Lanczos cluster expansion for non-extensive systems

A cluster expansion of the Lanczos recursion for non-extensive systems is developed based on the plaquette expansion for extensive systems, in which an auxiliary scaling parameter, Ω, plays the role of volume and introduces extensivity into the problem. Connected Hamiltonian moments of the non-extensive system are computed and introduced into the plaquette expansion in the usual way with Ω. The extensive energy is calculated for increasing orders of the expansion in 1/Ω and the ground state and mass gap of the finite few body problem recovered in the limit Ω → ∞. This new non-perturbative method is applied to the case of N bosons interacting harmonically in one dimension and the ground state energy and mass gap in the vacuum sector are calculated exactly.     

Pyrimido[5,4-b]quinolines. II. Reactions at the heterocyclic ring-carbon and nitrogen atoms

10-Chloro-7,8-dimethylpyrimido[5,4-b]quinolin-2,4(1H,3H)dione (I) was unreactive toward ammonia but it reacted with 2 molecules of n-butylamine, presumably via Dimroth-type ring-opening and closure, to give the N₃-butyl, N₁₀-butylamino derivative (IV). In similar reactions of 10-chloro-2,4-dimethoxy-7,8-dimethylpyrimido[5,4-b]quinoline (II) only the 4-meth-oxyl was displaced by either ammonia or n-butylamine. Alkyllithium reagents also displaced the 4-methoxyl as well as added to the 3,4 double bond of II to yield the corresponding gem-dialkyl substituted (C₄) derivatives; the C₁₀ chlorine remained unreactive. 2,4-Dimethoxy-7,8-di-methylpyrimido[5,4-b]quinoline-10-one (III) could be alkylated only in the form of the thallium salt. Treatment of the benzyl derivative of III with methylmagnesium bromide led only to the displacement of the 4-methoxyl by a methyl group.     

The Acoustic Limit for the Boltzmann Equation

The acoustic equations are the linearization of the compressible Euler equations about a spatially homogeneous fluid state. We first derive them directly from the Boltzmann equation as the formal limit of moment equations for an appropriately scaled family of Boltzmann solutions. We then establish this limit for the Boltzmann equation considered over a periodic spatial domain for bounded collision kernels. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that converge entropically (and hence strongly in L ¹) to a unique limit governed by a solution of the acoustic equations for all time, provided that its initial fluctuations converge entropically to an appropriate limit associated to any given L ² initial data of the acoustic equations. The associated local conservation laws are recovered in the limit. Accepted: October 22, 1999     

La relation de possion pour l'equation des ondes dans un ouvert non borne application a la theorie de la diffusion

We study in this note the solutions of two types of hyperbolic systems of conservation laws with oscillating data. The first one (a2?2 system) has only one linearly degenerate eigenvalue. Using the results of R.J.Di Perna related to genuinely nonlinear fields, one can describe the propagatrion of oscillations (which appear in only one direction) with an integro differential system for which one of the two unknowns is a field depending, of y ? ]0,1[, in addition of x and t. The second system is a linearly degenerate 3?3 system. We apply theory of compensated compactness, due to L. Tartar and F. Murat and, in the same way as above, we show that the initial oscillations can propagate; this propagation is then described witha a relaxed system of 3 unknowns     

Diffusion approximations for the scattering of resonance-line photons: Interior and boundary layer solutions

Resonance-line scattering in static low density media with large optical thickness has a diffusive behavior in both space and frequency because photons belonging to the Lorentzian wings of the line may be scattered almost monochromatically a very large number of times. This diffusive behavior holds on frequency scales and spatial scales, χ_c and τ_c, much larger than the scales associated with one elementary scattering of a wing-photon.A method developed for diffusion approximations in neutron transport theory, suitably generalized to handle diffusion in frequency space, is applied to the case of conservative scattering in a bounded medium with interior sources and zero incoming radiation. The method is to separate the line radiation field into an interior part and a boundary layer part which goes to zero in the interior. Each part is expanded in terms of a small parameter ?, which is the ratio of the mean free-path at frequency χ_c to the characteristic spatial scale τ_c.It is shown that the leading term in the interior asymptotic expansion is isotropic, zero on the boundary, and obeys a space and frequency diffusion equation. In the boundary-layer expansion, the leading term is of order ? and is a solution to a monochromatic transfer equation in a semi-infinite, plane-parallel medium. The emergent radiation field is shown to be of order ? and proportional to the gradient of the interior solution at the boundary. Its angular dependence, in the case of isotropic scattering in the atom frame, is given by the Ambartsoumian H-function. A comparison is presented between numerical solutions of the full transfer equation and asymptotic solutions. Non-conservative scattering and time-dependent problems are briefly discussed.     

Diffusion approximation for billiards with totally accommodating scatterers

We study the 2D motion of independent point particles colliding with a periodic array of circular obstacles. The interaction between the particles and the obstacles is described by a total accommodation reflection law. Assuming that the array of scatterers has finite horizon, the density of particles is approximated by the solution of a diffusion equation in the long-time and large-scale regime. The proof relies on a multiscale asymptotics and gives the order of approximation.     

Finite time analyticity for the two and three dimensional Kelvin-Helmholtz instability

The well-posed property for the finite time vortex sheet problem with analytic initial data was first conjectured by Birkhoff in two dimensions and is shown here to hold both in two and three dimensions. Incompressible, inviscid and irrotational flow with a velocity jump across an interface is assumed. In two dimensions, global existence of a weak solution to the Euler equation with such initial conditions is established. In three dimensions, a Lagrangian representation of the vortex sheet analogous to the Birkhoff equation in two dimensions is presented.This work was performed while C.B. was visiting the Dept. de Mathématiques, Nice     

Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

We consider a modification of the three-dimensional Navier–Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as e^|k|/k_d{{{\rm e}^{|k|/k_{\rm d}}}} at high wavenumbers |k|. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than e^{-C(k/k_d) ln(|k|/k_d)}{{{\rm e}^{-C(k/k_{\rm d})\,{\rm ln}(|k|/k_{\rm d})}}} for any C < 1/(2 ln 2). The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with C = C _* = 1/ ln 2. The same behavior with a universal constant C _* is conjectured for the Navier–Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier–Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.     

Setting and Analysis of the Multi-configuration Time-dependent Hartree–Fock Equations

In this paper, we formulate and analyze the multi-configuration time-dependent Hartree–Fock (MCTDHF) equations for molecular systems with pairwise interaction. This set of coupled nonlinear PDEs and ODEs is an approximation of the N-particle time-dependent Schrödinger equation based on (time-dependent) linear combinations of (time-dependent) Slater determinants. The “one-electron” wave-functions satisfy nonlinear Schrödinger-type equations coupled to a linear system of ordinary differential equations for the expansion coefficients. The invertibility of the one-body density matrix (full-rank hypothesis) plays a crucial rôle in the analysis. Under the full-rank assumption a fiber bundle structure emerges and produces unitary equivalence between different useful representations of the MCTDHF approximation. For a large class of interactions (including Coulomb potential), we establish existence and uniqueness of maximal solutions to the Cauchy problem in the energy space as long as the density matrix is not singular. A sufficient condition in terms of the energy of the initial data ensuring the global-in-time invertibility is provided (first result in this direction). Regularizing the density matrix violates energy conservation. However, global well-posedness for this system in L ² is obtained with Strichartz estimates. Eventually, solutions to this regularized system are shown to converge to the original one on the time interval when the density matrix is invertible.