrium States in the Thermostatted Periodic Lorentz Gas

We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external electric field E as well as a Gaussian thermostat which keeps the speed |v| constant. We show that despite the singular nature of the SRB measure its projections on the space coordinates are absolutely continuous. We further show that these projections satisfy linear response laws for small E. Some of them are computed numerically. We compare these results with those obtained from simple models in which the collisions with the obstacles are replaced by random collisions.Similarities and differences are noted.

nlin.CD↗
arXiv2011

Phase diagram of a generalized ABC model on the interval

We study the equilibrium phase diagram of a generalized ABC model on an interval of the one-dimensional lattice: each site $i=1,...,N$ is occupied by a particle of type $\a=A,B,C,$ with the average density of each particle species $N_\a/N=r_\a$ fixed. These particles interact via a mean field non-reflection-symmetric pair interaction. The interaction need not be invariant under cyclic permutation of the particle species as in the standard ABC model studied earlier. We prove in some cases and conjecture in others that the scaled infinite system $N\rw\infty$, $i/N\rw x\in[0,1]$ has a unique density profile $\p_\a(x)$ except for some special values of the $r_\a$ for which the system undergoes a second order phase transition from a uniform to a nonuniform periodic profile at a critical temperature $T_c=3\sqrt{r_A r_B r_C}/2π$.

cond-mat.stat-mech↗
arXiv2011

Checkerboards, stripes and corner energies in spin models with competing interactions

We study the zero temperature phase diagram of Ising spin systems in two dimensions in the presence of competing interactions, long range antiferromagnetic and nearest neighbor ferromagnetic of strength J. We first introduce the notion of a "corner energy" which shows, when the antiferromagnetic interaction decays faster than the fourth power of the distance, that a striped state is favored with respect to a checkerboard state when J is close to J_c, the transition to the ferromagnetic state, i.e., when the length scales of the uniformly magnetized domains become large. Next, we perform detailed analytic computations on the energies of the striped and checkerboard states in the cases of antiferromagnetic interactions with exponential decay and with power law decay r^{-p}, p>2, that depend on the Manhattan distance instead of the Euclidean distance. We prove that the striped phase is always favored compared to the checkerboard phase when the scale of the ground state structure is very large. This happens for J\lesssim J_c if p>3, and for J sufficiently large if 2<p<=3. Many of our considerations involving rigorous bounds carry over to dimensions greater than two and to more general short-range ferromagnetic interactions.

cond-mat.stat-mech↗