Energy-level statistics of model quantum systems: Universality and scaling in a lattice-point problem

We investigate the statistics of the numberN(R, S) of lattice pointsnZ ², in an annular domain (R, w)=(R+w)A\RA, whereR, w>0. HereA is a fixed convex set with smooth boundary andw is chosen so that the area of (R, w) isS. The statistics comes fromR being taken as random (with a smooth density) in some interval [c ₁ T,c ₂,T],c ₂>c ₁>0. We find that in the limitT the variance and distribution of N=N(R; S)–S depend strongly on howS grows withT. There is a saturation regimeS/T, asT, in which the fluctuations in N coming from the two boundaries of are independent. Then there is a scaling regime,S/Tz, 0<z<, in which the distribution depends onz in an almost periodic way going to a Gaussian asz0. The variance in this limit approachesz for genericA, but can be larger for degenerate cases. The former behavior is what one would expect from the Poisson limit of a distribution for annuli of finite area.     

Heterogenization of a chiral bis(oxazoline) catalyst by grafting onto silica

[reaction: see text] We report here the first heterogenization of a bis(oxazoline) ligand on an inorganic (silica) surface. The activity and enantioselectivity of this new material as a catalyst for the Diels-Alder reaction were checked, and it was shown that under certain conditions enantioselectivities similar to those of the homogeneous catalyst are reached. It was also shown that under these conditions the catalyst can be recycled without loss of activity or selectivity.     

Thermophysical properties of oil shale minerals

The thermal diffusivity values of eight minerals that are commonly associated with oil shales of the Green River formation have been measured by the laser flash technique. Data are presented in the temperature range 25–500°C for quartz, dolomite, calcite, plagioclase, analcite, pyrite, potassium feldspar and low albite. A comparison of the thermal conductivities of some of these minerals, calculated from experimentally measured thermal diffusivity and density, with the experimental values reported in the literature reveals good agreement. Trends in the variation of thermal diffusivity with temperature and anisotropic effects in thermophysical parameters are discussed from the point of view of grain boundary effects in these polycrystalline mineral aggregates.     

Direct measurement of the radiative lifetime of the A2Σ+(υ′ = 0, K′ = 1, J′ = 32) state of OH and OD

The radiative lifetime of the A²Σ⁺(υ′ = 0, K′ = 1, J′ = 3/2) state of OH and OD has been directly measured by following the decay of fluorescence excited by light from a frequency doubled dye laser. Stern-Volmer extrapolation of the results to zero pressure gave τ(OH) = 788 ± 13 ns and τ(OD) = 754 ± 12 ns.     

A simple sum rule for total radiative decay rates in diatomics

The quantity (ν³μ_e²), which occurs in expressions for the total radiative decay rate for an excited vibronic state of a diatomic, is evaluated as a simple expectation value for the initial state. In test calculations this expectation value agrees with the exact sum over states within ≈0.2%.     

Existence,uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations

We study positive solutions of the Dirichlet problem: u(x)+f(u(x))=0,xD ⁿ ,u(x)=0,xD ⁿ , whereD ⁿ is ann-ball. We find necessary and sufficient conditions for solutions to be nondegenerate. We also give some new existence and uniqueness theorems.Research supported in part by NSF Contract Number MCS 80-02337     

Transport properties of the Lorentz gas: Fourier's law

We investigate the stationary nonequilibrium (heat transporting) states of the Lorentz gas. This is a gas of classical point particles moving in a region gL containing also fixed (hard sphere) scatterers of radiusR. The stationary state considered is obtained by imposing stochastic boundary conditions at the top and bottom of , i.e., a particle hitting one of these walls comes off with a velocity distribution corresponding to temperaturesT ₁ andT ₂ respectively,T ₁ <T ₂. Letting be the average density of the randomly distributed scatterers we show that in the Boltzmann-Grad limit,,R 0 with the mean free path fixed, the stationary distribution of the Lorentz gas converges in theL ¹-norm to the stationary distribution of the corresponding linear Boltzmann equation with the same boundary conditions. In particular, the steady state heat flow in the Lorentz gas converges to that of the linear Boltzmann equation, which is known to behave as (T ₂-T ₁)/L for largeL, whereL is the distance from the bottom to the top wall: i.e., Fourier's law of heat conduction is valid in the limit. The heat flow converges even in probability. Generalizations of our result for scatterers with a smooth potential as well as the related diffusion problem are discussed.Research supported in part by NSF Grant no. Phy 77-22302.On leave of absence from the Fachbereich Physik der Universität, München. Work supported by a DFG fellowship.     

An alternative multipolar expansion for intermolecular potential functions

We have derived a new multipolar expansion for intermolecular potential-energy functions with applications in molecular physics, theoretical chemistry, and mathematical physics. The new formulation employs a separation of radial and angular terms with a simple index structure that leads to computational efficiency and ease of physical interpretation. For the case of the Coulomb interaction, we compare the present formulation with two conventional multipole expansions: the Cartesian tensor and the irreducible spherical tensor expansions. The new formalism leads to efficient numerical algorithms that are useful for general applications beyond intermolecular potentials. In addition to the electrostatic Coulomb interaction, we illustrate the formalism with applications to special function theory and a bipolar expansion involved in potential theory.